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Gödel's incompleteness theorem states that: "if a system is consistent, it is not complete." And it's well known that there are unprovable statements in ZF, e.g. GCH, AC, etc.

However, why does this mean that ZF is consistent? What does "relatively consistent" actually mean?

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I'm not sure "relatively consistent" is a technical term. There are things called "relative consistency results", but they are relative results about consistency, not results about relative consistency. –  Henning Makholm Aug 16 '12 at 15:36
Your statement of the incompleteness theorem is wrong. The system must also be assumed to be sufficiently powerful to allow you to encode arithmetic in it. (You probably know this, but other visitors to the site might get confused.) –  Harald Hanche-Olsen Aug 16 '12 at 15:52

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We have to hedge quite a bit in what you said. First, there are many complete theories: the theory of all groups of order 7 is complete, as an example. However, Gödel showed that any "nice enough" consistent first-order theory capable of encoding basic arithmetic is incomplete. (There is a complete theory of number theory, usually denoted $\mathrm{Th}(\mathbb{N})$, which consists of all formulas that are true in the standard model of number theory; unfortunately, this is not a "nice enough" theory -- it's impossible to program a computer to tell the axioms from the non-axioms!)

Also, these statements that you gave, (G)CH, AC,, are not provably unprovable in ZF. However, it is provable that they are only provable in ZF if ZF is itself inconsistent.

This is what a relative consistency result means: it is a result that says the consistency of one theory implies the consistency of the other. Kurt Gödel proved that if ZF is consistent, then so if ZFC+V=L (and also that V=L implies GCH). Paul Cohen proved that if ZF is consistent, then both ZF+$\neg$AC and ZFC+$\neg$CH are consistent. More fundamentally, ZF implies the consistency of PA. These are fundamentally relative consistency, and cannot be improved to straight consistency results. This stems from Gödel's Second Incompleteness Theorem, which basically implies acceptance of the consistency of ZF(C) is an article of faith (though one I'm happy to believe in).

I recommend that you look at the first page of George Boolos's "Gödel's second incompleteness theorem explained in words of one syllable" (Mind, vol.103, pp.1-3). It is a quite entertaining look at the meaning of Gödel's Second Incompleteness Theorem.

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Relatively consistent means that if some other system is consistent, then so is the given system. For example, ZF is relatively consistent with ZF-Foundation (and vice versa), and relatively consistent with ZFC (and vice versa). For a unidirectional example, the axioms of Peano arithmetic are relatively consistent with ZF. However, the consistency of Peano arithmetic does not entail the consistency of ZF. (Thanks again, Henning.)

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Doesn't one usually say "equiconsistent" in these cases? –  Henning Makholm Aug 16 '12 at 15:38
Yes, though off the top of my head I couldn't think of a consistency result that only went one way. –  Cameron Buie Aug 16 '12 at 15:43
If ZF is consistent, then PA is consistent? –  Henning Makholm Aug 16 '12 at 15:45
/facepalm/ Thank you, Henning. –  Cameron Buie Aug 16 '12 at 15:58
"If a system is not complete, then it is consistent." Seeing that complete usually means "proves $\sigma$ or $\lnot\sigma$ for every sentence $\sigma$" and that consistent usually means "does not prove every sentence $\sigma$," a system that is not complete must be consistent. There are variations but, in any case, I don't think your second sentence exactly says what you meant to say. –  François G. Dorais Aug 16 '12 at 17:02

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