non-set theoretic foundations of concrete mathematics...1.2 concrete mathematics here is an...

Non-Set Theoretic Foundationsof Concrete Mathematics

N J H Wontner

CCDDissertation on a Mathematical Topic

For the Degree of MMathPhil in Mathematics and Philosophy

Mathematical InstituteUniversity of Oxford

Hilary Term 2018

Abstract

A standard way to provide foundations for ‘everyday’ areas ofmathematics is through set theory, particularly ZFC. We describe al-ternative foundations.

Our primary focus are so-called ‘parasitic’ categorical foundations.Certain restricted set theories (and hence for much of concrete math-ematics) have already been founded this way, starting in 1964, withLawvere’s with the Elementary Theory of the Category of Sets [19].Mac Lane and Moerdijk construct a model of a bounded ZF-Foundationson categorical foundations, using well pointed topoi with Choice an aNatural Number Object [22]. We give details to this proof and discusssome options regarding Replacement.

We also investigate the possibility of topological or algebraic (grouptheoretic) foundations and compare these to the successful categorytheoretic foundations for standard set theories.

Finally, we consider more direct foundations for concrete areaswithout detouring through set theory, concentrating on direct in-terpretations of permutation group theory, basic analysis, and somegraph theory.

1

Acknowledgements

First and foremost, I would like to thank my supervisor, Dr. RolfSuabedissen. Without him, this work simply would not exist. I amvery grateful to Eva Levelt for her help in proofreading, and her math-ematical nous. My thanks also extend to Hannah Bavcic and the LadyMargaret Hall mathematicians for their advice and support on all mat-ters mathematical and stylistic. I thank my mother for her radicallydifferent take on the academic process, and my father for helping mekeep down to earth. I express my gratitude for Prof. James Studd,who has strongly influenced my critical approach, and appreciate hisuseful advice on presentation. I thank Zoe Chatfield for her continu-ing support on all fronts, her proofreading, and for coaxing this workinto completion. Finally, I would like to say how lucky I am to havesuch a supportive group around me, who made writing a pleasure, nota hardship.

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Contents

Contents

1 Introduction 51.1 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Concrete Mathematics . . . . . . . . . . . . . . . . . . . . . . 51.3 Set Theory as a Foundation . . . . . . . . . . . . . . . . . . . 6

2 Categorical Foundations 82.1 Categories and Topoi . . . . . . . . . . . . . . . . . . . . . . . 82.2 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 Set Theory, Categorically . . . . . . . . . . . . . . . . . . . . . 20

2.3.1 Extensionality . . . . . . . . . . . . . . . . . . . . . . . 212.3.2 Empty Set . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.3 Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.4 Union . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.5 Power Set . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.6 Foundation . . . . . . . . . . . . . . . . . . . . . . . . 312.3.7 Schema of Separation . . . . . . . . . . . . . . . . . . . 322.3.8 Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3.9 Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.3.10 Schema of Replacement . . . . . . . . . . . . . . . . . 372.3.11 Taking Stock: Replacement At All? . . . . . . . . . . . 40

2.4 Algebra, Categorically . . . . . . . . . . . . . . . . . . . . . . 422.4.1 Clarification . . . . . . . . . . . . . . . . . . . . . . . . 422.4.2 Internal Theory of Permutation Groups . . . . . . . . . 43

2.5 Analysis, Categorically . . . . . . . . . . . . . . . . . . . . . . 462.6 Discrete Mathematics, Categorically . . . . . . . . . . . . . . . 47

3 Topological Foundations 483.1 Interpreting ZFC . . . . . . . . . . . . . . . . . . . . . . . . . 493.2 Topological Axioms . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2.1 0 Space, 1 Point Space and Non-Degeneracy . . . . . . 513.2.2 Finite Limits . . . . . . . . . . . . . . . . . . . . . . . 513.2.3 Finite Colimits . . . . . . . . . . . . . . . . . . . . . . 513.2.4 Sierpinski Space . . . . . . . . . . . . . . . . . . . . . . 533.2.5 Omega . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2.6 Countable Products and Powers . . . . . . . . . . . . . 543.2.7 Indiscrete Space and Discretisation . . . . . . . . . . . 55

3

Contents

3.2.8 Separation . . . . . . . . . . . . . . . . . . . . . . . . . 563.2.9 Unit Interval . . . . . . . . . . . . . . . . . . . . . . . 573.2.10 Arbitrary Products and Sums . . . . . . . . . . . . . . 603.2.11 A Grab Bag of Advanced Constructions . . . . . . . . 60

3.3 Function Spaces: Analysis & Algebra . . . . . . . . . . . . . . 613.3.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 613.3.2 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.4 Discrete Mathematics . . . . . . . . . . . . . . . . . . . . . . . 633.4.1 Graphs Through Closed Sets . . . . . . . . . . . . . . . 633.4.2 Graphs Through Quotients and Glue . . . . . . . . . . 63

4 Algebraic Foundations 654.1 Interpreting Algebra . . . . . . . . . . . . . . . . . . . . . . . 654.2 Basic Entities . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3 Graphs and Matrices . . . . . . . . . . . . . . . . . . . . . . . 66

5 Conclusion 67

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1 Introduction

1.1 Foundations

For our purposes, a foundation for an area of mathematics is a combinationof a given basic theory, with a language and ontology, and an attempt to‘interpret’ (provide a truth-preserving translation to) the concrete theory inthat language and ontology.

A candidate foundational theory must be natural and believable - it willbe no good to take all theorems as axioms! For example, we expect thefoundation to be equi-consistent with ZFC, and hence prove the consistencyof the reduced theory.

So, a foundation should:

1. Stipulate some basic metaphysical kind (set, category, etc.).

2. Stipulate some believable(!) axioms on how these kinds are to interact.

3. Interpret the area for which it claims to provide a foundation.

4. Preserve the direction of plausibility.

1.2 Concrete Mathematics

Here is an unhelpful definition: ‘concrete’ means not abstract. A la MacLane, we mean something like “theory for applications rather than for itsown sake”. A rough characterisation is the mathematics practised by mostworking mathematicians in any research institute. Suitable restrictions (e.g.excluding those working in the foundations of mathematics!) will need to beapplied.

We exclude the higher reaches of set theory. Though these are not ex-plicit investigations into the foundations of mathematics, they are intimatelyconnected with axiomatisations. We will treat group theory, Hilbert Spaces,etc. as concrete, despite their apparent abstractness, as they are not explic-itly foundational enquiries. These limits might expand in the future (andpossibly, some present parts will be excluded [9]). We are concerned withtoday’s concrete mathematics.

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1.3 Set Theory as a Foundation


The canonical foundation of mathematics is set theory. What exactly con-stitutes ‘set theory’ is more controversial. ZFC is very widespread, but nottotally universal.

We are not principally concerned with the legitimacy of particular ax-ioms. For us, the significant property of ZF(C) is that much of ‘everyday’mathematics has already been modelled within it. We have canonical settheoretic interpretations for 〈x, y〉, functions, N, Q, R, groups, rings, fields,graphs, geometric objects, games, etc. One can have a rich mathematicalcareer entirely within mathematics which ZFC has founded.

The power of ZFC to provide a near comprehensive foundation for con-crete mathematics will motivate our investigation of parasitical foundations :if we can provide an interpretation of set theory, particularly (unbounded1)ZFC, we can then interpret the vista of mathematics already interpreted byset theory.

A candidate parasitic foundation includes two pieces of information:

1. An interpretation of the primitive notions of set theory.

2. A structure whose model proves the axioms of the set theory.

The standard primitive notions of set theory are as follows:

Definition 1 (Membership). x ∈ y iff x is a member of y

Definition 2 (Sethood). X is a set iff there is a y s.t. X ∈ y ∨ y ∈ X

If we assume that the ambient universe is composed of sets alone, ‘Set’could be an implicit notion. However, when the quantifiers range morewidely, we need a definition to pick out ‘exactly the sets’.

1This may be overly rich, as we may not need (very much) unboundedness for most ofconcrete mathematics.

6


The second condition on an interpretation of ZFC is that the model forour structure satisfies the ZFC axioms [18], which are:

ZFC 1 (Extensionality). ∀x∀y ∀z[(z ∈ x↔ z ∈ y)↔ x = y]

ZFC 2 (Empty Set). ∃x ∀a¬(a ∈ x)

ZFC 3 (Pairing). ∀x ∀y ∃z(x ∈ z ∧ y ∈ z)

ZFC 4 (Union). ∀f ∃a∀y ∀x[(x ∈ y ∧ y ∈ f)→ x ∈ a]

ZFC 5 (Power Set). ∀x ∃y ∀z[z ⊆ x→ z ∈ y]

ZFC 6 (Foundation). ∀x [∃a(a ∈ x)→ ∃y(y ∈ x ∧ ¬∃z(z ∈ y ∧ z ∈ x))]

ZFC 7 (Schema of (Bounded) Separation). Let φ be a (∆0) formula in thelanguage of ZFC, LZFC, with all free variables amongst x, z, w. Then:∀z ∀w ∃y ∀x[x ∈ y ↔ (x ∈ z ∧ φ(x))]

Here, bounded (∆0) means φ has only bounded quantifiers, e.g. ∀x ∈ aor ∃x ∈ a for some set a (i.e. relativised).

ZFC 8 (Infinity). ∃x [∅ ∈ x ∧ ∀y(y ∈ x→ y ∪ y ∈ x)]

ZFC 9 (Choice). ∀x [∅ /∈ x→ ∃f : x→⋃x ∀a ∈ x (f(a) ∈ a)]

ZFC 10 (Schema of Replacement). Let φ be a LZFC-formula with all freevariables amongst x, y, a, w. Then:∀a∀w

[∀x(x ∈ a→ ∃!y φ(x, y))→ ∃b ∀x

(x ∈ a→ ∃y(y ∈ b ∧ φ(x, y))

)]BZFC refers to bounded ZFC (∆0 as in ZFC7).

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2 Categorical Foundations

The project of giving set theory categorical foundation is rooted in Lawvere’sseminal “Elementary Theory of the Category of Sets” [19]. Significantly, wedo not need any ambient set theory for category theory (e.g., Tsukada [34]).So a parasitic categorical foundation will not necessarily be circular.

We largely follow the construction of Mac Lane and Moerdijk (MLM)of a model for BZFC [22], a parasitic categorical foundation of concretemathematics. We add details to a proof that a well pointed topos (WPT)with a natural number object (NNO) and Choice has a proper subclass of(equivalence classes of) ‘trees’ which satisfies the translation of BZFC above.

For this, we interpret Membership. However, we cannot rely on an ambi-ent set theory of objects of the category. Instead, Membership is interpretedas a particular claim about morphisms in our topos.

So, we describe an interpretation of set theory solely in terms of functionsbetween objects2. This is known as a structural set theory, as opposed to thematerial set theory of the classical interpretation of ZFC.

2.1 Categories and Topoi

The language of category theory has the following primitive predicates:

1. ob(X), indicating X is an object in the category.

2. m(f), indicating f is a morphism.

3. (f, g, h), indicating that f composed with g is h.

4. dom(X, f), indicating X is the domain of f .

5. codom(Y, f), indicating Y is the codomain of f .

We could rewrite all the categorical sentences as formulae using thesepredicates, but we shall be lighter on notation for clarity.

Most of the following definitions are from MLM.

2Tsukada’s ‘Pan-Categorism’ [34].

