7/29/2019 Con Math the 122512

1/70

1

A DIVINE CONSISTENCY PROOF FOR

MATHEMATICSby

Harvey M. Friedman*

Distinguished University Professor EmeritusMathematics, Philosophy, and Computer Science

The Ohio State UniversityColumbus, Ohio 43210December 25, 2012

* This research was partially supported by the JohnTempleton Foundation grant ID #36297. The opinionsexpressed here are those of the author and do notnecessarily reflect the views of the John TempletonFoundation.

Abstract. We present familiar principles involving objectsand classes (of objects), pairing (on objects), choice(selecting elements from classes), positive classes(elements of an ultrafilter), and definable classes(definable using the preceding notions). We also postulatethe existence of a divine object in the formalized sensethat it lies in every definable positive class. ZFC (evenextended with certain hypotheses just shy of the existenceof a measurable cardinal) is interpretable in the resultingsystem. This establishes the consistency of mathematics

relative to the consistency of these systems. Measurablecardinals are used to interpret and prove the consistencyof the system. Positive classes and various kinds of divineobjects have played significant roles in theology.

1. T1: Objects, classes, pairing.2. T2: Extensionality, choice operator.3. T3: Positive classes.4. T4: Definable classes.5. T5: Divine objects.6. Interpreting ZFC in T5.

7. Interpreting a strong extension of ZFC in T5.8. Without Extensionality.

INTRODUCTION

This work came about from our interactions at andreflection on two conferences hosted by the John TempletonFoundation. The first of these meetings was held in honor

7/29/2019 Con Math the 122512

2/70

2

of the 100th birthday of Kurt Gdel, in Vienna, in 2006.The second was held in honor of the 100th birthday of SirJohn Templeton, in Heidelberg, in 2012.

Interactions at the second of these meetings have

previously led to [Fr12b], where a basic mathematicalproperty of a mathematical object in a mathematicalstructure, immediately suggested by familiar ideas aboutGod, is shown to hold within some artificially createdmathematical structure. Specifically, there is a relationalstructure of finite relational type with a uniqueundefinable element, in the usual sense of first orderpredicate calculus with equality.

At the Vienna meeting, there was discussion of Gdel'sformalization of earlier ontological arguments for theexistence of God. This relied heavily on modal logic, butwhat we found particularly striking was the use of"positive properties", which has a substantial history andgoes back at least to Leibniz.

From a theological standpoint, it is perhaps natural toview the attribute of positive as a facility that God hascreated and given the world access to, perhaps in order tohelp direct us into "positive behaviors".

It is clear that Gdel was using "positive properties" as,mathematically speaking, an ultrafilter on properties. In

fact, at least implicitly, he was using "positiveproperties" as an ultrafilter on extensions of properties.I.e., whether a property is positive depends only on whatobjects it holds of. For discussion of ultrafilters andpositivity, see section 3 below.

It occurred to us that perhaps this highly intriguingultrafilter, viewed as an ultrafilter on classes ofobjects, can be used to prove the consistency ofmathematics.

At the time, we just didn't see how to get such ambitiousmathematical mileage out of this "positivity ultrafilter",at least in any simple basic conceptual way. This goalseemed particularly remote since the positivityultrafilter, as discussed by Gdel and implicitly byothers, is what is called a "trivial ultrafilter". I.e., anultrafilter consisting merely of the classes containingsome given special point.

7/29/2019 Con Math the 122512

3/70

3

In particular, following Gdel, and going back at least toLeibniz (see [Go95], and also [An90], [Le23], [Le56],[L369], [Le96], [Lo06], [Op95], [Op96], [Op06], [Op12],[So87], [So04]), a class of objects is positive if and onlyif it contains God. And trivial ultrafilters are, as the

name suggests, mathematically trivial. So it would appearthat one cannot expect to do anything substantial,mathematically, with the positivity ultrafilter.

Interactions with scholars from theology at the secondmeeting rekindled our interest in trying again to dosomething powerful with this mathematically trivialpositivity ultrafilter.

It occurred to us that if we take God out of the class ofall objects, treating God as exceptional, but keeping thepositivity ultrafilter, pruned to be over the class ofobjects excluding God, then the positivity ultrafilter isno longer trivial. In fact, it is a nontrivial ultrafilterover the class of all objects without God.

It is well known that in the context of set theory, certainkinds of ultrafilters are enough to prove the consistencyof mathematics, as formalized by the usual ZFC axioms. Inparticular, a nontrivial countably complete ultrafilterserves this purpose.

