On Monday, July 29, 2019 at 5:34:39 AM UTC-5, Bruno Marchal wrote: > > > On 28 Jul 2019, at 23:42, Lawrence Crowell <[email protected] > <javascript:>> wrote: > > > > 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]> >> 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.” > > > > The set N = {0, 1, 2, …} is trivially recursively enumerable (can be > generated by a digital machine/program). It is the range of the identity > function. > > Once a function is computable, or once a set can be computably generated, > we usually say that the function (an infinite object) is (partially) > computable, or that the set is (semi)-computable. A function can be said to > compute its extension. > > A function is NOT computable when there is no algorithm capable of giving > output on some input where it is defined. > > I have shown that the function deciding if a code compute a total or a > strictly partial function is (highly) not computable, although well defined > if we accept the excluded principle (as we do in classical (non > intuitionist) computer science, and as we have to do when we do theology, > given that a theology is a highly non constructive notion (provably so for > the theology of a machine (by definition: the study of the true > propositions (and subset of true propositions) on the machine, provable or > not by that machine). > > Bruno >
Numbers are computable, but the entire set Z of integers is not. The conscious being or human makes the inductive leap from the successors of 0 that the set of integers is an infinite set. 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/37c2dfd9-b1d1-46a2-abab-bf4924061070%40googlegroups.com.
