On Mon, Aug 26, 2019 at 7:10 AM Bruno Marchal <[email protected]> wrote:
> > On 26 Aug 2019, at 03:54, Jason Resch <[email protected]> wrote: > > > > On Sunday, August 25, 2019, Bruce Kellett <[email protected]> wrote: > >> On Mon, Aug 26, 2019 at 11:03 AM Jason Resch <[email protected]> >> wrote: >> >>> On Sunday, August 25, 2019, Bruce Kellett <[email protected]> wrote: >>> >>>> On Sun, Aug 25, 2019 at 11:03 PM Bruno Marchal <[email protected]> >>>> wrote: >>>> >>>>> On 25 Aug 2019, at 14:01, Bruce Kellett <[email protected]> wrote: >>>>> >>>>> On Sun, Aug 25, 2019 at 9:39 PM Bruno Marchal <[email protected]> >>>>> wrote: >>>>> >>>>>> On 25 Aug 2019, at 10:10, Bruce Kellett <[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. 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]. 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.
