On Sunday, July 28, 2019 at 4:42:40 PM UTC-5, Lawrence Crowell 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." > > LC >
Of course in programming "infinite structures" are not uncommon: e.g. *SMT Solving for Functional Programming over Infinite Structures* Bartek Klin, Michał Szynwelski University of Warsaw https://www.mimuw.edu.pl/~szynwelski/nlambda/nlambda.pdf *We develop a simple functional programming language aimed at manipulating infinite, but first-order definable structures, such as the countably infinite clique graph or the set of all intervals with rational endpoints. Internally, such sets are represented by logical formulas that define them, and an external satisfiability modulo theories (SMT) solver is regularly run by the interpreter to check their basic properties.* *Our goal is a set of programming idioms that would hide from the programmer as much as it is possible the fact that she or he is dealing with infinite sets presented by first-order formulas rather than with finite sets presented by enumerating their elements.* *The language is implemented as a Haskell module.* @philipthrift > > -- 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/3107fa72-5460-4117-96e6-a8f71468753e%40googlegroups.com.