8


Definition 3 (Limit & Colimit). Let I, C be categories. Let F : I → C be afunctor.

A limit is an object limIF in C and some morphisms fi : limIF → F (i)for each i in I such that for every morphism g : i → j in I, F (g) fi = fj,and the object is universal with this property.

I.e. for every object W compatible with projections πi : W → F (i)there is a unique morphism φ : W → limIF , so the arrows in the followingcommute:

W

limIF

F (i) F (j)

∃!φ

πi

πj

fj

fiF (g)

A special case is the binary product, where the object X×Y is a limit,equipped with projections into each co-ordinate

W

X X × Y Y

∃!

πY

πY

The colimit is the dual notion of the limit, i.e. is a limit in the oppositecategory Cop which has the same objects but each morphism is ‘reversed’.

Definition 4 (Diagram). A diagram is a collection of objects and mapsbetween those objects, such that if two maps compose, then the compositionis in the diagram.

Definition 5 (Finite Limits). A category has finite limits if it has all limitsover diagrams with finitely many objects and morphisms.

Definition 6 (Terminal Object). A terminal object in a category C is anobject 1 of C such that for any object x of C, there is a unique morphism! : x→ 1.

9


Definition 7 (Initial Object). An initial object in a category C is an object0 such that for any object x of C, there is a unique morphism ! : 0→ x

Definition 8 (Global Element). A global element of an object A in a categoryC with a terminal object 1 is a morphism g : 1→ A.

Observe that this does not involve the set-theoretic notion of membership.

Definition 9 (Mono). A morphism f : X → Y in a category is a monomor-phism (mono) if for every object Z and every pair of parallel morphismsg1, g2 : Z ⇒ X then

(f g1 = f g2) =⇒ (g1 = g2)

Definition 10 (Epi). A morphism f : X → Y in a category is an epi-morphism (epi) if for every object Z and every pair of parallel morphismsg1, g2 : Y ⇒ Z then

(g1 f = g2 f) =⇒ (g1 = g2)

Lemma 1 (Inverse). [29] If f : Y → X is a mono and g : Z → X is amorphism, then there is a pullback3 g−1(Y )4 of the following diagram.

g−1(Y )

Y Z

X

pY

pZ

f

g

Definition 11 (Inverse Image). The above pullback is called the inverse im-age of f : Y → X under g, written g−1(Y ).

3Binary limit of such pairs of morphisms.4In Set, g−1(Y ) ⊆ Z

10


Definition 12 (Subobject). A subobject of an object X in a category C isan equivalence class of monos into X, i : Y → X. A mono j : Z → X is inthe same equivalence class as i if each factors through the other, i.e. if thereis k : Y → Z such that i = j k.

Y Z

X

i

∃k

j

Subobjects are structurally similar to subsets. A subset is a canonicalnormal form of all the monos in an isomorphism class. Without such acanonical form, we need an independent way of classifying these isomorphismclasses. Informally, we say that Y is a subobject of X, written Y 4 X.

Definition 13 (Subobject Classifier). In a category C with finite limits,a subobject classifier is a mono true : 1 → Ω, such that for every monom : U → X in C, there is a unique morphism χ(U,m) : X → Ω that forms acommutative (pullback) square:

U 1 U 1

=

X Ω X Ω

m true m true

!χ(U,m) χ(U,m)

Informally, the subobject classifier determines the characteristic functionχ(U,m) of a subobject [U

m→ X], as the unique function such that U is aninverse image of “true ∈ Ω” under χ(U,m) [29].

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Definition 14 (Cartesian Closed). A category with finite products is Carte-sian closed if it is closed with respect to the Cartesian monoidal structure.

More concretely, the category C is (Cartesian) closed if every functorF : × Y has a right adjoint for all objects in the category. So for allX, Y ∈ C there is an H(X, Y ) ∈ C (the internal hom) and a morphismεX,Y : H(X, Y )×X → Y such that for all Z ∈ C and f : Z×H(X, Y ) thereis a unique φf : Z ×X → Y satisfying εX,Y (φf × idX) = f

H(X, Y ) H(X, Y )×X Y

Z Z ×X

εX,Y

∃!φf φf×idXf

So for any pair of objects X, Y there is an object in C which ‘acts like’the object of morphisms m : X → Y .

Definition 15 (Elementary Topos). An elementary topos, E, is a categorywhich has finite limits, is Cartesian closed, and has a subobject classifier, Ω.

It is ‘elementary’ as the axioms are first order.

Definition 16 (Power Object). For an object X in C (with finite limits), thepower object is an object P(X) in C with a morphism ∈X : X × P(X) → Ωsuch that for any object Y in C and mono f : X × Y → Ω, there is a uniquemorphism φ : Y → P(X) such that the following commutes:

Y X × Y

P(X) X ×P(X) Ω

φ 1×φf

∈X

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Lemma 2. A category is an elementary topos iff it has (all) finite limits and(all) power objects.

This is a standard result5, see e.g. MLM.

Observe that P(1) = Ω.

Lemma 3. A topos has all finite colimits [32].

Definition 17 (Well-Pointedness). A topos is well-pointed (WPT) if

1. 1 is not initial (non-degenerate)

2. For f, g : A⇒ B, f = g iff fx = gx for every global element x of A.

Definition 18 (NNO). A natural number object (NNO) on a topos E is anobject N of E with arrows

1O→ N

s→ N

such that for any object X of E with arrows x and f such that

1x→ X

f→ X

then there exists a unique h : N → N such that the following commute

1 N N

X X

O

x

s

!h !h

f

Clearly, N is unique up to isomorphism.

Definition 19 (Choice). A category C has Choice if every epi splits, i.e. ife : X → Y in C is epi, then there is a morphism s : Y → X such that e s= idY .

5Set motivates this equivalence. ΩSet = t, f = 2 (up to equivalence), and true :1 → 2 is the constant t-function. So we have a natural bijection between the morphismsm : X × Y → t, f and the morphisms g : Y → t, fX and then a natural bijectionto the morphisms k : Y → P(X). Generalising, P: E → E is such that ∀X,Y ∈ E ,HomE(X × Y,Ω) ∼= HomE(Y,P(X)), hence the equivalence.

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Lemma 4 (MLM). In a WPT E

1. if A, B are subobjects of X, then A 4 B iff every global element p :1→ X which factors through A factors through B

2. if X 6= 0 then the unique map f : X → 1 splits.

3. α : X → Y is epi iff ∀p : 1→ Y ∃q : 1→ X (αq = p)

4. if α : X → Y and B 4 Y such that every q : 1 → X is such that αqfactors through B, then α factors through B.

This concludes the basic definitions required. We give details to MLM’sconstruction.

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Idea for Main Proof

We interpret set theoretic claims as claims about morphisms of a par-ticular kind of object in E , trees. Sets correspond to (isomorphism classesof) trees. Membership corresponds to being (isomorphic to) a subtree deter-mined by a “second layer” node of the tree. Being a member of a membercorresponds to being isomorphic to a node in the “third layer”, etc.

etc. · · · · · · · · ·

nodes corresponding to elements of elements · · · • · · ·

nodes corresponding to elements · · · • · · ·

the ‘root’ •

For example [24], the tree of S = ∅, ∅ is:

•

•

• •

•

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2.2 Trees

2.2 Trees

The following can be formalised thoroughly in Mitchell-Benabou language(as in MLM), we use a variant language, LT .

Informally, “t : 1→ T” is the translation of “t can be found ‘somewherein’ T” (the membership tree). Similarly, “there is a mono i : S → T” is thetranslation of “S ⊆ T”.

Definition 20 (Tree). A Tree in E is an object T of E with a subobject ofP(T × T ), called a binary relation, R 4 T × T with certain properties. Wewrite x ≤ y := R(x, y) for x, y in T .

A tree is a pair 〈T,≤〉, where R is an ‘element’ of P(T × T ), i.e. is amono R : 1E → P(T × T ) which we interpret as R 4 T × T . We abbreviatethis to T , when the ordering is clear.

The tree ordering, R =≤T : 1 → P(T × T ) has properties (axiomatisedbelow) like having a base point, the ‘root’, and branches extending out ofthis point forming a second layer, the ‘points’, etc.

We cannot make use of any ambient set theory in this construction. Sowe cannot take the usual notion of binary relations in category theory, whichwould be a functor on pairs (i.e. sets) of objects in a category.

However, E has finite limits, so for each T , T×T is in E . By definition 16,E has power objects, so P(T × T ) is in E . So take R to be a ‘subset’ of T×T ,i.e. an ‘element’ of P(T × T ).

Throughout, “t ∈ T” represents a global morphism t : 1→ T , and t ≤ scorresponds to a morphism (t, s) : 1E → P(T × T ).

Tree 1 (Poset). R defines a reflexive partial order ≤ on T

I.e. for all t, s, r ∈ T : t ≤ t; t ≤ r ∧ r ≤ s →≤ r; t ≤ s ∧ s ≤ t → t = swhich factors through R.

Tree 2 (Root). There is an arrow 0T : 1E → T such that 0T ≤ t for allt : 1E → T .

Formally, there is an arrow 0T : 1E → T such that for any arrow α : 1E →T , the pair 〈0T , α〉 : 1E → T × T factors through the subobject R 4 T × T .

16

2.2 Trees

Definition 21. ↓ t := x|x ≤ t.

Lemma 5. ↓ t is an object of E, and ↓ t 4 T .

Proof. We are interested in R(., t) = s ∈ T : R(s, t)

Informally, ↓ t is the image of the relation R at a particular point, R(t).

Formally, note R 4 T × T , so R : T1 × T2 → Ω, where T = T1 = T2.

E is Cartesian closed, so has internal homs. Observe that hom(X ×Y,C) ∼= hom(X,ZY ). Hence, we can ‘uncurry’ R, i.e. R : T1 × T2 → Ω

corresponds to R : T2 → (T1 → Ω).

So R (t : 1→ T2) =: R(t) : T1 → Ω

E has inverse images of diagrams with monos.

(R(t))−1(true)

1 T1

Ω

true

R(t)

Observe that (R(t))−1(true) 4 T1 satisfies the definition of ↓ t.

Tree 3 (Downwards Closure). For all t : 1 → T , ↓ x is linearly ordered bythe restriction of the relation ≤ of T.

17

2.2 Trees

Lemma 6 (Restrictions). For S 4 T , the restriction ≤T∣∣S

is also in E.

Proof. Informally, E has ‘internal intersections’, so for T 4 S, ≤T∣∣S

corre-sponds to (≤T ) ∩E (S × S), in E .

Formally, S 4 T so there is a mono m : S → T , and a mono m × m :S × S → T × T .

≤T 4 T × T , so there is a mono i :≤T → T × T .

We then take inverses:

i−1(S × S)

S × S ≤T

T × T

m×mi

Observe that i−1(S × S) 4≤T is the restriction.

Remark. ≤T∣∣S

may not (tree) order S, e.g. 〈S,≤T∣∣S〉 may violate Tree 6.

Tree 4 (Well-Founded Down). For all S 4 T , S 6= 0, there exists a y : 1→ Swhich is ≤-minimal in S.

In a WPT this is equivalent to: for every subobject S 4 T , such that S isnot initial, there is a y : 1→ S such that for all z : 1→ S, if 〈z, y〉 : 1→ T×Tfactors through R, then y = z. Here, y is the ≤-minimal element.

