Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

In Kunen's Set theory (2011), he says that there is a finitistic proof that $ $Con$(ZF^-)\implies $Con$(ZF)$. He also mentions elsewhere that if $\Theta$ is at least as strong as finitistic reasoning and we have a finitistic proof of $ $Con$(\Lambda)\implies $Con$(\Gamma)$ then:

$\Theta \vdash $Con$(\Lambda)\implies $Con$(\Gamma)$

Now my question is when he says there is a finitistic proof of $ $Con$(ZF^-)\implies $Con$(ZF)$, does he mean that $\Theta \vdash $Con$(ZF^-)\implies $Con$(ZF)$ for any $\Theta$ at least as strong as finitistic reasoning (like $BST^-$)?

My confusion is this: Understanding an Easy Relative Consistency Proof

Which leads me to believe that the $\Theta$ needs to be stronger than $BST^-$ in this particular case.

If this is the case, then what's the reason Kunen is not more clear on this point, ie why doesn't he explicitly say: "We have shown that $ZF^- \vdash $Con$(ZF^-)\implies $Con$(ZF)$"

Any help is appreciated, thanks!

share|improve this question

1 Answer 1

up vote 2 down vote accepted

Look at the third and fourth paragraphs of the answer you linked to. There Andres Caicedo gives a syntactic argument, which looks finitistic to me. What exactly leads you to believe that you need something stronger than finitistic reasoning for this argument?

share|improve this answer
    
Thanks for the reply. I've reread Andres' explanation again. So what he's saying is one way we can show $ZF^- \vdash $Con$(ZF^-)\implies $Con$ (ZF)$ is by using the Completeness Theorem, and we can do so since basic model theory can be formalized in $ZF^-$ and so we can prove the Completeness Theorem in $ZF^-$. Or he says we can work in a weaker system $\Theta$ (such as $BST^-$ or one that can carry out finitistic reasoning?) and show that $\Theta\vdash $Con$(ZF^-)\implies $Con$ (ZF)$ by the syntactic argument which he gives. Am I on the right path here? –  user52534 Sep 11 '13 at 21:28
    
@user52534 Right. To work in a weak system, ignore the part about the completeness theorem (which works only in stronger systems) and concentrate on the syntactic argument. –  Andreas Blass Sep 11 '13 at 21:48

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.