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> Suppose it wasn't consistent. That means the model you're investigating isn't ZF+Con(ZF). It's ZF+Con(ZF)+Not(Con(ZF)) because you've added the extra axiom a

> Suppose it wasn't consistent. That means the model you're investigating isn't ZF+Con(ZF). It's ZF+Con(ZF)+Not(Con(ZF)) because you've added the extra axiom as part of your proof by contradiction. Obviously you trivially get not(Con(ZF)) from this model, let alone anything else via the principle of explosion. ZF+Con(ZF)+Not(Con(ZF)) is not consistent and it shouldn't surprise you that you can use it to trivially derive contradictory results. Ultimately, Godel's second incompleteness theorem says that you can't prove the consistency of a formal system within that system. The Wikipedia article is dense and awful to read but has a reasonably succinct statement of the theorem. https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Second_incompleteness_theorem https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_... EDIT: "mind blowing material" - on that we can both agree. This is something I really wish I got to study more back in uni. Further editing to clean up some minor errors.

ikeboy 9y ago
>It's ZF+Con(ZF)+Not(Con(ZF)) because you've added the extra axiom as part of your proof by contradiction. No. It's ZF+Con(ZF)+not(con(ZF+con(ZF)). Which is, in fact consistent (assuming ZF is).
- kirrent 9y ago
  Of course! Thankyou, that makes me much more comfortable. ZF+Con(ZF)+not(con(ZF+con(ZF)) is an example of what's sometimes called the self hating theory, which is known to be consistent if ZF+Con(ZF) is consistent (you said much the same thing in your original comment). As it's known to be consistent, you can't apply the principle of explosion to produce Not(Con(ZF)). EDIT: Obviously I'm not really the sort of person to talk about this stuff with you because of how much of a neophyte I am. Perhaps this lecture would make it clearer? http://www.scottaaronson.com/democritus/lec3.html http://www.scottaaronson.com/democritus/lec3.html In particular, the excerpts: The Second Incompleteness Theorem establishes what we maybe should have expected all along: that the only mathematical theories pompous enough to prove their own consistency, are the ones that don't have any consistency to brag about! If we want to prove that a theory F is consistent, then we can only do it within a more powerful theory -- a trivial example being F+Con(F) (F plus the axiom that F is consistent). But then how do we know that F+Con(F) is itself consistent? Well, we can only prove that in a still stronger theory: F+Con(F)+Con(F+Con(F)) (F+Con(F) plus the axiom that F+Con(F) is consistent). And so on infinitely. (Indeed, even beyond infinitely, into the countable ordinals.) On the other hand, again by the Second Incompleteness Theorem, ZF can't prove its own consistency. If we want to prove Con(ZF), the simplest way to do it is to posit the existence of infinities bigger than anything that can be defined in ZF. Such infinities are called "large cardinals." (When set theorists say large, they mean large.) Once again, we can prove the consistency of ZF in ZF+LC (where LC is the axiom that large cardinals exist). But if we want to prove that ZF+LC is itself consistent, then we need a still more powerful theory, such as one with even bigger infinities.

ikeboy 9y ago

>It's ZF+Con(ZF)+Not(Con(ZF)) because you've added the extra axiom as part of your proof by contradiction. No. It's ZF+Con(ZF)+not(con(ZF+con(ZF)). Which is, in fact consistent (assuming ZF is).

kirrent 9y ago
Of course! Thankyou, that makes me much more comfortable. ZF+Con(ZF)+not(con(ZF+con(ZF)) is an example of what's sometimes called the self hating theory, which is known to be consistent if ZF+Con(ZF) is consistent (you said much the same thing in your original comment). As it's known to be consistent, you can't apply the principle of explosion to produce Not(Con(ZF)). EDIT: Obviously I'm not really the sort of person to talk about this stuff with you because of how much of a neophyte I am. Perhaps this lecture would make it clearer? http://www.scottaaronson.com/democritus/lec3.html http://www.scottaaronson.com/democritus/lec3.html In particular, the excerpts: The Second Incompleteness Theorem establishes what we maybe should have expected all along: that the only mathematical theories pompous enough to prove their own consistency, are the ones that don't have any consistency to brag about! If we want to prove that a theory F is consistent, then we can only do it within a more powerful theory -- a trivial example being F+Con(F) (F plus the axiom that F is consistent). But then how do we know that F+Con(F) is itself consistent? Well, we can only prove that in a still stronger theory: F+Con(F)+Con(F+Con(F)) (F+Con(F) plus the axiom that F+Con(F) is consistent). And so on infinitely. (Indeed, even beyond infinitely, into the countable ordinals.) On the other hand, again by the Second Incompleteness Theorem, ZF can't prove its own consistency. If we want to prove Con(ZF), the simplest way to do it is to posit the existence of infinities bigger than anything that can be defined in ZF. Such infinities are called "large cardinals." (When set theorists say large, they mean large.) Once again, we can prove the consistency of ZF in ZF+LC (where LC is the axiom that large cardinals exist). But if we want to prove that ZF+LC is itself consistent, then we need a still more powerful theory, such as one with even bigger infinities.