However, to straightforwardly state that the positivity

ultrafilter is a nontrivial countably complete ultrafilter,together with the mathematical infrastructure needed tostate and make use of this, is far too technically brutalto be of fundamental philosophical or theological meaning.

So we encountered the following challenge. To findfundamental properties of the positivity ultrafilter which,in the context of a conceptually very minimal mathematicalinfrastructure, has sufficient power to construct a modelof ZFC, and perhaps more.

This paper offers our response to this challenge. Here is asuccinct six part description of our framework.

1. Objects and classes of objects, linked by membership.This is arguably viewed as within our capacity to clearlyimagine, as it is an accepted useful working component ofmodern mathematics. It should be noted that in modernmathematics, objects are usually restricted to objects with

7/29/2019 Con Math the 122512

4/70

4

a definite mathematical purpose, rather than a completelygeneral notion of object. See section 1.

2. Pairing of objects - a combining of any two objects intoone. This is generally viewed as within our capacity to

clearly imagine. E.g., the pair of two objects can be takento be the idea of having the first followed by the second.In various guises, this is also an accepted useful workingcomponent of modern mathematics. See section 1.

3. A choice operator CHO that picks an element out of eachnonempty class of objects. Formally, x A CHO(A) A,where CHO(A) is read "chosen element from A". This appearsto be beyond our capacity to clearly imagine. I.e., wedon't have an understanding of how to go about making thechoices. However, this continues to be an accepted usefulworking component of modern mathematics. See section 2.

4. The positivity attribute on classes of objects (thepositivity ultrafilter). This seems definitely beyond ourcapacity to clearly imagine. I.e., we don't have anunderstanding of definite criteria for positivity. Thenotion of ultrafilter is also an accepted useful workingcomponent of modern mathematics. But a specific preferredultrafilter, particularly on the entire universe ofobjects, has not appeared as a generally accepted componentof modern mathematics. See section 3.

5. The definability attribute on classes of objects,reflecting the standard notion of "class of objectsdefinable from the preceding concepts 1-4 withoutparameters". We have a clear understanding of this notion,relative to the primitives to which it is being applied(here, in 1-4 above). This is an essential feature ofmodern mathematical logic, and also has useful interactionswith several branches of mathematics - especially real andcomplex algebraic geometry. It is exemplified by Tarski'sformal treatment of truth in formalized languages. Seesection 4.

Thus the positivity attribute is the one feature above thathas such a deep philosophical and theological meaning. Theadditional axiom that creates the vast logical power, isthe Divine Object axiom.

Recall that there does not exist an object which lies inall positive classes, as we treat God as exceptional and

7/29/2019 Con Math the 122512

5/70

7/29/2019 Con Math the 122512

6/70

6

this second approach, we have both the "Real World" and thewider "Supernatural World". Thus we have a more complexframework in that we have two sorts of objects. But anumber of axioms are simplified. In particular, in thissecond approach, we do not need the positivity ultrafilter.

So a tradeoff is emerging, where we use one sort of objectand the positivity ultrafilter (this is what we do here),or two sorts of objects without the positivity ultrafilter(this is what we will do in [Fr13]).

We view [Fr11], [Fr12a], [Fr12b], [Fr13], as part of awider program which we call Concept Calculus. In ConceptCalculus, we aim to identify groups of informal conceptsthroughout the informal and semiformal intellectuallandscape, and formulate fundamental transparent principleswhich are of sufficient power to interpret the usualformalizations for mathematics and beyond. These provideformal interpretations of mathematics, consistency proofsof mathematics, and relative consistency proofs ofmathematics, as in Theorems 6.27 and 7.34.

1. T1: Objects, classes, pairing.

In sections 1-5, we present, in cumulative stages, theaxioms of our theory T5 that we use to interpret ZFC insection 6. The reader can dispense with this backgroundmaterial and go directly to section 5. T1 - T4 admit

familiar standard models constructed well within ZFC, but T5does not.

In this section, we focus on a particularly simple system T1that provides the basic apparatus needed to support thesubstantial and flexible use of objects and classes ofobjects required for the interpretation of ZFC.

The two sorted T1 is very similar to the two sorted systemZ2 of second order arithmetic. The main difference is thatin Z2, the objects are nonnegative integers, with arithmetic

operations taken as primitives. In T1, the objects are notconstrained in any way, and are meant to include allobjects whatsoever, as reflected in type 0 of the usualRussell theory of simple types (in modern formulation). Theclasses of objects are reflected in type 1 of the usualRussell theory of simple types (in modern formulation).

