some statements weaker than wor

Some statements weaker than WOR

Liuzhen Wujoint work with Ralf Schindler and Liang Yu

Institute of MathematicsChinese Academy of Sciences

October 22, 2016

Liuzhen Wu (AMSS) Some statements weaker than WOR October 22, 2016 1 / 21

Axiom of Choice Background

The statement of Axiom of Choice

Throughout this talk, we assume ZF, the Zermelo-Frankel set theory, as theground theory. Axiom of Choice has numerous equivalent forms in literature, all ofwhich can be treated as ”the” statement of Axiom of Choice.

Definition (Axiom of Choice)

The Axiom of Choice is refer to one of the following statements:

1 For any set X of nonempty sets, there exists a choice function f defined on X .

2 (Well-ordering Principle WO) Every set can be well-ordered.

3 (Zorn’s lemma) Every non-empty partially ordered set in which every chain(i.e. totally ordered subset) has an upper bound contains at least onemaximal element.



Consequences of Axiom of Choice: holy side

Followings are some consequences of AC:

1 Boolean prime ideal Theorem.

2 Tychonoff’s theorem in topology

3 Hahn-Banach Theorem.

4 Any consequence of Zorn’s Lemma appeared in Algebra textbook.

5 Any “correct” statements need AC while you are not aware of.



Consequences of Axiom of Choice: evil side

Followings are some consequences of AC:

1 Existence of nonmeasurable set.

2 Banach-Tarski Paradox.

3 Any “wrong” statements follow from AC and you know it.



Typical construction base on AC

The construction of Vitali subset V of [0, 1]:

For reals in [0, 1], sayx ∼ y

whenever x − y is rational. ∼ is an equivalence relation on [0, 1].

[X ]: the equivalence class of x .

By AC, we can choose one element out of each equivalence class. Thus there is aset V ⊂ [0, 1], with the property that for each x , there exists a unique y ∈ V anda unique rational number r such that x = y + r .

V is not Lebesgue measurable.



The consistency of AC

Theorem (Godel)

Con(ZF )→ Con(ZF + AC )

By the inner model method.



The independency of AC

Theorem (Cohen)

Con(ZF )→ Con(ZF + ¬AC )

By the forcing and symmetric model method.



AC is true

The reasons:

The good consequences are true. The bad consequences are forgivable.

AC is a powerful tools.

(For set theorists)AC holds in any canonical inner model of ZF up to now.

A simple explanation of the last item:Such inner models are “constructible”. In these models we “construct” ACmanually.



AC is false

The reasons:

The bad consequences are too annoying to be true.

There should be correct ways to obtain good consequences of AC.

(For set theorists) ¬AC holds in some canonical model of ZF .

or

ZF is false.

Remark: No set theorist judges the validity of ¬AC by forcing models.



Some canonical models without AC

1 Any models of ZF + AD.

2 Chang’s model under many measurable cardinals.

3 L[Vλ+1] under j : L[Vλ+1]→ L[Vλ+1].

A common phenomenons: They all need large cardinals. This is not bycoincidence. Indeed, the recent advance in inner model theory suggests if thecurrent inner model program ultimately fails, then probably the right inner modelabsorbs all large cardinals need to be compatible with ¬AC .



A few words about AD

Definition (Axiom of Determinacy)

Every two-player game of perfect information of length ω, in which the playersplay naturals, is determined.

Theorem (Mycielski-Swierczkowski)

AD implies any subset of reals are Lebesgue measurable.

Corollary

AD implies ¬AC .


Axiom of Choice Statements weaker than Axiom of Choice

Statements weaker than Axiom of Choice

Now we see it is reasonable to separate consequences of AC from AC.

Theorem (Halpern, LoS and Ryll-Nardzewski, Feferman)

The followings does not implies Axiom of Choice:

1 Boolean Prime Ideal Theorem.

2 Hahn-Banach Theorem.

3 The existence of non-measurable set.

Theorem (Kelley, Blass-Halpern)

The followings are equivalent to Axiom of Choice:

1 Tychonoff’s Theorem.

2 The existence of Hamel Base for any vector space.


Axiom of Choice Statements weaker than Axiom of Choice

A folklore open problem for decades

What happen when we only deals with the mathematical object actually appearsin the mathematics studies? Or, is it true that nice local consequences of AC doesnot necessarily entail corresponding local version of AC?

