Comments on: construction of the natural numbers: work of god or work of man? http://www.evolution-nextstep.com/archives/2881 take the next step Fri, 05 Dec 2008 13:22:05 +0000 http://wordpress.org/?v=2.6.5 By: David http://www.evolution-nextstep.com/archives/2881#comment-6002 David Fri, 30 Jun 2006 06:58:16 +0000 http://www.evolution-nextstep.com/archives/2881#comment-6002 "One needs concepts outside the realm of ZF, which cannot themselves be proven from within ZF." But, if I am not very much mistaken, this 'supersystem' would itself be subject to Goedel's Theorems, thus a 'super-supersystem' would be needed to prove the consistency of the supersystem. Or am I wrong? “One needs concepts outside the realm of ZF, which cannot themselves be proven from within ZF.”

But, if I am not very much mistaken, this ’supersystem’ would itself be subject to Goedel’s Theorems, thus a ’super-supersystem’ would be needed to prove the consistency of the supersystem. Or am I wrong?

]]> By: j.d. http://www.evolution-nextstep.com/archives/2881#comment-5991 j.d. Fri, 30 Jun 2006 02:04:26 +0000 http://www.evolution-nextstep.com/archives/2881#comment-5991 This can be proven, but with great difficulty, as Godel's incompleteness theorems say that the consistency of Zermelo-Frankel cannot be proven from within Zermelo-Frankel. One needs concepts outside the realm of ZF, which cannot themselves be proven from within ZF. There are several statements that are undecidable statements from ZF; if any one of them are true, then Godel's theorem gives you your proof of consistency. It would be more accurate to say "believed" instead of "known". ZFC is generally accepted to be consistent. However, it has been known for some time that if ZF is consistent, then so is ZF with the Axiom of Choice. This can be proven, but with great difficulty, as Godel’s incompleteness theorems say that the consistency of Zermelo-Frankel cannot be proven from within Zermelo-Frankel. One needs concepts outside the realm of ZF, which cannot themselves be proven from within ZF. There are several statements that are undecidable statements from ZF; if any one of them are true, then Godel’s theorem gives you your proof of consistency.

It would be more accurate to say “believed” instead of “known”. ZFC is generally accepted to be consistent.

However, it has been known for some time that if ZF is consistent, then so is ZF with the Axiom of Choice.

]]> By: David http://www.evolution-nextstep.com/archives/2881#comment-5971 David Thu, 29 Jun 2006 18:44:09 +0000 http://www.evolution-nextstep.com/archives/2881#comment-5971 "We shall assume the validity of the standard (Zermelo-Frankel with choice) set theory, which is known to be a consistent logical system as discussed previously." How do we know that? “We shall assume the validity of the standard (Zermelo-Frankel with choice) set theory, which is known to be a consistent logical system as discussed previously.”

How do we know that?

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