For presentations of Z2, see [Si99], p. 4, andhttp://en.wikipedia.org/wiki/Second-order_arithmetic. We

7/29/2019 Con Math the 122512

7/70

7

follow the treatment in [Si99] using the two sortedlanguage for nonnegative integers and sets of nonnegativeintegers, with 0,1,+,,

7/29/2019 Con Math the 122512

8/70

7/29/2019 Con Math the 122512

9/70

9

by x. Obviously we can argue that x cannot be the same as xfollowed by x, but perhaps more convincing is that xfollowed by x, and x followed by x followed by x are ofdifferent lengths, and therefore are distinct. (Theexistence of at least one object x is normally considered

to be a logical truth).

3. L1 Comprehension. This is the standard class existenceprinciple. The objects and classes of objects, with L1Comprehension, constitutes the first two stages of a commonversion of the Russell theory of simple types. L1Comprehension alone has the trivial model with exactly 1object and exactly 2 classes. The usual way of achievingstrength is to add an axiom of infinity, involving theexistence of an ordering on objects with a certainproperty. Or alternatively, to use the inclusion relationamong classes as the ordering. Under these approaches, weneed to use classes of classes of objects, or even classesof classes of classes of objects, or alternatively, binaryrelations on classes.

In contrast, here we use only objects and classes ofobjects, with the Pairing axiom doing not only the work ofthe axiom of Infinity, but also enough work to avoid havingto use higher types or binary relations. Also, Pairing hasthe explicit interpretation as discussed above.

In sections 6 and 7 we will need only L5 Comprehension

where has only a few class quantifiers. In this paper, wewill not go into just how many quantifiers are needed, butpreliminary indications are that two suffice. There is alsothe matter of how many quantifiers over objects are needed,which is also beyond the scope of this paper.

We will consider fragments of L1 Comprehension, and comparethem with corresponding fragments of Z2. These fragments ofComprehension are finitely axiomatizable.

The following result goes back to Tarski, but we include a

proof here. It is of independent interest, and allows us toavoid a number of details when establishing finite and nonfinite axiomatizability. However, in section 3 we cannoteasily take this route, and use another method to establishTheorem 3.6.

LEMMA 1.1. Let S1,S2 be two theories with commutinginterpretations 1,2. I.e., for all sentences of S1, 2(1()) is provable in S1. Then the interpretations 2,1

7/29/2019 Con Math the 122512

10/70

10

are commuting for S2,S1. In addition, S1 is finitelyaxiomatizable if and only if S2 is finitely axiomatizable.If K is a finite axiomatization of S1 (S2) then 1(K) (2(K))is a finite axiomatization of S2 (S1).

Proof: Let S1,S2,1,2 be as given. Let

be a sentence ofS2. We want 1(2()) provable in S1. It suffices to

show that 2(1(2())) = 2() 2(1(2())) isprovable in S2. This follows from the hypothesis applied to2().

Let K finitely axiomatized S1. Let S2 prove . Then S1proves 2(). Hence K proves 2(). Therefore K 2() isprovable from nothing. Hence 1(K) 1(2()) is provablefrom nothing. Therefore 1(K) proves . QED

LEMMA 1.2. T1 is interpretable in Z2.

Proof: We take the objects for T1 to be Gdel numbers ofterms in 0 and the binary function symbol P. We take theclasses for T1 to be the sets in Z2. QED

THEOREM 1.3. T1 and Z2 have commuting interpretations.

Proof: Let 1 be the interpretation of T1 in Z2 given by theproof of Lemma 1.2. We now give the interpretation 2 of Z2in T1.

We have to develop arithmetic in T1. By the first line ofPairing, fix a b. Set 0 = P(a,b). By the second part ofPairing, 0 is not of the form P(v,v).

We say that A is good if and only if 0 A (v A)(P(v,v) A). Let be any class consisting of theobjects lying in every good A. Let 1 be P(0,0). Obviouslyv,w P(v,v) 0 (P(v,v) = P(w,w) v = w). Also ifA , 0 A, and (v A)(P(v,v) A), then ,A have thesame elements (because A is good). So P(v,v), v , servesas an inductive successor function on .

Using P, we have full access to ternary relations on , andhence also binary functions on , including comprehensioninvolving quantification over them. So we can develop +,,