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Let $P$ be a claim. There are two questions.

Is it true in ZFC that $P$?

And the other is:

Is it true that $P$?

What is difference between these two questions?

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Are you aware that there are axiom systems other than ZFC, and that sometimes statements are true in one but not true in the other? If you are aware of this, then I don't understand your question. – Zev Chonoles Jul 8 '13 at 0:40
The latter question is more or less ludicrous. Any proof of a claim will be based on assumptions at some level. The second is unanswerable. The first may not be. – Cameron Buie Jul 8 '13 at 0:41
@Paul: No, not always... – Zev Chonoles Jul 8 '13 at 0:48
I assume "true" (without qualifications) means "true in the standard model", which in this case is the class $V$. This, at least, seems the standard usage in set theory. (A different question, of course, is how we can go about verifying truth. Typically, we do this by simply proving the statement via arguments in principle formalizable in $\mathsf{ZFC}$, and most of the time this is what authors mean. Naturally, one could mean other things, for example, if one accepts the existence of large cardinals.) – Andrés E. Caicedo Jul 8 '13 at 1:14
I will quote from Wikipedia :) about something that in also described in many "introduction to proofs" textbooks: "As practised, a [mathematical] proof is expressed in natural language and is a rigorous argument intended to convince the audience of the truth of a statement." – Carl Mummert Jul 8 '13 at 1:21
up vote 5 down vote accepted

That depends entirely on what theoretical assumptions are silently being made (e.g., by being implicit in the context) in the second question. If the person asking the second question is working in $\mathsf{ZFC}$, then there is no difference. If the working theory is $\mathsf{ZF}+\neg\mathsf{AC}$, on the other hand, the two questions are very different. And the background theory need not be based on $\mathsf{ZF}$ at all; some alternatives are listed here.

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I upvoted this, but then I changed my mind, since it lacks the (imo illuminating) pedantry of distinguishing between a theory and its models (provability and truth) – Ben Millwood Jul 8 '13 at 1:20

I suspect that "true in ZFC" is intended to mean "provable in ZFC." The relationship between that and just plain "true" is more a philosophical question than a mathematical one. Let me explain two of the many possible philosophical positions, in the hope that they give an idea of the range of possibilities.

(1), a version of formalism: In mathematics, we have no way of knowing things except by proving them. We accept, more or less arbitrarily, ZFC as the standard basis for our proofs. So when something is proved in ZFC, we are willing to say that it is true. It makes no sense to speak of truth apart from provability. Therefore, "true" simply means provable in ZFC.

(2), a version of Platonism: There is a world of abstract entities, including the sets and related entities studied in mathematics. To say that a mathematical statement is true is to say that it corresponds to the actual situation in this ideal world. We have only a partial understanding and intuition of the ideal world; much of that understanding is summarized in the ZFC axioms, so anything provable from those axioms is known to be true. But there are other truths that we may not be able to prove and, indeed, may not have any other way of knowing. Therefore, "true" is a considerably broader notion than provability in ZFC.

To see how these viewpoints work out in a nontrivial situation, consider the fact (provable in ZFC) that, if ZFC is consistent, then it neither proves nor refultes the continuum hypothesis (CH). A formalist (or, more precisely, someone who subscribes to the version of formalism that I've described) would conclude that, if ZFC is consistent, then CH is neither true nor false. (He might reconcile this conclusion with the law "p or not p" of classical logic by declaring CH to be meaningless.) And he might have a problem with the issue of consistency of ZFC, because, if ZFC is consistent then this fact is not provable in ZFC.

A Platonist would begin by brushing aside the clause "if ZFC is consistent"; it's obviously consistent because it's true. (So the consistency of ZFC is one example of truth beyond what is provable in ZFC). He knows that CH either holds or doesn't hold in the world of sets, but he probably doesn't know which way it goes. So he would be happy to find new axioms that are clearly true (as the ZFC axioms are) and decide CH.

I think I've stated both the formalist and the Platonist positions in fairly extreme forms, so I wouldn't be surprised to get comments from actual formalists and actual Platonists objecting that I've unfairly distorted their positions. I hope such comments would include corrections of the distortions, so as to build a more accurate picture of the positions that people actually take.

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(Of course, both the formalist and platonist views are getting more complicated now with the push for the multiverse picture.) – Andrés E. Caicedo Jul 8 '13 at 1:27

The terminology in the question isn't used correctly.

There is a standard definition of true in a model given in every logic text. For example, the statement "$x^2 = -1$ has a solution" is false in the real numbers (one model where the truth value of that statement is defined) and is true in the complex numbers (another model where the truth value is defined).

When there is an intended interpretation of a theory, we say that a sentence of that theory is simply true if the sentence is true in the intended interpretation. Not every theory has an intended interpretation, but most set theorists have historically believed that ZFC does. One way that we know that a sentence is true is to prove it, but the truth or falsity is independent of our ability to produce a proof.

Similarly, if the words of a statement have normal English meanings, the statement is true disquotationally if it is true as an English sentence. For example, "the word 'car' has three letters" is true in this sense. For a mathematical example, "the number 5 is a prime natural number" is also disquotationally true based on the usual English meanings of the word that we learn long before we learn advanced mathematics.

The phrase true in ZFC is an abuse of language. A statement can be provable in ZFC, disprovable in ZFC, or independent of ZFC, but it cannot be "true" in ZFC. Most likely what the question intends is "provable in ZFC".

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Most likely "true on $\mathsf{ZFC}$" is shorthand for "true in models of $\mathsf{ZFC}$", which of course is the same as "provable in $\mathsf{ZFC}$". I would consider this standard abuse of language, and probably wouldn't even notice it when reading. And as you say, "true" means "true in the standard (or intended) model of set theory" (which of course leads to the question of what this can possible mean, but that, as Andreas points out, is more a philosophical issue). – Andrés E. Caicedo Jul 8 '13 at 1:22
@Andres: in advanced contexts, I agree it is common to see people say things like "the law of the excluded middle is true in classical logic, but not constructive logic", and similar abuses of language. But in student writing, it often indicates the student hasn't separated the syntactic notion of provability from the semantic notion of truth. So I think that if someone seems to be confused about what true in ZFC means, the most likely reason is they haven't mentally replaced it with "provable in ZFC". – Carl Mummert Jul 8 '13 at 1:27
Oh, sure. I just mean that when I see this in a paper, it doesn't even make me frown--as I said, I may not even notice it. But naturally, what is standard abuse for a colleague may be source of confusion for a student. – Andrés E. Caicedo Jul 8 '13 at 1:30

The first type of assertion is well-defined and the second is undefinable.

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However there are many questions are the second type in the paper. Are they all undefinable? – Paul Jul 8 '13 at 0:47
@Paul: Excuse me, in what paper? If you were reading a paper that led you to ask this question, is there any reason not to mention this fact in your post? – Zev Chonoles Jul 8 '13 at 0:50
I feel this answer is literally true but perfectly unhelpful. It would be much better if you would clarify in what sense you mean undefinable, because undefinability-wrt-some-theory is imo a much more specific notion than completely-inscrutable. – Ben Millwood Jul 8 '13 at 0:57
@BenMillwood, there isn't any well-defined theory with respect to which "truth of proposition P of ZFC" becomes well-defined parametrically in P (unless one retrenches to provability in a particular system as the definition of truth) and/or individually for each P. It is undefinable in as strong a sense as any that one could assign to the word 'undefinable'. – zyx Jul 8 '13 at 1:26

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