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My question is basically, does there exist a statement X independent of ZF such that ZF + X implies a statement P of first-order arithmetic, but ZF + not X implies not P?

Now X cannot be the axiom of constructibility due to Schoenfield's absoluteness theorem, which states that the axiom of constructibility, and thus its consequences like the axiom of choice and the continuum hypothesis, can't be used to prove any statement of first-order arithmetic that you couldn't already prove using ZF: http://en.wikipedia.org/wiki/Absoluteness#Shoenfield.27s_absoluteness_theorem

Also, there are examples like Con(ZF), but they're not really interesting, because obviously Con(ZF) is a true statement assuming that ZF is sound. So I'm specifically looking for statements X whose truth value cannot be deduced from the assumption that ZF is sound.

So perhaps a preliminary question should be, does there exist any statement independent of ZF which can prove statements of first-order arithmetic that ZF can't prove, but whose truth value does not follow from the assumption that ZF is sound?

Any help would be greatly appreciated.

Thank You in Advance.

EDIT: This is based on a question of mine from Math.Stackexchange:

EDIT 2: Just to clarify, when I said "does not follow from the assumption that ZF is sound", I was speaking metamathematically, I wasn't talking about reasoning within the language of ZF. I meant that we shouldn't be obliged to believe either X or its negation just because we believe that the axioms of ZF are true. The axiom of choice is an example of such a statement: if we accept that the axioms of ZF are true, that doesn't compel us to accept either the axiom of choice or its negation. But unfortunately, the axiom of choice doesn't have any new consequences for first-order arithmetic, outside of what ZF already allows us to prove. So I want a statement like that that does have consequences for first-order arithmetic.

Still, if people find it too informal to talk about metamathematical reasoning, we can make things more precise by defining the truth predicate of ZF within NBG set theory, as shown in theorem 1 of (Mostowski 1950). Or, if @JoelDavidHamkins is right and we can't define a truth predicate within NBG, I'm happy to define the truth predicate within Morse-Kelley set theory instead. But however we formalize it, I hope my intent is clear: I want a statement X such that mathematicians who agree that the axioms of ZF are true can still disagree about whether statement X is true.

EDIT 3: I've come to realize that consistency statements Con(T) don't satisfy what I'm trying to ask, because the only warrant of their independence from ZF is that we assume that the underlying theory T is in fact consistent. Because if the theory were inconsistent, then the falsehood of Con(T) could be proven in ZF and even in PA. The reason for that is that consistency statements are all Pi_1 statements, and every false Pi_1 statement can be proven wrong in PA and thus ZF. So I want the arithmetical statement P to be higher up in the arithmetical hierarchy than Pi_1.

Now, as @WillSawin has pointed out, it's easy to show that there are truths of first-order arithmetic that are independent of ZF + (All Pi_1 truths), because otherwise an oracle who could decide Pi_1 truths would be able to decide all truths of first-order arithmetic, which is obviously wrong. So we know that such a statement exists. But I would prefer to have a concrete statement, not just a proof that a statement exists.

And more importantly, while I do want P to be a statement of first-order arithmetic, I would prefer that X not be a statement of arithmetic. I want X to be a set-theoretic statement that's not an arithmetic statement, something like the axiom of choice, the continuum hypothesis, or a large cardinal axiom. The fundamental intent of my question is, can there be an unfalsifiable disagreement about which set theory is correct, which leads to an unfalsifiable disagreement about which statements of first-order arithmetic are correct? As noted above, Schoenfield's absolute theorem rules out the axiom of choice and the continuum hypothesis, and this answer to my Math.SE question suggests that large cardinals won't work either. So does anyone know any non-arithmetical set-theoretic statement, independent of ZF, which implies a non-Pi_1 statement of first-order arithmetic that's independent of ZF?