Tree 5 (Well-Founded Up). For all S 4 T , S 6= 0, there exists a w : 1→ Swhich is ≤-maximal in S.

This seems only to be required to prove Foundation, in § 2.3.6.

Definition 22 (Tree Morphism). A morphism of trees β : T → T ′ is amorphism T → T ′ in E which preserves the root 0T and the relation ≤T , i.e.β(0T ) = 0T ′ and t ≤T s→ β(t) ≤T ′ β(s).

Tree mono, epi, iso, endo, and automorphisms are defined similarly.

Tree 6 (Rigid). The only (tree) automorphism α : T → T is the identity.

18

2.2 Trees

Definition 23 (Node). A node of T is a global element t : 1→ T in E.

Definition 24 (Branch). A note t determines a subtree, its upwards closure↑ t = x|t ≤ x, called a branch.

Claim. ↑ t is an object and tree.

The proof is similar to Lemma 5, this time we are interested in R(t, .) =s ∈ T |R(t, s).

R 4 T × T , so R : T1 × T2 → Ω. As E is Cartesian closed, ‘uncurry’ toform R : T1 → (T2 → Ω).

So R (t : 1→ T1) : T2 → Ω

E has inverses, so we take the inverse image of true : 1→ Ω:

(R(t))−1(true)

1 T2

Ω

true

R(t)

So (R(t))−1(true) =: ↑ t is an object of E .

↑ t is ordered by the restriction ≤T∣∣↑t. So it inherits all of its tree structure

from the underlying tree 〈T,≤〉. So ↑ t is isomorphic to a tree (treating t asthe root).

Lemma 7. If a tree T satisfies Tree 1-6 then so does any branch of T

Definition 25 (Covering). A node t covers a node s iff t < s and there isno node u such that t < u < s.

Definition 26 (Point). A node covered by the root of T is called a point.

Definition 27 (Ancestor). For t a node of T , t 6= 0, there is a unique6 pointa(t) of T (i.e. a(t) : 1→ T ) such that 0 < a(t) ≤ t. We call a(t) the ancestorof t.

Observe: if t is a point, then a(t) = t.

6By Tree 3 and 4.

19

2.3 Set Theory, Categorically


Translation Scheme

We can now provide a systematic translation scheme between the ordinarylanguage of ZFC LZFC , or L=,∈, and our language of trees, LT .

Translation 1 (Set). “a is a set” is translated as “Ta is a tree”, using distincttrees where necessary.

Translation 2 (Membership). “a ∈ b” is translated as “there is an isomor-phism Ta ∼= ↑p for some point p in Tb” where “∼=” is a tree isomorphism.

By rigidity and Lemma 7, p is unique.

Translation 3 (Set Equality). “a = b” is translated as “Ta ∼= Tb” where“∼=” is a tree isomorphism.

Observe, a = b iff ↑0a ∼= ↑0b.

Translation 4 (Quantifiers). The quantifier “∀x” is translated as “for alltrees T”.

Translation 5 (Connectives). “¬” is translated as “¬”. “∧” is translatedas “∧”.

By the construction of LZFC , these translations are sufficient to interpretany formula. Standard abbreviations (∃, ∨) of LZFC correspond to standardabbreviations in LT .

We can also translate in the opposite direction, interpreting primitivenotions of category theory (in particular, of trees) e.g. “morphism”, “object”and “composition”. Set theoretic morphisms will be translated as (primitive)category theoretic morphisms in the obvious way.

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Big Theorem. A WPT with a NNO and Choice models BZFC - Replace-ment.

Proof Plan.

We construct models for BZFC in the theory of trees. We translate claimsabout sets into claims about trees, then verify that they hold in E . Clearly, itis sufficient to verify that any such model satisfies the translation of ZFC1-10.

In particular, letting S be a collection of isomorphism classes of (rigid)trees in E , we prove the translations of BZFC1-9 holds of S, and we discussReplacement.

We can ‘combine’ parts of trees in various ways7 to form an object in E .By defining a new order relation on the resultant object we obtain a furthertree. This legitimises several methods we use, like ‘gluing together’ differenttrees, and manipulating the ‘levels’ of a tree (e.g. connecting covered nodes tothe node ‘two beneath them’). Where necessary, we more formally describewhy these methods are legitimate.

2.3.1 Extensionality

∀x ∀y ∀z[((z ∈ x← z ∈ y) ∧ (z ∈ x→ z ∈ y))↔ x = y]

We then translate this into a claim about trees:

In E , for any trees T , T ′ there is a (tree) isomorphism T → T ′ iff

1. for any point p of T there is a point p′ of T ′ with ↑p ∼= ↑p′

and

2. for any point p′ of T ′ there is a point p of T with ↑p ∼= ↑p′

Proof. ( =⇒ ) By definition, a tree isomorphism α : T → T ′ sends points pof T to points α(p) of T ′ with ↑ p ∼= ↑α(p). In particular, for each point p,(1) holds, and for each point p′, (2) holds.

(⇐= ) This requires a lemma.

7The only basic constructions are finite limits and the power object, but E is closedunder a range of other constructions.

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Lemma 8. For a tree T , the points of T together form an object, PT in E.

Informally, this object consists of the ‘first layer’ of nodes. We constructthe ≤-preimage of a subobject 0T ⊆ T , which is the ‘intersection’ of ≤Twith π−12 (0T).

Proof. Recall from Lemma 6 that for each t ∈ T , ↓ t = (R(t))−1(true).

Define f : T → P(T ) as f(t) = (R(t))−1(true) 4 T2. Observe that forglobal elements t, s of T , t ∈ f(s) iff t ≤ s. So f is the function from a nodeto its predecessors.

Observe that the points of T are the nodes which are covered by 0T , soPT is the preimage of 0T.

Define the inclusion i : 0T → P(T ), a mono, and then take theinverse image, f−1(0T), of the following diagram:

f−1(0T)

0T T

P(T )

if

So, the f -inverse image of the representative of 0T, a subobject of T ,is the object of points, PT , if it exists. E has finite limits, so it has pullbacks,so it has inverse images of monos. So this inverse image exists. Hence PT isan E-object.

Proof of Extensionality Continued.

Suppose (1) holds. Fix p. By rigidity, the p′ such that ↑p ∼= ↑p′ is unique,and the isomorphism α :↑p→↑p′ is also unique (so we do not need Choice).So we define a morphism f : p 7→ p′ in E from TP to TP ′ taking each point toits unique counterpart. The subtrees ↑ p, ↑ p′ are isomorphic, so there is anisomorphism αp :↑p→↑p′ for any T -points.

Exactly similarly, we then construct an isomorphism ‘the other way around’.By interchanging T and T ′, (2) implies there is a morphism g : TP ′ → TP ,g : p′ 7→ p, such that for each p′, there as in isomorphism βp′ : ↑p′ →↑g(p′).

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↑p, ↑p′ are trees. By rigidity, f is the inverse of g. Hence αp is the inverseof βf(p) for each p.

So, we extend the (tree) isomorphism between the object of points to anisomorphism on the trees i : T

∼→ T ′ by:

i(t) = f(t) for t a point in Pi(t) = αp(t) where p is the ancestor of t.

So these are trees from the same equivalence class, and hence they rep-resent the same set.

Finally observe that i 4 T ×T ′ is in E as i = f∣∣PT∪E π(X) = f

∣∣PT

+π(X)

where X 4 PT × T × T ′ is the subobject consisting of triples 〈p, q, r〉 forq ≤↑p r, and π : PT ×T ×T ′ → T ×T ′ is the appropriate projection (so π(X)represents

⊕p∈PT

αp∣∣↑p).

2.3.2 Empty Set

The (abbreviated) translation reads:

“there is a tree T such that for all trees S it is not the case that there is apoint p of T such that ↑p ∼= T .”

Proof. 0 ∈ E , so there is a tree T0 in E consisting of just the root 0, with thetrivial ordering 0 × 0. T0 has no points, so there is no ↑ p for any S to beisomorphic to. So we can interpret the set ∅ as T0.

T0 : •(0)

Hence, the translation holds of T0.

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2.3.3 Pairing

Translation: for all trees T , S there is a tree P whose only points p and qare such that there are tree isomorphisms ↑p ∼= T and ↑q ∼= S

Proof. There are two cases.

Firstly, suppose T 6= S. Consider the object P ′ consisting of the trees T ,S ‘side by side’, whose roots are 0T , 0S, along with a new root 0P ′ to whichthe two roots are now connected (i.e. 0T and 0S are points of P ′).

· · · · · · · · · · · ·

0T 0S

0

Formally: E has finite colimits. So, colim(S, 1) = S+1 is in E . By takingfurther colimits, construct P = T + S + 1.

Observe that 1 only has one ‘element’ (with the usual gloss): 0 ∈ 1.

Define a partial order on this object:

1. 0 ≤P r for all r in P (including 0T and 0S, the roots of the original twotrees)

2. t ≤P t′ in P iff t ≤T t′ in T

3. s ≤P s′ in P iff s ≤S s′ in S

So ≤P =≤T + ≤S +(0 × 0) + (0 × (T + S)) 4 (T + S + 1)× (T +S + 1). These partial relations are disjoint so the coproduct corresponds tothe internal union. The ordering ≤P is E-union, and so is an object of E .

We then check that P is a tree. Verifying the tree axioms hold is a simpleexercise: P inherits almost all structure from T and S, e.g. being well-founded up and rigidity (rigidity motivates the case distinction). The new

24


root 0 is the only extra case for downwards closure and well-founded down,but clearly these two still hold. Hence P is a tree.

By construction, P only has two points, 0S and 0T . So, there are no fur-ther elements of the point-collection to represent a member in the representedset.

↑0S ∼= S, and ↑0T ∼= T , with the obvious tree morphism: mapping everynode to ‘itself’8. Hence, P satisfies the translation above.

Secondly, suppose S = T . The proof is similar. Consider the objectP = T + 1. The intuition here is that we add an extra root ‘before’ theoriginal root:

1. 0 ≤ r for all r in T .

2. t ≤ t′ in P iff t ≤ t′ in T .

Exactly similar arguments show that P is a tree, in particular P is rigid,and satisfies the translation above.

2.3.4 Union

Translation: For all trees F there is a tree A such that for all trees T andS, if there is a point p of S and q of F such that ↑ p ∼= T and ↑ q ∼= S thenthere is a point r of A such that ↑r ∼= T .

Proof. Informally, we ‘delete’ the layer of points, and take the nodes coveredby the points to be our new points (as these covered nodes correspond to themembers of the points, which themselves correspond to the members of theoriginal set).

Formally, observe that this object, A, is in E for similar reasons to above:instead of considering the object of points, we consider the object of nodescovered by points, along with their generated subtrees.

Order A by ≤A, the restriction ≤F∣∣A

, which is in E by Lemma 6. ≤Fsatisfies Tree 1-5, so ≤A satisfies them.

8The node from the original tree (S or T ) which it is mapped to in P .

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However, A may not satisfy Tree 6, as members of members of our originalset could be identical. For example x, x, y ‘doubles up’ on x amongstthe nodes covered by the points. Then A has the ‘swap’ automorphism takingeach point corresponding to one x, p to the point corresponding to the otherx, q. This is an automorphism which is not the identity. This violates Tree6, rigidity.

· · · · · · · · · · · · · · · · · ·

0x 0x 0y 0x 0x 0y

01 02 =⇒

0 0

∼= ∼=

To correct for this, we identify any copies.

