On Sunday, July 28, 2019 at 5:09:56 AM UTC-5, Bruno Marchal wrote: > > > On 27 Jul 2019, at 20:42, Lawrence Crowell <[email protected] > <javascript:>> wrote: > > On Saturday, July 27, 2019 at 8:38:12 AM UTC-5, John Clark wrote: >> >> All that assumes that infinity exists for any meaningful use of the word >> “exists” and as far as I know nobody has ever found a infinite number of >> anything. Mathematics can write stories about the infinite in the language >> of mathematics but are they fiction or nonfiction? >> >> John k Clark >> >> > Infinity is not a number in the usual sense, but more a cardinality of a > set. Infinity has been a source of trouble for some. I work with Hilbert > spaces that have a form of construction that is finite, but where the > finite upper limit is not bounded ---- it can always be increased. This is > because of entropy bounds, such as the Bekenstein bound for black holes and > Bousso bounds on AdS, that demands a finite state space for local physics. > George Cantor made some set theoretic sense out of infinities, even a > hierarchy of them. This avoids some difficulties. However, I think that > mathematics in general is not as rich if you work exclusively in finitude. > Fraenkel-Zermelo set theory even has an axiom of infinity. The main point > is with axiomatic completeness, and mathematics with infinity is more > complete. > > > Mechanism provides an ontological finitism (what exists are only 0, s(0), > s(s(0)), …), but it explains why those finite objects will believe > correctly in some phenomenological infinite (already needed to get an idea > of what “finite” could mean. > The infinite is phenomenologically real, but has no ontology. > > No first order logical theories can really define the difference between > finite and infinite. Even ZF, despite its axiom of infinity is not able to > do that, in the sense that it too has non standard model, in which we can > have a finite number greater than all the “standard” natural numbers 0, > s(0) … > > I am not sure why you say that adding an axiom of infinity makes a theory > more complete. There are sense it which it only aggravate incompleteness. > > Once a theory is rich enough to define and prove the existence of a > universal machine, that theory becomes essentially undecidable (which means > that not only it is undecidable, but it is un-completable: all the > effective consistent extensions are undecidable. > > Bruno > > I am not a set theory maven particularly. I only know the basic things and some aspects of advanced topics I have read. The recursive function is to take 0 and "compute" s(0) and then ss(0) and so forth. The entire set is recursively enumerable and the idea that given 0 and computing s(0) one has ss^n(0) = s^{n+1}(0) is induction. That this leads to a countably infinite set is recursively enumerable and that is not something one can "machine compute." I think this is this "extension."
LC > > > > Richard Feynman talked about Greek mathematics, the axiomatic formal > systems of mathematics, and Babylonian mathematics that is set up for > practical matters. I have no particular preference for either, and think it > is interesting to switch hats. > > LC > > >> >> On Sat, Jul 27, 2019 at 7:36 AM Lawrence Crowell < >> [email protected]> wrote: >> >>> On Thursday, July 25, 2019 at 10:02:39 PM UTC-5, Bruce wrote: >>>> >>>> On Fri, Jul 26, 2019 at 12:48 PM John Clark <[email protected]> wrote: >>>> >>>>> When I was younger I read a lot of science fiction, I don't do it so >>>>> much anymore and technically I didn't do it this time either but I did >>>>> listen to a audio book called "We Are Legion We Are Bob" it's the first >>>>> book of the Bobiverse trilogy and I really enjoyed it. You can get a free >>>>> 5 >>>>> minute sample of the book here: >>>>> >>>>> We Are Legion (We Are Bob): Bobiverse, Book 1 >>>>> <https://www.amazon.com/We-Are-Legion-Bob-Bobiverse/dp/B01L082SCI/ref=tmm_aud_swatch_0?_encoding=UTF8&qid=&sr=> >>>>> >>>>> It tells the story of Bob, a young man who has just sold his software >>>>> company for a crazy amount of money and decides that after a decade of >>>>> hard >>>>> work he's going to spent the rest of his life just goofing off. On a whim >>>>> he signs with a Cryonics company to have his head frozen after his >>>>> death and then just hours later while crossing the street to go to a >>>>> science fiction convention is hit by a car and dies. Five subjective >>>>> second >>>>> s later he wakes up and finds that a century has passed and he's been >>>>> uploaded into a computer. This is all in the opening chapter. >>>>> >>>>> Parts of the story are unrealistic but parts of it are not, I think it >>>>> was Isaac Asimov who said it's OK for a science fiction writer to >>>>> violate the known laws of physics but only if he knows he's doing it, and >>>>> when Dennis Taylor, the creator of Bob universe, does it at one point >>>>> with >>>>> faster than light communication it's obvious that he knowns it. And I >>>>> can't >>>>> deny it makes for a story that is more fun to read. I have now read (well >>>>> listened) to all 3 Bob books and, although parts are a little corny and >>>>> parts a little too Star Trek for my taste, on the whole I greatly enjoyed >>>>> them all. They're a lot of fun. >>>>> >>>>> The only other novel I can think of that treats the subject of >>>>> uploading with equal intelligence is "The Silicon Man". >>>>> >>>>> The Silicon Man by Charles Platt >>>>> <https://www.amazon.com/Silicon-Man-Cortext-Charles-Platt/dp/1888869143> >>>>> >>>>> John K Clark >>>>> >>>> >>>> Consider any of the earlier novels by Greg Egan, the Australian hard >>>> science fiction write based in Perth, WA: particularly "Permutation City" >>>> (1994). >>>> >>>> Bruce >>>> >>> >>> I had this idea of a science fiction story of where minds are stored in >>> machines in order to "eternally" punish them. The idea is that if a million >>> seconds in the simulated world is a second in the outer world then one can >>> in effect construct a near version of eternal hell-fire. The setting is a >>> world governed by complete terror. Then Egan came out with Permutation >>> city, which explores a similar set of ideas. >>> >>> The problem with the idea of putting minds into machines is that >>> machines can run recursive functions or algorithms, but in a number system >>> such as Peano's we make the inductive leap that the successor of any number >>> can't be the same number or zero in all (infinite number) cases. We can >>> make an inference from a recursively enumerable set. I would then think >>> that the idea of putting minds into machines, or robotic consciousness, is >>> at this time an unknown, maybe an unknowable, proposition. >>> >>> LC >>> >>> -- >>> 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/44ed303e-5650-430b-b255-bc28392194ae%40googlegroups.com >>> >>> <https://groups.google.com/d/msgid/everything-list/44ed303e-5650-430b-b255-bc28392194ae%40googlegroups.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] <javascript:>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/1c518015-9dc2-47a8-968c-3b6c8eed1594%40googlegroups.com > > <https://groups.google.com/d/msgid/everything-list/1c518015-9dc2-47a8-968c-3b6c8eed1594%40googlegroups.com?utm_medium=email&utm_source=footer> > . > > > -- You received this message because you are subscribed to the Google Groups "Everything List" group. 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