On Monday, July 29, 2019 at 5:27:55 AM UTC-5, Philip Thrift wrote: > > > > 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 >
The paper looks rather dense, but I save it and maybe I will get to it at some point. It though looks as if they have implemented something that appears to give infinite strings. I doubt this acts as a hyper-Turing machine. 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/b2f7d0de-bc46-4660-a302-013a6eed5764%40googlegroups.com.