Lemma 9 (Non-Rigid Trees). A tree T is non-rigid iff there is a node t ofT which covers two non-identical nodes x and y, such that ↑x ∼= ↑y.

Proof. (⇐= ) If such x, y exist, then there is an automorphism swap : T → Tsuch that swap : x 7→ y, swap : y 7→ x, leaving other nodes fixed.

swap 6= id. So T is non-rigid.

( =⇒ ) If T is non-rigid there is an automorphism a 6= id.

T is Well-Founded Down, so there is a ≤-least x such that a distinct y issuch that x = a(y). As a is an automorphism ↑x ∼= ↑y.

Also, x 6= 0, so x is covered by a node t, which must be fixed by a byminimality of x.

Proof of Union Continued.

The only possible failure of rigidity in F is in the ‘second layer’ of nodes.The layers above are rigid as F is rigid.

Define SF , the ‘second layer’-object of F , similar to in Lemma 8. DefineR 4 SF × SF , an equivalence relation such that s ∼ t iff ↑s ∼= ↑ t.

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Note that R is in E for similar reasons to the above.

We then take A′ to be the object consisting of a (chosen) representativefrom each equivalence class.

Claim. SF/R =: A′ is an object of E

Quotient objects are appropriate colimits. E has colimits, so has quotientobjects [22] [33]. In particular, for R(a, b) (iff ∃α : a ∼= b), we construct:

R

SF SF/R

R

Define ↑ A′ := 1 + A′ + B, where B = 〈s, t〉|s in A′, t 6= 0↑s in ↑ s 4T 4 A′ × T is an object of E (details are as in § 2.3.5). ↑A′ has a root anda copy of each branch above the representative from each equivalence class.Order ↑A′ by:

1. 〈s, t〉 ≤ 〈s′, t′〉 iff s = s′ and t ≤ t′ in ↑s

2. s ≤ 〈s, t〉

3. For s, s′ in A′, s ≤ s′ iff s = s′

4. 0 ≤ t for all t

This ordering is in E for standard reasons.

Exactly similarly to A, ↑ A′ satisfies Tree 1-5. By construction, ↑ A′ isrigid. So it is a tree. So this tree witnesses the translation.

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2.3.5 Power Set

Translation: For any tree T there is a tree S such that for all trees R, if thereis a mono i : R → T then there is a point pR in S such that R ∼= ↑pR, i.e. ifR injects into S, then there is an isomorphism between R and a branch of S.

Proof. Informally, we construction Pow(T ) like so: let P be the object ofpoints of T .

• Pow(T ) has a root, 0.

• For each ‘subset’ s ⊆ P , Pow(T ) has a point S.

• Above the point S is a layer of nodes consisting of a point pS for eachpoint pS in s.

• Each pS determines a subtree which is (isomorphic to) ↑pS, the branchof the point to which it corresponds.

· · ·

· · · pS · · ·

· · · S · · ·

0

↑pS

other subsets

More formally: let P(P ) = ΩP in E be the power object of the pointsP 4 T . Let a(x) be the ancestor of x. Define:

B = 〈t, S〉|t 6= 0 in T, S ⊆ P and a(t) ∈ S

Claim. B is in E , and B 4 T ×P(P ).

Consider the membership map, ∈P : P ×P(P )→ Ω.

Taking inverses, ∈−1P (true) = 〈t, S〉|t ∈P SNow observe that a induces a map a× id : T × P(P ) → P × P(P ), and

let g :=∈P (a× id) : T ×P(P )→ Ω.

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g−1(true)

true T ×P(P )

P ×P(P )

Ω

a×id

g

∈P

So: g−1(true) = 〈t, S〉|t ∈P a−1(S).

‘Intersecting’ with the object 〈t, S〉|t 6= 0, S ⊆ P yields B.

By construction, B 4 T ×P(P ).

Pow(T ) := B+P(P )+1, this coproduct is in E , with ‘elements’ 〈t, S〉 ∈ B,S ∈ P(P ) and 0 ∈ 1.

Define a partial order on these.

1. 〈t, S〉 ≤ 〈t′, S ′〉 iff S = S ′ and t ≤ t′ in T

2. S ≤ 〈t′, S ′〉 iff S = S ′

3. S ≤ S ′ iff S = S ′

4. 0 ≤ S for all S ∈ Pow(T )

Informally, ≤= (0×0)t (0× (B+P(P )))t idP(P )tX tY where:

• X ∼= X 4 (≤T ) × P(P ) such that 〈〈t, t′〉, S〉 ∈ X iff t ≤T t′ ∧a(t), a(t′) ⊆ S.

• Y 4 P(P )×B such that 〈S, 〈S ′, t〉〉 ∈ Y iff S = S ′.

So, for standard reasons, ≤ in E .

Clearly, the objects ‘in’ P(P ) are points of Pow(T ), as they are (≤-)beloweverything (by 2.), except for the root (by 4.).

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Let tree S represent a subset of the set represented by T (written S ⊆setT ), i.e. R ∈T S ⇒ R ∈T T . By the definition of tree-membership, for a

point p′ in S, there is a unique point p in T such that ↑p ∼=↑p′. Let S be thecollection of such points in T . S is a point of Pow(T ), as it is in P(P ).

S ∼= S and S ∈T P(P ) so S is realised by a point in Pow(T ). So Pow(T )has the required properties in the translation.

Claim. Pow(T ) is a tree.

Tree 1-6 are mostly clear. Two sketches are provided.

Well Founded Up.

T is a tree, so Well-Founded Up.

The ordering on Pow(T ) is ‘bounded’ by the ordering on B (by 1.), itselfa subordering of T (as the ordering is induced by the ordering on the secondco-ordinate).

So Pow(T ) is Well-Founded Up.

Rigid.

There are 3 forms of subtree in Pow(T ):

1. ↑0 = Pow(T )

2. ↑S where S ⊆set P is a P(P )-point (as discussed above)

3. ↑〈t, S〉 where a(t) ∈ S

So, again we apply Lemma 9 to each of the forms of subtrees.

Suppose Pow(T ) is not rigid, then there are x1, x2, nodes in Pow(T ),such that x1 ∼= x2, which are covered by the same node y. Cases:

• If x1 = 0, then x2 ∼= x1 iff ↑x1 = Pow(T ) =↑X2 iff x2 = 0. So x1, x2are non-distinct .

• If xi is of the form Si then ↑S1∼=↑S2. So, by rigidity of T , x1 = S1 =

S2 = x2, .

• If xi = 〈ti, Si〉, as xi are covered by one node (the branch determinedby one a(t), by definition of B), we deduce S1 = S2. xi are isomorphic,so also t1 ∼= t2. By the rigidity of T , t1 = t2, .

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• The only other case, wlog, is x1 = S and x2 = 〈t, S ′〉, both covered bya node y. But by construction, subset S is covered by 0 and 〈t, S ′〉 byeither S ′ 6= 0 or 〈t′, S ′〉 6= 0 (by definition of 0), .

So there is no witnessing pair as in Lemma 9. So Pow(T ) is rigid.

2.3.6 Foundation

Translation: For any tree T , such that T 6∼= 0 (i.e. not initial) there is a treeS which is isomorphic to a branch of a point of T such that there is no treeR which is isomorphic to a branch of a point of T (↑ pT ) and a branch of apoint of S (↑pS). I.e.:

(?) ∀T 6= 0∃p0 ∃S (S ∼= ↑p0 ∧ ∀pT ∀pS (↑pT 6∼= ↑pS))

Proof. Consider the (sub)set of T -nodes

N = t node of T |∃ point q of T : ↑ t ∼= ↑q

Clearly any T -point is in N . T 6∼= 0, so N is not empty.

T is Well-Founded Up, so there is a point m of N which is maximal bythe ordering ≤, which is at least as high as the points.

Let r be the point of T such that there is a tree isomorphism on thesubtrees α :↑m→↑r.

1. If no T -node covered by r belongs to N , then r represents a set whichis disjoint from x, i.e. satisfies (?).

2. For contradiction, suppose otherwise. Then there is a node t coveredby r, and T -point, s, such that there is a tree isomorphism β :↑s→↑ t

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• • •

• α(t)

m • • •

· · · • t • • •

· · · r s

0

α

α

β

Then α β is also a tree isomorphism, giving ↑ s ∼= ↑ t ∼= ↑α(t). So α(t)is a node in N . Isomorphism α shows that t > r, so α(t) > m, so m is notmaximal. .

2.3.7 Schema of Separation

Schematic Translation: For any trees S and T there is a tree R such thatfor any point p of S, p is isomorphic to a point of R iff φT (p, T ) holds.

φT (S, T ), the translation of the set theoretic φ(x, y), is some sentence builtup in LT , in terms of points and tree isomorphisms. For example “u ∈ x...”is translated “nu covered by px...” etc. This existence and uniqueness of sucha translation can be proved by induction of complexity of sentences in LZFC[22].

One subtlety is the translation of the quantifiers from φ into φT . This ispossible as φ is ∆0: the quantifiers of φ are restricted to some set anyway.So, we can translate these LZFC-quantifiers as quantifiers ranging over points(or nodes) of S, etc.

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Proof. Write P for the object of points of S in E . There is a well-definedE-object9 ([22] §IV.5)

F = p ∈ P |φT (↑p, S).

Define ↑ F as the tree composed of ↑ p 4 T , for each F -point p, as in§ 2.3.4.

There is a Pow(S)-point, p, such that ↑p ∼= ↑F (as, externally, we knowF ⊆ S). ↑F is isomorphic to a subtree, which is isomorphic to a tree T . Bytransitivity, ↑F ∼= T . So ↑F satisfies the tree axioms.

Finally, the tree ↑ F represents a set f which witnesses the L(B)ZFC

sentence:∀x ∈ s (φ(x, y) ⇐⇒ x ∈ f)

We can then reinterpret this sentence as one of LE , where parameter sets sand y are interpreted as the trees S and T representing each respective set.Hence ↑F is a tree which witnesses the translation of the relevant instanceof the schema.

2.3.8 Infinity

Translation: there is a tree T such that the there is a branch which is iso-morphic to the empty tree ↑ p0 ∼= T0 (i.e. where T0 has no points) and forany branch ↑p there is branch ↑q of T such that ↑q ∼= (↑p)+.

(S)+ is the tree which has all of the points (and their covered branches) ofS, with an extra point: the root of S, and its covered branch (and ↑0S = S).

· · ·

0S · · ·

0S+

S

S

9Lawvere suggests that separation in E is driven by the subobject classifier ([11] §4.8).

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Claim. S+ is a tree.

S+ is the union of a coproduct (1 + S) ∪ S, so is in E .

The define ordering ≤+:

1. 0S+ ≤+ 0S

2. t1 ≤+ t2 iff ti ∈ S1 and t1 ≤S t2

3. t1 ≤+ t2 iff ti ∈ S2, ti 6= 0j, and t1 ≤S t2

Clearly, ≤+ is in E , so S+ is a tree.

Proof of Infinity. As in ZFC, we drive our intuition by considering a treerepresenting each natural number. T0, the zero-tree, represents 0 ∈ N. Forall n ∈ N, (Tn)+ represents n+ 1, where Tn represents n. As expected, (Tn)+

has n+ 1 nodes (as m 6= n→ Tm 6∼= Tn).

· · ·

0S · · ·

0T0 0S+ · · ·

0

S

S

further points

For each node, if ↑pn ∼= Tn, label the node n. So the 0 of TS+ , the tree ofthe successor of S, is labelled nS + 1, where nS is the number label of 0S.