Question

Does the existence of Hamel Base for R over Q implies WOR, the existence ofwell-ordering of reals?


Hamel Base for R/Q and AC

Theorem (Schindler-Wu-Yu)

Con(ZFC )→ Con(ZF + there is a Hamel Base for R/Q + ¬WOR).

R/Q is the Q-vector space with domain R, equipped with the usual + and ×operation.


Hamel Base for R/Q and AC The proof

Framework

1. Start from Cohen-Halpern-Levy model N = HOD(O)L[G ] over L.

2. Working in N, we construct a forcing PH whose generic filter is a RN/Q HamelBase. Let GH be the corresponded generic set and FH be the correspondedmaximal antichain.

3. In N[GP ], define NP = HOD({FP} ∪ RN[GP ])N[GP ].

4. We show that NP and N have the same reals and there is no well ordering ofreals in NP .



Cohen-Halpern-Levy model over L

Cohen-Halpern-Levy model is the simplified version of Cohen’s first ¬AC -model.It is defined as

N = HOD(g)L[G ].

Notation:G is the Add(ω, ω)-generic filter over L.Add(ω, ω): the forcing add ω many Cohen reals.g : the set of Cohen reals add by G .HOD: the class of hereditary ordinal definable sets.

Theorem (Cohen,Feferman, Halpern-Levy)

In N, the following is true:

1 ZF + ¬WOR holds.

2 Boolean Prime Ideal Theorem.

3 Every locally countable Σ12-equivalence relation has a transversal. In

particular, there is a nonmeasurable set.



Decomposition of reals of N

Fact

Any reals in N is in some L[gi0 , . . . , gin ] for some gi0 , . . . , gin in g .

In N, the reals admits the following decomposition: R =⋃

I∈ω<ω1

WI , where

WI = RL[gI ] \⋃

J(I WJ .

Lemma

The decomposition defined above lifts to a direct sum system {V I}I∈ω<ω1

for R/Q.



The forcing PH

p ∈ PH if p : ω → RN is a total function so that

1 For any I ∈ ω<ω1 , ran(p) ∩WI is linear independent over span(⋃

J(I WJ) i.e,all linear combination of ran(p) ∩WI is not in U|I |,

2 (Termspace Preparation) For any gI , p � gI ∈ L[gI ], where p � gI is a functionfrom ω defined as:

(p � gI )(n) =

{p(n) : p(n) ∈ L[gI ]0 : Otherwise.

p ≤ q if ran(p) ⊇ ran(q).



The forcing PH

Lemma

PH has the followings property:

1 PH does not add any new reals.

2 PH does not add a well-ordering of R.

3 PH adds a Hamel-Base for R/Q.

The proof relies on the following lemma:

Lemma (Homogeneity of PH under Add(ω, ω))

For any condition (p, b) ∈ Add(ω, ω) ∗ PH , φ a sentence in the forcing languagewith its parameters being names for ordinals, gI for some I , and for R and FH .Assume that p ∈ Add(ω,O) and p forces that b ∈ L[gI ]. Now if there is some(q, c) < p with q � Add(ω, I ) = p and q c � I = I decides φ, then (p, b) decidesφ.

The proof uses the termspace and Prikry type forcing argument.



Generalizations

Theorem (Wu-Yu)

In Cohen’s model, for any locally countable Σ21-binary relation R, there is a

maximal discrete set for R.

Theorem (Wu-Yu)

For any ZF provable Π12-preorder P,

Con(ZFC )→ Con(ZF + there is a maximal antichain of P + ¬WOR)

Theorem (Wu-Yu)

Assume Con(ZF+AD), then there is a ZF + ¬AC model in which for any ZFprovable second order definable, locally countable binary relation R, there is amaximal discrete set for R.


Question

Question

Can we separate the following statements

1 There exists a Hamel base for R/Q.

2 There exists a Luzin set.

3 There exists a Siepinski Set.

Thank you for your attention!


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