> On 29 Jul 2019, at 13:06, Lawrence Crowell <[email protected]> > wrote: > > 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.
? All numbers in N, Q, and Z are computable. For the real numbers, it is different and more subtle, but usually we represent the (computable) real number by total computable functions from N to N. N, Z, and Q are equivalent with respect of computability theory. They are all equivalent to V_w (V_omega), the set of rank smaller than w (omega, the least infinite ordinal), which is the theory of finite sets. > The conscious being or human makes the inductive leap from the successors of > 0 that the set of integers is an infinite set. They do that in set theory, not in arithmetic, except at the meta-level in case work on arithmetic (as opposed to work in arithmetic). (Z, +, *) and (Q, +, *) are representable in (N, +, *) , like the SK-combinators, and all universal machineries, are representable in (N, +, *). Bruno > > 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] > <mailto:[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 > > <https://groups.google.com/d/msgid/everything-list/37c2dfd9-b1d1-46a2-abab-bf4924061070%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]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/BE858227-90D6-46C5-8AAA-7670280EA2F4%40ulb.ac.be.