By construction, every node is labelled (by some natural number), andthere is a strictly decreasing sequence, cn, of natural numbers labelling thenodes linking a node n to its ancestor.

By construction, these labelling sequences are unique. So we can identifyeach node with their sequence.

E has a NNO, denoted N. A sequence, cn, can be viewed as a functionf ∈ Hom(N,N) = NN such that ∀n ∈ N, f(n) > f(n+ 1), where minn ∈

34


N : f(n) = 0 is the length of the sequence. This requires a notion of ‘min’on an NNO, which can be defined10.

f represents the sequence 〈f(0)− 1, ..., f(n− 1)− 1〉.

Define the relevant inductive-object as:

T = f ∈ NN : ∀n ∈ N((f(n) 6= 0 =⇒ f(n) > f(n+ 1))

The ordering is defined by:

f ≤ g iff ∀n ∈ N(f(n) 6= 0 =⇒ g(n) ≤ f(n)

Informally, f ≤ g iff the f -sequence is an initial segment of the g-sequence.Clearly, this is a partial order on an object in any topos with an NNO.

Finally, T is an object and tree in E :

• (N) ∈ E and is Cartesian closed, so contains NN, so T is an object andT 4 NN (e.g. by Separation).

• T clearly satisfies Tree 1-3.

• Non-empty subobjects of T are ordered by the restriction of the orderon T , which is a strict order on N, so there will be both a ≤-minimaland ≤-maximal element (i.e. Tree 4 and 5).

• We observed that m 6= n =⇒ Tm 6∼= Tn. If there were a non-trivialautomorphism on T , some trees Tn and Tm representing distinct n andm would be isomorphic, . So T is rigid.

10One approach is to observe that morphisms S : N → N correspond to morphismsP(S) : P(N) → (P(N) ∈ Hom(P(N),P(N)). So, for A ⊆ N , i.e. A : 1 → P(N), define↑A :=

⋃n∈N snA : 1 → P(N), which exists, as E has limits. So define least(A) := a iff

∀b ∈↑A,∀n ∈ N, Snb 6= a. I.e. no finite applications of S to another object in ↑A yieldsa.

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2.3.9 Choice

Aim: model the construction of a function from a set to its union such thatf(A) ∈ A.

Translation: Let T 6∼= 0 be a tree. There is a function f : T →⋃T ,

such that for any T -point p, with ↑ p 6∼= 0, there is a ↑ p-point q such thatf(↑q) ∼= ↑p.

Proof. Let S = t node in T : there is a point covering t. Observe that Sare the points of

⋃T .

Let A : S → P be the ancestor map from S to the T -points, P .

Suppose f, f ′ : P → X are parallel morphisms to some object X suchthat f A = f ′ A : S → X. So f = f ′ on the ancestors of S. f(n) 6= f ′(n)implies n is an uncovered node, i.e. ↑ n ∼= 0. But we deliberately excludedsuch points, so f = f ′. So A is epi.

By Lemma 4 (3), A is epi iff for every p : 1 → P there is a q : 1 → Ssuch that Aq = p. I.e. A is epi iff for every p ∈ P there is a q ∈ S such thatA(q) = p.

E satisfies (topos-)Choice. So, there is a morphism σ : P → S such thatA σ is the identity morphism idP . So σ is an arrow from the P , the pointsof T , to S, the points of

⋃T .

This is most of the way to a choice function. Informally, it is a map frommembers (points) of T to members of members of T .

Using σ we define a tree representing a (set theoretic) function f : X →⋃X. ZFC-functions are sets of ordered pairs. Informally, take a copy of each

pair-trees 〈p, σ(p)〉, remove their respective roots, and attached them all toa new root (or identify all their roots).

Since E models the ZFC axiom Pairing, Extensionality, and Comprehen-sion, we can construct the (ordered-)pair-tree 〈p, σ(p)〉.

The object of points of the function tree K is collection of all such pairs,and is an object and subobject of (P × S) (as in § 2.3.5).

Finally, F = ↑ K, where ↑ K is the tree of pairs 〈↑ p, ↑ σ(p)〉. We canconstruct this as ↑ K = 1 + K + B, where B 4 T × T is an appropriateobject, as in § 2.3.5.

36


F is ordered by the appropriate ordered-pair ordering, and then by theinherited ordering on each of the copies of ↑p, ↑σ(p).

As ever, F inherits most of its structure from the branches from which itis made. Each branch is a subtree, and hence a tree, so the tree axioms holdfor F .

2.3.10 Schema of Replacement

Translation: let Φ be the LT translation of LZFC sentence φ, with free (tree)variables amongst x, y, T, w, so S is not a free variable. Then:

∀T∀w, if, for all T -points pT there is a unique tree (equivalence class) tsuch that Φ(pT , t, w, T ) then there is a tree S such that for all T -points, pT ,q is a S-point iff Φ(pT , q, w, T ).

Replacement is slightly more complicated, e.g. see Goldblatt [11]. Ouraim so far has been to show that for any ‘honest’ set S in M (our model ofset theory), S corresponds to a tree in E .

However, MLM (particularly Mac Lane) were strongly against the inclu-sion of a Replacement axiom, and it is notably absent from their proof. TheMLM axioms of E are too weak to prove Replacement [22]. Lawvere’s originalETCS omitted it as well [19].

To model all of BZFC, we diverge from the MLM construction.

Method 1: Assume Replacement

Osius’ account is motivated by considering the construction so far [28].Infinity holds if(f) the topos has an NNO. Exactly similarly, Choice holdsif(f) our topos has Choice11. Osius suggests that Replacement follows thesame pattern:

(RepT) If Φ(M,C) is a LE -formula, and A,B,C are objects, then thefollowing is an axiom:

∀A ∃B ∀(A M→ Ω) (∃C Φ(M,C) =⇒ ∃C(C 4 B ∧ Φ(M,C))).

Where C 4 B means ∃f : C → B which is 1-1 [24].

11E ’s power object does other work besides proving Power Set

37


This closely resembles the following (ZF-equivalent) form of Replacement(in LZFC):

∀a ∃b ∀m ∈ a(∃c φ(m, c)→ ∃c ∈ b φ(m, c))

Theorem 1. 12 (RepT) =⇒ Replacement holds in the E-model.

Proof. (RepT) is deliberately chosen to make this trivial. It is ‘stronger’than (though ZF-equivalent to) Replacement, as the function Φ need notbe 1-1, so (RepT) is more like a schema of Collection [18]. As E proves(ZF-)Separation, we infer Replacement from Collection.

Method 2: Transitive Inclusion

Rather than BZFC, Mitchell13 [26] considers a set theory, Z1, which ex-cludes Replacement in favour of:

(M) Every set is contained in a (least) transitive set.

This is strictly weaker than Replacement, as Vω+ω is stratified, whilstReplacement implies that every set (in the model) lies in a transitive set.However, there are good reasons to want the weaker (M). In particular,(M) implies ∈-induction ([9] §6.3.6).

Theorem 2. (M) holds in the E-model.

Proof Sketch. A set X is transitive iff x ∈ y ∈ X =⇒ x ∈ X. We constructa tree representing the smallest transitive set T ⊇ X.

We define a process A : E → E which adds a copy of the t-points ‘next to’t, for each t ∈ T . A is iterated on T (representing X), so that the ‘limit’ treeis fixed under A. (In the first iteration, the second layer nodes are adjoinedas points. To ensure closure, the new second layer, including the old thirdlayer, needs to be adjoined).

Formally, define B 4 T × T as B := 〈t, t′〉 : t ∈ T, t′ ∈↑ t.B + T + 1 is ordered:

1. 0 ≤ S for all S in B + T + 1, for 0 ∈ 1.

2. t ≤ t′ iff t≤T t′ for all t, t′ ∈ T12[28] §9.513Goldblatt has a similar approach [11] §12.4.

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3. (t, s) ≤ (t, s′) iff s≤S s′ for t, s, s′ ∈ T

4. (t, r) ≤ (t′, s) iff t≤T t′ for r, s, t, t′ ∈ T

Define an equivalence relation ∼ as t ∼ t′ iff α : ↑ t ∼= ↑ t′ is a treeisomorphism. ∼ identifies any node and a ‘copy’ of itself. E has limits,colimits, and quotients, so contains B + T + 1 and the transitive closure ofT , TC(T ) := (B + T + 1)/ ∼. TC(T ) is ordered by the inherited quotient-ordering.

Lemma 9 ensures rigidity, as the only possible non-rigid branches are thebranches of points from the original tree. But the original tree is rigid, sothere are no witnessing pairs of points.

TC(T ) inherits its further tree structure from the underlying trees. Thisensures that the induced ordering on the equivalence classes is a tree ordering.

Method 3: Image of Trees

Finally, McLarty emphasises the Cantorian idea behind Replacement,that the image of a set under a function is a set [24]:

(II) ∀E-relation R(x, Y ) of arrows x to sets Y in E , ∀A, if∀x ∈ A ∃!Sx (R(x, Sx)) then there is a set S such that

∃f : S → A ∀x ∈ A(f−1(x) = Sx).

So if each x ∈ A (i.e. x : 1→ A) is assigned a unique (equivalence class)Sx by R, then there is a set S and an arrow f : S → A such that for any Sxwith R(x, Sx), there is a map i : Sx → S such that Sx is the pullback of Aunder f , i.e. the set Sx is the f -inverse image of x.

Sx

1 S

A

i

x

f

So S =∐

x Sx, so we have a tree representing the set Sx|x ∈ A.(II) has a tree translation, (IIT ), with a tree TY representing each set Y ,

etc.

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Theorem 3. (IIT) =⇒ Replacement holds in the E-model.

Proof. Method: (IIT ) =⇒ (RepT)

RTP: ∀A ∃B ∀(A M→ Ω) (∃C Φ(M,C) =⇒ ∃C(C 4 B ∧ Φ(M,C))).

Let R(x, Y ) be the E-relation:

(Φ(x, Y ) ∧ ∀Z(Φ(x, Z) =⇒ ∃f : y1−1−→ Z ) ∨ (Y ∼= ∅ ∧ ¬∃ZΦ(x, Z)).

An epi f : y → Z exists, as minimal elements are unique up to isomor-phism and agree up to isomorphism with the minimal elements from anotherset Z where F (x, Z) holds, see [24].

So: ∅ has a 1-1 function from C to B, and the inverse image of x ∈ Aalong any f : S → A has a 1-1 function g : f−1(x)→ S.

Let B = S. Then (RepT) follows from R by (IIT ).

2.3.11 Taking Stock: Replacement At All?

If we really want full Replacement, we stipulate Osius’ (RepT). This isthe approach for modelling Infinity and Choice. Why is Replacement anydifferent?

One standard reason for excluding Replacement is that it may only beessential for ‘artificial examples’, which are legitimated by ZFC, but are notused in concrete mathematics, e.g. examples from the higher reaches of settheory.

For example, the first non-trivial time replacement fails is in Vω+ω [18],Vω+ω ` ZFC-Replacement. This already seems large for concrete mathemat-ics. Almost any part of concrete mathematics takes places in a universe nolarger than Vω1 , so for most concrete purposes we might only need countablereplacement [14].

