May I ask whether you might be user PyRulez? (Your name appears in the email of his user profile.) If so, I’d like to use your real name in my post, unless you would prefer I didn’t.

]]>Let me write explicitly how to apply Kleene’s theorem:

We define the program $f(e)$ such that:

1. It searches for the proof that $e$ does not enumerate exactly the sequence $s$.

2. Upon finding such a proof, $f(e)$ (not $e$ yet!) enumerates this sequence.

Then by the theorem we end up with a program $e$ that searches for the proof about itself and then behaves in a forbidden way.

But if $e$ finds such a proof for some sequence $s$, then $T$ is inconsistent. So in fact $e$ will not find such a proof in $T$. That means, on the other hand, that $T$ is consistent with the fact that the program $e$ enumerates any sequence $s$ (since no proof of contrary has been found). So there is a model $M$ of $\PA$ in which the program $e$ enumerates exactly $s$.

This is basically the same what you wrote, but for some reason rephrasing it made

it easier for me to understand. Maybe it will help some less experienced readers like me 🙂

About your remarks here, the way I think about it is that we seem to have a variety of foundational frameworks in set theory and mathematics, and one can seem sometimes to interpret those frameworks within one another, either by going to definable inner models or forcing extensions, as you say. The gold standard for two such foundational theories to be “equivalent” or just-as-good in foundational matters, would be bi-interpretation. And so I find it extremely interesting that there basically are no nontrivial instances of bi-interpretation amongst the strengthenings of ZF.

]]>it was nice seeing you the other day. and this is a nice result!

having said this, however, let me just say that it doesn’t really have much to do with what you say is shocking. in fact i am not sure why it is shocking for the reasons you outlined. it could be shocking for some other reasons, but to me, i don’t see the connection.

for one thing, when one speaks of bi-interpretability one usually allows forcing extensions…

but more importantly, i really do not think the formal version of the bi-interpretability claim is of any mathematical significance whatsoever. bi-interpretability claim, to me at least, consists of set of problems that, if solved, would imply deep connection between various seemingly different areas of set theory.

for instance, MM(c) can be forced over models of AD_R+Theta is regular (Woodin). MM implies that there are canonical models of AD_R+Theta is regular (Trang) and of LSA (myself and Trang).

the real problem here is whether one can show that MM implies Woodin’s (*), or whether MM can hold in a generic extension of some model of determinacy. these will be true pillars of pure set theory, i think.

as far as philosophy behind this stuff goes, i think that it is the least interesting aspect of all of it. but if one cares, let me just say that to me, math is more like science, i am doing math like a scientist would do science, i draw pictures, i do examples, i find relations between the objects i investigate, and i generalize. i understand that others maybe doing math differently, but if one has this sort of view of math then the point just is that your experience might lead you to believing that axiom A is true, while another might be led to believing that axiom B is true, and this we have witnessed in physics. when i think about set theory, i have to really try hard to think in a way that determinacy fails…so for me, choice is never given. unlike Saharon who always thinks that choice is given. this is a result of what we have been doing separately, but it is hardly the case that we don’t understand each other….

then the substantial claim is that it is only a psychological illusion that A and B are disjoint, provided that both explain sufficiently many experiences, people who believe in A can understand and appreciate what people who believe in B do and vice a versa, at least the A person can understand why the B person sees the mathematical universe the way he or she sees it and vice a versa. there are precise conjectures that when resolved could support this view. i think the formal version of this view is not interesting, at least to me.

it is like Englsih vs Armenian, i believe that there is no formal translation that preserves full meaning between English and Armenian, but given enough time and exposure, i can explain the true meaning of every Armenian expression to someone who grew up in the states. That might require me putting the person in an Armenian environment for a long enough time, but the person will eventually get it (he or she has to also be open to learning the meaning of Armenian expressions). i am not sure what the formal version of such a claim is, and i am not sure if it is interesting to formalize it, it is certainly important to have words-for-words kind of translations, but as we all know, this doesn’t really do it.

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