> On 26 Aug 2019, at 15:28, Jason Resch <[email protected]> wrote: > > > > On Mon, Aug 26, 2019 at 7:10 AM Bruno Marchal <[email protected] > <mailto:[email protected]>> wrote: > >> On 26 Aug 2019, at 03:54, Jason Resch <[email protected] >> <mailto:[email protected]>> wrote: >> >> >> >> On Sunday, August 25, 2019, Bruce Kellett <[email protected] >> <mailto:[email protected]>> wrote: >> On Mon, Aug 26, 2019 at 11:03 AM Jason Resch <[email protected] >> <mailto:[email protected]>> wrote: >> On Sunday, August 25, 2019, Bruce Kellett <[email protected] >> <mailto:[email protected]>> wrote: >> On Sun, Aug 25, 2019 at 11:03 PM Bruno Marchal <[email protected] >> <mailto:[email protected]>> wrote: >> On 25 Aug 2019, at 14:01, Bruce Kellett <[email protected] >> <mailto:[email protected]>> wrote: >>> On Sun, Aug 25, 2019 at 9:39 PM Bruno Marchal <[email protected] >>> <mailto:[email protected]>> wrote: >>> On 25 Aug 2019, at 10:10, Bruce Kellett <[email protected] >>> <mailto:[email protected]>> wrote: >>>> The mathematical structure might describe these things, but descriptions >>>> are not the things they describe. >>> >>> I think you confuse the mathematical structure, and the theory describing >>> that mathematical structure. Those are very different things. >>> >>> I think that is exactly the mistake that you make all the time. >> >> Where? I don’t remind you ever show this. >> >> I have said it many times. A mathematical structure is an abstract human >> construct. Such a structure might go some way towards describing physical >> reality, but the map is not the territory. >> >> >> Bruno is talking about the territory and I think you are confusing it with >> Bruno talking about the map. To be clear, axioms in math are just theories >> to explain the mathematical reality, >> >> Using the word "reality" here just begs the question. Arithmetic (or >> mathematics) is nothing more than the product of its axioms. Proofs from the >> axioms may not capture all that one might regard as "truth", but that is >> really beside the point. Using the word "truth" is just as fraught as using >> the term "reality" -- question begging. >> >> Any system of axioms can only prove a finite number if bits of Chaitin's >> constant. More powerful systems can prove more bits of it, but no system is >> capable of proving endless bits of it. So where does this number belong? >> It's complete set of digits are not decidable under any system of axioms. >> It's not the product of any system if axioms. > > > Calude mentions an interesting theorem by Solovay. There is a universal > machine U such that ZFC cannot compute *any* bit of its Chaitin-Omega number. > Not even the first bit. I guess this used ZFC + some strong axiom (Hmm… like > the arithmetical soundness of ZF probably). That Universal machine U is not > predictible at all by ZFC, yet, its behaviour is arithmetically > deterministic. Assuming ZFC arithmetically sound (which I find very > plausible). > > > Interesting. Is the idea to make a Turing machine that first must generate > some proof before it makes its first step at processing the input program, > and so long as the proof can be found it will make progress (but if no proof > exists under some theory) then that theory can't prove the machine is > universal.
I should find the paper. I think that in this case, ZFC can prove U to be Universal, and still cannot compute any bit of its Omega number. But may be you are right. I am not sure. I guess the proof involved diagonalization/second-recursion-theorem, and that ZFC involves itself in the definition of the machine U. I have tried to find the paper on the net, without success. It is the paper: R. M. Solovay. A version of Ω for which ZFC can not predict a single bit, in C.S. Calude, G. P ̆aun (eds.). Finite Versus Infinite. Contributions to an Eternal Dilemma, Springer-Verlag, London, 2000, 323–334. Bruno > > Jason > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected] > <mailto:[email protected]>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/CA%2BBCJUh-VF6aSS%2BKx2OnVtbEShQJ9UhhR%2B%2BPyjB-fYPE7jwnOQ%40mail.gmail.com > > <https://groups.google.com/d/msgid/everything-list/CA%2BBCJUh-VF6aSS%2BKx2OnVtbEShQJ9UhhR%2B%2BPyjB-fYPE7jwnOQ%40mail.gmail.com?utm_medium=email&utm_source=footer>. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/A28A6E9B-6A34-4D58-8BC9-FBE3B3BEB034%40ulb.ac.be.