(CountRep) ∀φ ∈ Form(LZFC) if Free(φ) ⊆ X, x, y, w, then∀X∀w[∀n(n ∈ ω =⇒ ∃!yφ) =⇒ ∃X∀x(x ∈ ω =⇒ ∃y(y ∈ X ∧ φ))]

Whilst Vω+ω does not satisfy (CountRep), Vω1 does. This could be a‘small’ model for parts of concrete mathematics.

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Success?

The strongest alternatives to the MLM construction would be a categor-ical foundation of full ZFC, including unbounded Separation and Replace-ment. However, this may be overly strong for concrete mathematics. ZFCwithout Replacement, or with Countable Replacement might be sufficient.

If E does not model (CountRep), then we could stipulate an axiomsimilar to (RepT), which only models countable replacement, the obvious“Countable-RepT”.

So far, we have stayed close to set theory (with non-set theoretic foun-dations in mind). More radically, we discuss direct categorical foundations,without reference to set theory. We outline several starting points and ap-proaches. A more full investigation would be worthwhile.

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2.4 Algebra, Categorically


Concrete mathematics uncontroversially contains some algebraic structures,e.g. (at least) theories of finite groups and finite dimension vector spaces (forlinear algebra). We can then build rings from groups, etc. We focus on asketch of part of group theory.

Encoding a Finite Group in ZFC

Consider a finite group G = (g1, ..., gn). Take an n-size set, e.g. n ∈ On.Let i ∈ n represent gi. Define a 2-place multiplication function M(−,−) :n2 → n such that M(gi, gj) 7→ gi ·G gj. We then encode G as the pair〈n,M〉 ∈ V .

〈n,M〉 is not unique, but demonstrates that we can encode the algebraicstructure through sets. How do we directly interpret finite groups categori-cally?

2.4.1 Clarification

Problem one: What is the Goal?

The standard categorical definition of a group14 generates a particulargroup, not the category of groups, Grp. We are searching for a rich category,C, with some categorical conditions (like E), and a translation scheme fromthe language of group theory LGT to the language of the category LC :

T : LGT → LCsuch that for a group theoretic theorem, φ, T (φ) is true, and can be verified.

E.g., a cyclic group Cn (n > 1) witnesses the (LGT -)formula:

φn = ∃G(∃g ∈ G(gn = 1g∧g1 6= 1G∧...∧gn−1 6= 1G)∧∀h ∈ G(h = g∨...∨h = gn))

We aim to translate this sentence and verify its translation.

Problem two: What is the Group Theory?

We first must characterise group theory. The existence of certain groupsdepends on the ambient set theory of the group theory. This can be seen im-mediately in a naturally arising algebraic problem, the Whitehead problem:

14A group G is the Hom-collection of a groupoid CG with one object [1].

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is there an abelian group A which is not free such that Ext(A,Z) = 0? She-lah [30] proved that this is independent of ZFC: assuming ¬CH and Martin’sAxiom, such a group exists, whilst V = L implies there is no such group. Sothere is a genuine question about what constitutes group theory.

Given our primary interest in concrete mathematics, we restrict our grouptheory from its full abstract strength to the theory of finite permutationgroups.

Aim: to construct a rich category which:

1. contains Sn for each n ∈ N,

2. is closed under:

(a) subgroups,

(b) products,

(c) quotients (by normal subgroups),

(d) automorphism groups.

So we aim to prove that the collection of arrows is closed under the appro-priate interpretation of these construction.

2.4.2 Internal Theory of Permutation Groups

Instead of parasitism, we look at the internal theory of permutation groupsin E : there is a ‘natural’ group theory based on the automorphism object forthe object corresponding to each natural number.

Cyclic 2-Group

Starting small, we construct C2 the cyclic 2-group, before generalising.We can think of C2 as:

C2f1 f0

where f0 f1 = f1 = f1 f0 and f0 f0 = f0 = f1 f1.Informally, consider 2 = colim(1, 1). We define the subcollection of

Hom(2, 2) of the morphisms which are mono and epi (invertible endomor-phisms), the automorphism object, Aut(2),.

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Theorem 4. For n ∈ N, Aut(n) represents the permutation group Sn and isin E.

Proof Sketch. n+m and Hom(X, Y ) are objects of E .

Aut(X) 4 Hom(X,X) is unique such that ∀f : 1→ Aut(X), evX (f ×1X)15 is mono and epi.

Hence, Aut(X) is an object in E .

Clearly, the identity map 1X is mono and epi, so belongs to Aut(X)

Associativity The internal composition map, cX : Aut(X) × Aut(X) →Aut(X), is such that:

XX ×XX ×X c×1X−→ XX ×XevX,X−→ X (?)

(Aut(X)× Aut(X))× Aut(X) Aut(X)× (Aut(X)× Aut(X))

Aut(X)× Aut(X) Aut(X)× Aut(X)

Aut(X)

∼=

cX×id id×cX

cXcX

By (?), the above diagram commutes, so multiplication is associative.

Commutativity The appropriate diagram for ‘multiplication by 1’ also com-mutes:

Aut(X) Aut(X)× Aut(X)

Aut(X)× Aut(X) Aut(X)

q×1

11×q cX

cX

15evX,Y : Y X ×X → Y corresponds to the identity map, id : Y X → Y X .

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where q is the (unique) constant morphism q : Aut(X)1→ 1E

q→ Aut(X).

Inverses (with respect to cX) [2], the following commutes:

Aut(X)× Aut(X) Aut(X) Aut(X)× Aut(X)

Aut(X)× Aut(X) Aut(X) Aut(X)× Aut(X)

〈id,id〉

id×inv id inv×id

〈id,id〉

cXcX

These diagrams interpret the familiar equations:

cX(cX(X, Y ), Z) = cX(X, cX(Y, Z))

cX(X, 1) = x = cX(1, X)

cX(X, inv(X)) = 1 = cX(inv(X), X)

So, for a, b ∈ Aut(X), the composite map ab lies in Aut(X), its inversea−1 (i.e. ‘aop’) lies in Aut(X) [4], and 1X is in Aut(X).

Corollary 1. Aut(2) represents C2(= P2).

One can verify that there are exactly 2 global elements of Aut(2), andthat the morphisms of Aut(2) ‘work’ as expected for C2.

Further research is needed to find a subtopos that is closed under thenatural group construction (e.g. subgroups, products, etc.). Constructingthe maps f : Aut(X)→ Aut(Y ) which represent group homomorphism (i.e.respect composition) would be particularly significant.

45

2.5 Analysis, Categorically

2.5 Analysis, Categorically

A minimal requirement on the direct encoding of analytic structures is anencoding of R. Better would be encoding some stronger structures, e.g.Hilbert Spaces, but only some proper subclass of function spaces is necessaryfor concrete mathematics.

We outline a direct encoding of Z, and leave open whether the subtoposis closed under natural analytic constructions.

The encodings of Z, Q, and R are reminiscent of standard set theoreticencodings [24]. For example Z can be represented as a certain equalizer(limit) of functions from a product of the NNO to itself:

〈m,n,m′, n′〉|m+ n′ = m′ + n → (N×N)× (N×N)

N ∈ E and E is closed under finite limits, so this object is in E . This is anequivalence relation on N. The object of equivalence classes, Z, is a colimit(coequalizer) q : N×N→ Z under 〈m,n〉 = m− n.

I.e., for any f : Z → T , there is a f : N ×N → Z such that 〈m,n〉 ∼〈m′, n′〉 =⇒ f(m,n) = f(m′, n′). So, the following commutes:

N×N Z

T

q

ff

If i ∈ Z corresponds to 〈m,n〉 then f(i) = f(m′, n′). This ‘integer-object’,Z, is universally characterised in the style of NNO.

We could define addition and multiplication on Z, in the natural way, thenQ is a quotient object of Z ×N in a similar way. So too R is constructedvia Dedikind cuts [3]. Detailed constructions can be found in McLarty [24],Goldblatt [11], or Johnstone [16] §D4.7.

Significantly, the constructions are structurally very similar to those ofZFC. The difference is that ZF(C) defines e.g. quotients and powersetsuniquely via elements, whilst a topos theoretic foundation is only ever definedup to isomorphism.

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2.6 Discrete Mathematics, Categorically

2.6 Discrete Mathematics, Categorically

The standard candidates for concrete discrete mathematics are finite combi-natorics and graph theory16; we comment on the latter. A graph consists ofa collection of edges and vertices, both must be encoded.

Given VG, some collection of appropriate ‘size’, we may interpret an (undi-rected) graph as a map G : VG → P (VG) which takes a vertex, v, to thecollection of vertices connected to v. This is (unsurprisingly) close to thenotion of a ‘tree’ in the E construction above, which employs graph theoreticnotions17.

This suggests that direct foundations of graph theory will be similar tothe E-foundation for BZFC. So, we might prefer the more complicated butcomprehensive foundation of BZFC, as opposed to a similar-but-piecemealfoundation for graph theory.

16Lawvere seems to think graph theory is not an analysis of a single category, but aseries of linked categories, which theorists move between, without adequate justification[20].

17A node of an arbitrary graph can have several ancestors.

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3 Topological Foundations

To counter the intuition that topological foundations are implausible (due totopology’s reliance on set theory), we analyse the candidate foundations sofar.

Each time we start with some primitives notions, some basic entities18,and some basic and ‘advanced’ constructions19, which allow us to build enti-ties from previous ones.

• In ZFC, the primitive notions are membership and set; the basic setsare ∅ and ω; the basic constructions are (perhaps) pairing, separation,and union; the advanced constructions might be choice functions andreplacing sets, etc.

• In E , the basic entities are 0, 1, and N, the basic constructions arethe finite limits, and the advanced construction is the power object (orequivalently, Ω).

Question Which topological basic entities, basic constructions, and ad-vanced constructions are required in order to provide a model for a rich settheory, and more specifically for a large fragment of concrete mathematics?

Method of Foundation We want to build a rich category, C, froma small collection of natural axioms which has a topological realisation. Acandidate plausible foundation must have:

1. enough morphisms, objects, and subobjects (i.e. continuous functions,spaces, and subspaces),

2. ways to build objects,

3. ways to specify subobjects.

18‘Basic entities’ correspond to unconditional existence claims: some entities are stipu-lated in our theory, rather than constructed from previously ‘known’ entities.

19The distinction between basic and advanced constructions may not be hard and fast.

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3.1 Interpreting ZFC

3.1 Interpreting ZFC

Attempt 1

Suppose C = top. Then Vα × Vα, 2 ∈ C. Further, we have the map

∈α: Vα × Vα → 2

So we can simply encode the ∈-relation as the preimage ∈−1α (1), whichgives the ∈-hierarchy on Vα, a model of ZF(C) for sufficient α.

As before, we can then parasitically found concrete mathematics via ZFC.Unlike with E , which only models BZFC, there are no boundedness issueshere.

However, this is unsatisfactory as a foundation. This only shows thatthere is a model for ZFC ‘somewhere in’ the category top. We needed priorknowledge of how top works. But we have not yet constructed top at all.

Instead, we want to stipulate some basic entities and constructions, andbuild a rich category which contains a model of ZFC.

Attempt 2

Suppose C contains all discrete spaces.

Informally, the discrete spaces have no significant internal structure (think-ing set theoretically, every singleton is open, so no point is special). So Chas a representative for each cardinal κ. So we want to find an ‘interesting’relation:

Eκ : κ× κ→ 2

with the ‘right’ ∈-structure. However, that C contains the discrete spacesdoes not give us enough information about the morphisms in the category.So we may not have such an ‘interesting’ relation, like ∈α above.