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Amazing. Two hours of waiting between cross posting your question. You should at least point out that this was posted on m.SE and provide a link. –  Asaf Karagila Aug 14 '13 at 3:19
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Regarding your third edit, how can you say that you want X to be not arithmetic, given that it must be provably equivalent to P? –  Joel David Hamkins Aug 15 '13 at 1:29
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As I mention in my comments on my answer, Con(KM+True $\Pi^0_1$) is such a statement. (And it seems to me that KM proves "ZF is sound".) –  Joel David Hamkins Aug 15 '13 at 1:51
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Each new edit makes the question more and more confused. And you are mistaken in your last comment. You may want to read the relevant sections of Metamathematics of first-order arithmetic, where most of this is covered carefully. –  Andres Caicedo Aug 15 '13 at 3:32
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"if @JoelDavidHamkins is right and we can't define a truth predicate within NBG" This is not a matter of opinion (and he's right). –  Andres Caicedo Aug 15 '13 at 5:02
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3 Answers 3

Since you want ZF+X to prove P and ZF+$\neg$X to prove $\neg$P, it follows that X is equivalent to P. So basically, you are asking for an arithmetic assertion that is independent of ZF, but which satisfies your requirement that it "not follow from the assumption that ZF is sound".

Of course, there are plentiful arithmetic assertions that are independent of ZF, provided that ZF is consistent, such as the Rosser sentence for ZF. But presumably this doesn't satisfy your requirement.

If large cardinals are consistent, then we get more examples. For example, Con(ZF + there is a measurable cardinal) is thought by many set theorists to be true, and in this case it is not provable in ZF, since it implies Con(ZF). But one cannot deduce that it is true just from the assumption that ZF proves only true arithmetic assertions, since if ZF is consistent, then it is consistent with ZF that this assertion is false, yet ZF has an $\omega$-model and hence is arithmetically sound. (This is because the assumption that ZF has an $\omega$-model has weaker consistency strength than a measurable cardinal.)

This reasoning shows that for any consistent theory T whose consistency strength is stronger than an $\omega$-model of ZF (and this includes all of the usual large cardinals), then Con(T) will be an example of a statement independent of ZF yet not deducible merely from the fact that ZF is arithmetically sound.

Update. You have explained that you want to find an arithmetic statement independent of ZF and not provable from the assumption that ZF is sound, where this is formalized in Kelley-Morse set theory.

Indeed, the soundness of ZF is formalizable in KM, since this theory proves the existence of a full satisfaction class for first-order truth. And so in KM we can formulate the assertion $S$ that says "every theorem of ZF is true". Indeed, S is a theorem of KM, which can be seen by induction on the length of proofs, since we have a truth predicate for first-order truth.

So the example sentence $P$ you seek could just be Con(KM). If true, this will be independent of ZF, and you cannot prove it from the assumption that ZF is sound, since then you could prove it in KM, which you cannot by the incompleteness theorem.

(But since you've made the move from ZF to KM, one might think you now want to consider statements P independent of KM, which are not deducible from the "soundness" of KM, and we would be back to the essence of the earlier objections, one level up.)

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Joel, nice example on the consistency of large cardinals. I feel that this example is still somehow "cheating" and misses the examples the OP is after; but I'm getting a vibe that maybe the OP doesn't know what he is looking for exactly either. –  Asaf Karagila Aug 14 '13 at 3:39
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I'm not sure exactly what you mean. Tarski's theorem on the non-definability of truth shows that we can have no formal concept of "true" that is expressible within our theory. So you'll have to be more precise about exactly what you mean. –  Joel David Hamkins Aug 14 '13 at 4:22
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@KeshavSrinivasan So you are using an informal metalanguage and assume a Platonic notion of sets and soundness of ZF means that these sets are a “model” of ZF? Well, if your notion of soundness is not defined by a formal system but a philosophical notion, then what is an “implication” of this kind of soundness? And how should we tell whether there is such an implication? (and even philosophically this notion is questionable) –  The User Aug 14 '13 at 8:55
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@Keshav, I'm sorry that you don't like the Con(KM) example, although I believe it does answer the question that you've asked. Indeed, it is my impression that you've received several thoughtful answers here to your various questions, as well as help in formulating your intention... –  Joel David Hamkins Aug 14 '13 at 21:45
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Well, I'm finding your question to be a moving target. For your latest version, in the previous comment, it seems that you can use Con(KM+true $\Pi^0_1$), meaning the consistency of the theory of KM with the collection of $\Pi^0_1$ sentences that happen to be true. Note that this consistency assertion is no longer $\Pi^0_1$, since the theory is not computable, but it is $\Pi^0_2$. –  Joel David Hamkins Aug 15 '13 at 1:42
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Let me, for a moment, follow Joel in taking "sound" to mean arithmetically sound, i.e., all arithmetical statements provable in ZF are true. That fact is expressible as a sentence $S$ in the language of set theory. Then Con(ZFC $+\ S$) is true but not deducible from the soundness assumption $S$.

