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It was very wrong of me to write that "nonstandard numbers may be incompatible". Perhaps, I did not wake actually up. If I only new how to edit comments.... What surprised me is how Joel managed to show that something is $provable$ by ruling out all the models where it is wrong. I did not expect this to be possible - but, as I said, I am not very good in models.
Joel, there is a gap. Applying the lemma, we may get $\phi$ provably equivalent to $\psi$. In this case $\phi\vee\psi$ is equivalent to $Con(PA)$. Then, the lenghts of nonstandard proofs are nonstandard numbers, which may be incompatible. So, we need another argument to show that $\phi\vee\psi$ can be proved.
Joel, there is a gap. Applying the lemma, we may get $\phi$ provably equivalent to $\psi$. In this case $\phi\vee\psi$ is equivalent to $Con(PA)$. Then, the lenghts of nonstandard proofs are nonstandard numbers, which may be incompatible. So, we need another argument to show that $\phi\vee\psi$ can be proved.
Daniel, Let me explain myself. Suppose for a moment that the question was "A continuous map of a good contractible compact topological space into itself has a fixed point" (Which is what I mean, actually). You gave an example, I did not accept it for the space is not good enough for me. Would you like this? I failed to foresee all the possible pathology: the fault is mine. But is it a good reason to create a new question? I doubt it. P.S. By the way, could anyone give me Bing's article? For now, I cannot get it from where I am.
Explanation: The statement should be of kind "Let $X$ looks like a ball. Then...". Definitely, Kinoshita's example does not look like a ball, for it is not locally connected. So, I added this condition to rule it out.