To ensure we do have enough morphisms (i.e. continuous maps), we needto stipulate strong axioms, e.g. § 3.2.8.

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3.2 Topological Axioms

Building TOP

Dow and Watson promisingly demonstrate that a few axioms suffice toguarantee that C = top in a surrounding universe of ZFC [7]. Internalisationof such a construction would allow a foundation of ZFC, as in Attempt 2.

1. C is closed under

(a) sums

(b) quotients

(c) finite products

2. ω + 1 ∈ C

3. ∀κ ∈ Card≥ω, ∃X ∈ C, ∃U ⊆ X open discrete subset with |X| = κwhich witnesses tightness κ exactly.

Theorem 5 (Dow & Watson).

If 1., 2., and 3. hold, then C = top.

Their construction essentially requires an ambient (external) set theory,especially in their use of quotients. So it is not fit for our purposes of in-ternally constructing top, and then interpreting ZFC. However, it provideshope that such an internal construction is possible.


Aim: provide a collection of internal axioms with which we can constructa subcategory rich enough to found areas of concrete mathematics, withoutconcessions to extra theory (e.g. ambient set theory).

We list some possible axioms, roughly in order of natural plausibility. Ifnecessary, the axioms could be formalised into LT , the language of topology,which has two unary predicates:

1. sp(X), being a space (i.e. object) in the category.

2. cm(f), being a continuous map in the category between spaces in thecategory.

There are also the obvious n-ary predicates of composition, domain, andcodomain.

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3.2.1 0 Space, 1 Point Space and Non-Degeneracy

Top 1. There is a 0 space in C, i.e. ∃0 ∈ C, such that for any object x ∈ C,there is a unique morphism ! : 0→ x.

Top 2. There is a 1-point space in C, i.e. ∃1 ∈ C, such that for any objectx ∈ C, there is a unique morphism ! : x→ 1.

Top 3. 0 6= 1.

3.2.2 Finite Limits

Top 4. C has finite limits

top has all finite (co-)limits so this is a reasonable axiom. We are par-ticularly interested in the binary product, Z = X × Y :

W

X X × Y Y

ff×g g

πY

πY

3.2.3 Finite Colimits

Top 5. C has finite colimits.

The direct sum Z = X + Y is a colimit:

X X + Y Y

W

ff+g

g

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Definition 28 (Discrete 2 Space). 2D := 1 + 1.

1D 0D

Figure 1: 2D

The preimage, f−1(1), under a function f : X → 2D is:

X f−1(1) 1

2D

f(f,id)

inc

Definition 29 (Clopen Subspace). For X,Z ∈ C, Z is a clopen subspace ofX, written Z ⊆ X clopen, iff ∃f : X → 2D such that f−1(0) = Z

Given limits and colimits, this definition makes sense, as 1, 2 ∈ C, andwe can take preimages.

Definition 30 (Clopen Complement). If ∃f : X → 2D (f−1(0x) = Y )(i.e. Y ⊆ X), X\Y = f−1(0y). By finite colimits, X\Y ∈ C.

Lemma 10. A finite discrete space is a finite sum of copies of the 1 pointspace.

Proof. Let N be the discrete space on n(∈ ω) elements.Let σ(n) =

⊕n1 1i.

A ⊆ σ(n) is a singleton iff A = 0i.∀i, 1i is discrete, so ∀i, A = 0i is open.A space is discrete iff all singletons are open.So σ(n) is a discrete space on n elements.So σ(n) ∼= N

So if 1 ∈ C and C has binary sums, C contains the finite discrete spaces.

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3.2.4 Sierpinski Space

Top 6. ∃S ∈ C such that ∃0S, 1S : 1 → S such that 0S 6= 1S, ∀t : 1 → St = 0S ∨ t = 1S, and ∀m : S → S, m(0S) = 1S =⇒ m(1S) = m(0S) = 1S.

1S 0S

Figure 2: S

Definition 31 (Open). Z ∈ C is (X-)open if ∃X ∈ C ∃f : X → S such thatf−1(1S) = Z.

3.2.5 Omega

Top 7. There is a space ω ∈ C such that any singleton of ω (global elementg : 1→ ω) is open and ω is infinite.

We can encode “being infinite” by stipulating that C has an NNO (e.g.ω itself!). Then ω is infinite means there are maps x : 1→ ω and f : ω → ω,and a map h : ω → N such that the following commutes:

1 ω ω

N N

x

O

f

h h

s

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3.2.6 Countable Products and Powers

It would be very useful to have countable products, e.g. in building theCantor Set, 2ω.

However, unlike with binary products, we cannot write down the obvi-ous formula (or diagram) characterising the rule: our logic is finitary so wecannot describe a countable limit, as we cannot write down infinitely manyquantifiers.

Nor is it clear that we can express a countable family of spaces, as wedo not have an ambient set theory. So we cannot treat this the family as a(set-)function of (set) omega.

Let’s ignore these worries for now, and write the following formula whichis not in LT :

∀I : C → ω,∃P →∏n∈ω

(I−1(n))

Formalising slightly:

Top 8. Suppose Xn are (somehow!) indexed by ω. Then ∃Z =∏

n∈ωXn ∈ Ci.e. if there are countably many maps fi : W → Xi, then there is a uniquemap (fi)ω : W →

∏Xi such that all(!) diagrams of the following shape

commute:

W

∏Xn Xi

(fi)ωfi

πi

Definition 32 (Cantor Set). The Cantor Set is the product of countablymany copies of 2D,

⋃i∈ω 2D,(i) = 2ω

This definition makes sense given 1 ∈ C, finite limits and countable prod-ucts.

Generally, countable (infinite) constructions are hard to formalise in LT .There is an option for countable (self-)powers :

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(Powers) ∀X ∃Z and a map Z × ω → X such that ∀W with P :W × ω → X there is a map f : W → Z such that following commutes:

W × ω Z × ω

X

f×id

Pπ

We could write Z = Xω.

The intuition is that P is a family of maps pn : ω → X such that pn(x) =P (x, n) and the π is a family of projections πn : Z → X such that πn(x) =π(x, n).

The diagram is not quite a limit, so (Powers) seems not to be impliedby C’s finite limits. However, (Powers) is similar to limits in spirit: Z is auniversal object of a more complicated (finite!) diagram.

Question: Can (Powers) be used to construct countable products orsums of spaces? If so, this seems good motivation for taking it is an axiom.

3.2.7 Indiscrete Space and Discretisation

Top 9. ∃2I ∈ C such that ∃0I , 1I : 1→ 2I such that 0I 6= 1I , ∀t : 1→ 2I t =0I ∨ t = 1I and ∀m : 2I → 2D, m(0) = m(1).

1I0I

Figure 3: 2I

Definition 33 (Subset). A ⊆ X iff ∃f : X → 2I (f−1(0I) = A).

55


Top 10. ∀X ∃D ∀Z(Z is D-clopen ↔ Z ⊆ X and Z ⊆ D ↔ Z ⊆ X).

I.e. S ⊆ D ⇐⇒ S ⊆ X, but every S ⊆ D is open.

Definition 34 (Equally Large). X, Y ∈ C are equally large iff the respectivediscretisations DX

∼= DY , i.e. there is a f : DX → DY such that f−1 : DY →DX is a morphism from C.

3.2.8 Separation

Separation lets us specify subspaces from spaces. It is not clear how strongthe separation axiom/construction should be. Its strength will determinewhich subobjects (i.e. subspaces) are in the category. There are two methods:

1. Separation-like constructions using previous axioms.

2. Stipulating new axioms.

If C is fulltop for X, Y (i.e. HomC(X, Y ) = Homtop(X, Y )) then manyspaces can be constructed by method 1. E.g.: open, closed, finite, and co-finite subspaces. However, there are major issues here:

• These proofs assume C is (somewhat) fulltop. However: we cannotwrite down a formula expressing this, there is no way to pick out topfrom within C. Indeed, C might believe C = top.

• We want more subspaces, beyond the open, closed, and (co-)finite sub-spaces of the spaces in C.

Method 2 is also flawed, as expressing a separation axiom is difficult.The naıve axiom “∀X ∈ C if Z is a subspace of X, Z ∈ C” is trivial, it isformalised:

∀X ∈ C ∀Z ∈ C(subspace(Z,X)→ Z ∈ C)

However, ‘externally’ we may ‘know’ Z has other subspaces. We reallywant ‘top-separation’:

(TopS) ∀X ∈ C if Z is a top-subspace of X, Z ∈ C.

Again we cannot express (TopS) from within LT . We are building top,we cannot rely on any set-theoretic knowledge of topology.

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Instead, we might stipulate a schema where each axiom constructs thesubspace which is characterised by a suitable formula in LT . However, thisis most easily expressed using a serious deviation into set theory, which isnot obviously avoidable.

3.2.9 Unit Interval

Many combinations of the above axioms only yield zero-dimensional spaces.

Lemma 11. A space generated from 1 by finite products and sums is zero-dimensional

Proof. By induction:1 is zero-dimensional (0 is a clopen basis).Suppose X, Y are zero-dimensional, so have clopen bases B,B′ respec-

tively.Then B ∪ B′ is a clopen basis for X + Y . So X + Y is zero-dimensional.B × B′|B ∈ B ∧ B′ ∈ B′ is a clopen basis for X × Y . So X × Y is

zero-dimensional.

However, we want ‘larger’ spaces, e.g. for founding analysis. It is notclear how to do this without some one-dimensional spaces, something that‘looks like’ [0,1] or R. One-dimensional spaces help found analysis, whilstalso simplifying founding graph theory (§ 3.4.2).

If [0, 1] cannot be constructed, we can stipulate [0, 1] ∈ C, e.g. using this(non-ordered) topological characterisation [35]:

Top 11. ∃I ∈ C such that I is separable, compact, connected, and for everyx ∈ X\a, b, the space X\x is not connected.

We then try to internalise these properties in LT .

1. Connected: I has no non-empty disjoint clopen cover. More fully: ¬∃non-trivial f, f ′, g, g′ : I → S such that f−1(1S) = (f ′)−1(0S) andg−1(1S) = X\f−1(1S) and (g′)−1(0S) = X\f−1(1S).

2. Separable: ∃f : I → 2D such that D = f−1(1), D and ω are equallylarge, and for all open sets O, O ∩D 6= ∅.

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The last condition can be encoded ∀f : I → 2D, if (f−1(D) ⊆ (0∨1) and (f−1(O) ⊆ (0∨1) then (f−1(O) ⊆ 0 ∧f−1(D) ⊆0) ∨ (f−1(O) ⊆ 1 ∧ f−1(D) ⊆ 1)

3. Internal compactness: ∀Y the projection π : X × Y → Y is closed.

Kuratowski’s Theorem says for T2 spaces, X is compact iff ∀Y , π :X × Y → Y is closed ([8] 3.1.16). But this internalisation has two issues.Firstly, we have not encoded what it means for a map to be closed, thoughpresumably we could do this.

More seriously, Engleking’s proof requires ambient set theory, e.g. itlooks at countable-properties of families of closed subsets. So Kuratowski’stheorem may not be an internal theorem of C.

Finally we note that these ‘internal’ properties may not correspond to theexpected ‘external’ version. So the internal [0,1] constructed might be verydifferent from the canonical unity interval.

Alternatively we could construct [0,1] as a quotient. Quotients are thecoequalizers of internal equivalence relations (congruences) [33].