Now let me suppose instead that you intended soundness to mean more than arithmetical soundness, say, for example, that all ZF-provable statements of higher-order arithmetic are true. That's also expressible as a sentence in the language of set theory, so the same idea as above still gives an example.

To escape from this sort of example, one needs to interpret "soundness" in a way that can't be expressed in the language of ZF. For example, it might mean that every sentence provable in ZF is true in the full universe of sets. Truth in the universe is not expressible in the language of ZF, so you don't get cheap examples as above. But by eliminating the cheap answer, this version of soundness raises questions about the very meaning of the question. What does it mean for something to "follow from" soundness, when soundness can't be expressed in the language of set theory? You'd need to explain (at least) what sort of deductions are intended by "follow".

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I am indeed talking about soundness in the sense of every theorem of ZF being true, or equivalently the axioms of ZF all being true. When I was talking about following from soundness, I was speaking metamathematically, I meant that there should be no way to conclude metamathematically that X or its negation is true from the assumption that ZF is sound. Or if you feel that talking about metamathematically reasoning is too informal, then here's a way to make it more precise: I think the truth predicate for ZF is expressible in NBG set theory, so we can talk about proving X using that. –  Keshav Srinivasan Aug 14 '13 at 4:53
    
@KeshavSrinivasan, NBG does not prove the existence of a first-order truth predicate, although KM does. In this case, you could use Con(KM), which if true, is independent of ZF and implies the consistency of a standard model of ZF, which is elementary in V. Thus, it is similar to the other examples we have here. –  Joel David Hamkins Aug 14 '13 at 12:12
    
@JoelDavidHamkins I just put a link in my question to a paper by Mostowski, which seems to show that the truth predicate of ZF is expressible in NBG. Am I interpreting it incorrectly? Also, about Con(KM), is it possible for ZF to be sound and yet KM to be inconsistent? In other words, if someone found an inconsistency in Morse-Kelly set theory tomorrow, would that mean that the axioms of ZF can't all be true? –  Keshav Srinivasan Aug 14 '13 at 18:44
    
To my (quick) reading of that theorem, Mostowski does not present a full satisfaction class, but rather a class that he proves works for any particular formula, as a scheme. This will result in merely a partial satisfaction class, as GBC will not in general prove that every formula is covered by it (and it cannot prove this, as this would mean GBC proves the consistency of ZF, which is impossible). If you google "partial satisfaction classes" you will find out more; most of the material is for models of arithmetic, but much of the theory carries over to set theory. –  Joel David Hamkins Aug 14 '13 at 18:51
    
@KeshavSrinivasan I have had a look at the paper: You are right, there is a truth predicate, but what you call “soundness” of ZF (i. e. the statement “every axiom of ZF is true”) cannot be proved in NBG. However, in an ω-model of NBG ZF is “sound”. –  The User Aug 14 '13 at 19:00
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Harvey Friedman has discovered a number of arithmetic (actually, $\Pi^0_1$) statements of combinatorics which are independent of $ZFC$. This is part of his ongoing research project, "Boolean Relation Theory" (see http://www.math.osu.edu/~friedman.8/pdf/BoolRelnThyNotes100601.pdf).

Now, since these statements follow from (in fact, are equivalent to) the consistency of large cardinal propositions, it's not clear that they are what you are looking for; but you may still find them interesting.

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In fact, it is clear that they are not. –  Andres Caicedo Aug 15 '13 at 2:00
    
@Andres: you're right, I didn't read the question carefully enough. If the OP says I should, I'll delete this answer. –  Noah S Aug 15 '13 at 2:11
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I don't think you should. This is an interesting and valuable observation, and illustrates a subtlety of the question. –  Andres Caicedo Aug 15 '13 at 2:38
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