X X/ ∼

Y

π

g∃!f

However, it is unclear whether we can internalise the necessary equiva-lence relations. This motivates another dubious axiom:

(TopQ) All top-congruences are C-congruences.

Lemma 12 (Quotient Construction: Cantor’s Staircase). [0, 1] = 2ω/Rwhere R is a specific equivalence relation.

Proof Sketch. Let R : 2ω×2ω → t, f identify eventually constant sequences(end points) with their ‘neighbour’, which is also eventually constant, e.g.(0, 1, 1, ...) ∼= (1, 0, 0, ...). Then [0, 1] ∼= 2ω/R.

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To drive the intuition of the proof, consider the ‘middle-third’ construc-tion of the Cantor Set, C ⊂ [0, 1], with sequences (di)ω, such that di ∈ 0, 2.Define the equivalence relation r : C × C → t, f as

r :∑di

3i→∑di/2

2i

Then r(2ω) = [0, 1].

E.g. q = (0, 2, 0, 2, ...) represents 1/4. So

r(q) =∑ 1

22i=∑

(1

4)i = 1/3 ∈ 2ω/r

Figure 4: Cantor’s Staircase [25]

So, if 2ω ∈ C, and C contains the congruence (e.g. (TopQ)), then (ahomeomorphic copy of) [0, 1] ∈ C.

However, [0,1] is not a subspace of 2ω. For example, subspaces of zero-dimensional spaces are zero-dimensional20. 2ω is zero-dimensional, [0,1] isnot (it is one-dimensional!) so it cannot be subspace of 2ω.

20Take a clopen basis for the superspace. The intersection of these sets with the subspaceis a clopen basis for the subspace [8].

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To generate [0,1] in this way, we require an axiom strong enough to lookat subsets of spaces, not just subspaces. It is unclear if the appropriateequivalence relation can be expressed in LT , without an ambient set theory.

Lemma 13. The Sierpinski set, S, is a quotient of [0,1], in the obvious way.

3.2.10 Arbitrary Products and Sums

As with countable products, it is not clear that either of these can be ex-pressed within the language, as (1) our logic is finitistic, so no formula hasinfinitely many quantifiers, and (2) we do not have an ambient set theory toe.g. recursively define an infinite limit.

Question If C contains large discrete spaces, e.g. ω, D of size |2ω|, canthese be used to ‘index’ large products in C?

3.2.11 A Grab Bag of Advanced Constructions

The following are natural results, but it is less clear whether they should beconstructions (i.e. axioms) of the foundation.

1. Hyperspaces

These correspond to the subobject classifier in the topos.

2. Gδ subspaces

3. Stone-Cech compactifications

4. βω

Lemma 14. The ω can be constructed from βω using a weak separationaxiom, in the obvious way.

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3.3 Function Spaces: Analysis & Algebra


As before, we search for direct encodings of certain areas of concrete math-ematics in a rich universe of topological spaces (that is closed under certainconstructions), rather than foundations which are parasitic on set or categorytheory.

3.3.1 Analysis

Analysis has strong conceptual ties to topology. A sticking point is functionalanalysis. Given an interpretation of the domain and codomain, categorytheory and set theory do well in interpreting function spaces. A topologicalfoundation may naturally interpret the domain and codomain (as spaces!),but may struggle to internally encode a function space.

(IFS) Given spaces X, Y ∈ C, the space HomC(X, Y ) = Y X ∈ C.

This axiom corresponds to a topos E being Cartesian closed. Carte-sian closure is equivalent to containing an internalisation of all hom-objects.(IFS) is the case with function spaces.

Lemma 15. Suppose 1, ω ∈ C, C has finite limits, and (IFS) holds in C.Then C contains 2ω.

Proof. By assumption, 1-point space, 1 ∈ C.So, by the binary sum construction 1 + 1 = 2D ∈ C.By assumption, ω ∈ C.By (IFS), Hom(2D, ω) = 2ω ∈ C

So we can construct some internal version of the Cantor Set. As with[0,1] in § 3.2.9, this internal 2ω may not be the canonical, external CantorSet.

However, if (IFS) does not hold in C, then these hom-objects do notwork like function spaces. In particular, in a function space we expect that:

(XY )Z ∼= XY×Z

i.e. the evaluation map and the composition map are continuous.

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This fails in general. For example, there is no topology on the space of(continuous) functions f : Q → R such that both the evaluation map andthe (point-wise) composition map are continuous ([8] §2.6.E).

To ensure that these maps are continuous on all of C, we need:

(LCT2) ∀X, X is locally compact21 and T2.

Of course, many top-spaces are not locally compact Hausdorff, but theE foundations also did not use all the objects of E . The absence of certainspaces from our chosen foundational subcategory is not necessarily problem-atic.

We simply note that it is unclear if (LCT2) subcategory has all the spaces(and morphisms!) required to interpret ZFC.

3.3.2 Algebra

To interpret algebraic structure, we need to express ‘addition’ of elements.We can do this by considering the function spaces of certain spaces. For ex-ample, taking the function space on a finite discrete space yields a topologicalanalogue of a finite permutation group.

Gartside and Smith, classify the (closed) subgroups of profinite groups,and show this space carries a natural (profinite) topology, which can oftenbe classified up to homeomorphism [10]. This classification can be used tointerpret these groups in a suitably rich subcategory of top. Founding theprofinite groups gives plenty of algebraic structure for concrete mathematics.

Theorem 6 (Gartside & Smith). Let G be a nilpotent pro-p group with weightw(G) ≤ ℵ1. Then the set of closed subgroups S(G) has a natural topologysuch that it lies in precisely one of the following four classes.

1. S(G) is a finite discrete space iff G is finite.

2. S(G) ∼= ωn+ 1 for n ∈ N iff G is virtually Zp.

3. S(G) ∼= Pe lczynski space22 iff G is finitely generated and h(G) > 1.

4. S(G) ∼= 2w(G) iff G is not finitely generated.

21Assuming we can internalise these properties in C.22A certain countably based profinite space ([10] pg3).

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3.4 Discrete Mathematics


3.4.1 Graphs Through Closed Sets

Definition 35 (Space of a Graph). The space (XG, T ) of a graph G is suchthat XG = V ∪ E and T is defined as: C ⊆ V ∪ E is closed iff ∀e ∈E ∩ C, ep(e) ∈ C (where ep(V )→ E takes a vertex to its end points).

This definition does preserve the right information. For example, considera graph:

1 2

3

4 5

The subcollection 2, 3, 5, (2, 3), (3, 5), (2, 5) ⊂ V ∪ E is represented bya closed subspace, so should represent a subgraph (i.e. all edges should haveinternal vertices), and it does.

2

3

5

The above lightly uses ambient set theory, but this seems translatable LT(e.g. intersection in § 3.2.9).

3.4.2 Graphs Through Quotients and Glue

In the topological representation of a graph, TG, each vertex is representedby a distinct point and each edge by a distinct arc, homeomorphic to [0,1].These are ‘glued’ together in a suitable way [13].

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Lemma 16. (TopQ) =⇒ finite TG ∈ C

Proof Sketch. We prove we can ‘glue’ the ‘0’s of finitely many copies of [0,1].

• [0, 1]i ∈ C.

• C has finite limits, so S =∑

[0, 1]i ∈ C

• Define top-congruence r : S → S by ∀x ∈ S\0i, r(x) := x, otherwiser(0i) := 0i.

• By (TopQ), r is a C-congruence.

S/r ∈ C has the required properties.

Graph theorists sometimes use the analytic structure of the edges of agraph, motivating this model. However, it relies on the dubious (TopQ).

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4 Algebraic Foundations

Set theory and category theory both provided rich and plausible foundationsfor mathematics. There is some hope for a topological foundation, too. Wenow turn to our final candidate foundation: algebra.

We suggest possible approaches for an algebraic foundation, and notesome limitations of these attempts. This is still very much in the exploratoryphase.

Question Which basic algebraic entities and constructions are requiredfor a model for a substantial fragment of concrete mathematics?

4.1 Interpreting Algebra

Concrete mathematics contains at least some algebra, so algebraic founda-tions have a head start. Presumably, the basic entities will be amongst thosewe hoped to found, e.g. groups, fields, etc. Hence, there is a trivial interpreta-tion of algebraic structure ‘from within’ algebra, the ‘identity’ interpretation.For example, if C2 is primitive, interpret C2 as itself.

4.2 Basic Entities

One plausible collection of basic entities would be the free group on n letters∀n ∈ N. Infinite groups seem necessary for interpreting analytic structure,and for each n for e.g. encoding functions on n variables. We can also buildthe richer algebraic objects (fields, rings) from groups.

Problem There is no way to take direct sums (or other colimits) ofgroups. This is a major restriction on expressiveness: most concrete theoriesmake use of limits and colimits. This seems an unavoidable problem of agroup theoretic foundation.

This doesn’t seem solvable by taking richer algebraic structures as thebasic entities: we can interpret this richer structure in the theory of groups,so the language cannot have more expressive power. For example, fields don’thave sums - they don’t even have (direct) products!

Generally, candidate algebraic subcategories have very few objects andextremely few morphisms.

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4.3 Graphs and Matrices

4.3 Graphs and Matrices

Take graph theory as an example for directly interpreting. We might rep-resent each graph by its adjacency or incidence matrix [12]. So a theory of‘binary matrices’ allows us to encode graphs.

1 3

2 4

This graph, G, could be represented by its adjacency matrix:0 1 1 11 0 0 11 0 0 01 1 0 0

For this kind of representation, our foundation must include:

1. A collection, S, |S| ≥ 2 (e.g. C2)

2. The general theory of matrices on a collection S.

Problem 2. seems too structurally complex to be a natural axiom/stipulation.

Problem There is no clear way to ‘connect’ graphs. Suppose we want toconnect two graphs at a base point, e.g. linking ‘3’ of once copy of G to ‘4’ ofanother copy. This requires an appropriate addition of matrices (of possiblydifferent size), in this case an 8 × 8 matrix. It’s not clear how to generallydescribe such an addition.

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5 Conclusion

Because ZFC has founded much of (concrete) mathematics, there has devel-oped a consensus that ZFC is ‘the’ foundation of mathematics (whatever thismeans). We have given some counter-evidence that rocks this boat.

ZFC is not alone in providing a foundation for concrete mathematics. Thetopos E is also a rich constructive arena for doing mathematics. It modelsmuch of BZFC, but also has its own natural interpretations for concretemathematics. We might say that where set theory formalises ‘in-ness’ (∈),category theory formalises ‘between-ness’ (→). Both are so simple, it maybe unsurprising we can derive a great deal from these notions.

Topos theory has a ‘distinctively foundational’ flavour. But there is hopein the use of more concrete areas as a base theory. Whilst algebraic foun-dations suffer from the apparent inability to express coproducts/sums, atopological foundation has some hope. Various axioms suggest that a richtopological theory could be internally constructed, to interpret concrete areasof mathematics, either parasitically or directly. There is fruitful foundationalwork to be done in these ‘more concrete’ areas, not just in basic, deliberatelyfoundational accounts.

These kinds of results may clear the air in terms of how we approachfoundations, and provide some reason to reassess the emphasis and focuson set theory as the basic language of mathematics. Instead, some kind of‘foundational pluralism’ might be advantageous, using the different founda-tions when it naturally interprets a concrete theory. Crucially, we know thatinter-interpretation is always possible, if necessary.

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References

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