On Saturday, March 7, 2020 at 6:07:26 AM UTC-6, Philip Thrift wrote: > > > > > This is about the *λ_ZFC* calculus, not the *λ calculus*. > > > λ_ZFC contains infinite terms. Infinitary languages are useful > and definable: the infinitary lambda calculus [10] is an example, and > Aczel’s > broadly used work [2] on inductive sets treats infinite inference rules > explicitly. > > @philipthrift > > I am aware of this, It is a bit like considering Peano arithmetic in a domain where the axioms of infinity and choice hold.
LC > > On Friday, March 6, 2020 at 5:25:13 PM UTC-6, Lawrence Crowell wrote: >> >> On Friday, March 6, 2020 at 5:57:34 AM UTC-6, Philip Thrift wrote: >>> >>> >>> >>> While programming/computing in (hypothetical) infinite domains is >>> interesting ... >>> >>> *Computing in Cantor’s Paradise With λ_ZFC* >>> https://jeapostrophe.github.io/home/static/toronto-2012flops.pdf >>> >>> how any of this relates *in any way* to physical reality (the *stuff of >>> nature *that is *actually around us* in the universe, vs. just some >>> theoretical, mathematical concoction someone may come up with) is dubious. >>> >>> (Things like consciousness is another thing, or subject: It may be >>> "beyond" Turing, bit in a way that has nothing to do with "super" or >>> "hyper" Turing or Cantor or Godel.) >>> >>> @philipthrift >>> >> >> λ-calculus is equivalent to Turing computation. In fact it is similar to >> Assembly language. It might be that some of these problems could be looked >> at according to λ-calculus. >> >> LC >> >> >>> >>> On Friday, March 6, 2020 at 5:40:08 AM UTC-6, Lawrence Crowell wrote: >>>> >>>> Szangolies [ J. Szangolies, "Epistemic Horizons and the Foundations of >>>> Quantum Mechanics," https://arxiv.org/abs/1805.10668 ] works a form >>>> of the Cantor diagonalization for quantum measurements. As yet a full up >>>> form of the CHSH or Bell inequality violation result is waiting. There are >>>> exciting possibilities for connections between quantum mechanics, in >>>> particular the subject of quantum decoherence and measurement, and Gödel’s >>>> theorem. >>>> >>>> If we think of all physics as a form of convex sets of states, then >>>> there are dualisms of measures p and q that obey 1/p + 1/q = 1. For >>>> quantum >>>> mechanics this is p = ½ as an L^2 measure theory. It then has a >>>> corresponding q = ½ measure system that I think is spacetime physics. A >>>> straight probability system has p = 1, sum of probabilities as unity, and >>>> the corresponding q → ∞ has no measure or distribution system. This is any >>>> deterministic system, think completely localized, that can be a Turing >>>> machine, Conway's <i>Game of life</i> or classical mechanics. A quantum >>>> measurement is a transition between p = ½ for QM and ∞ for classicality or >>>> 1 for classical probability on a fundamental level. >>>> >>>> What separates these different convex sets are these topological >>>> obstructions, such as the indices given by the Kirwan polytope. The >>>> distinction between entanglements is also given by these topological >>>> indices or obstructions. How these determine a measurement outcome, or the >>>> ontology of an element of a decoherent sets is not decidable. This is >>>> where >>>> Gödel’s theorem enters in. A quantum measurement is a way that quantum >>>> information or qubits encode other qubits as Gödel numbers. >>>> >>>> The prospect spacetime, or the entropy of spacetime via event horizon >>>> areas, is a condensate or large N-entanglement of quantum states then >>>> implies there is a connection between quantum computation and information >>>> accessible in spacetime configurations. These configurations may either be >>>> the Bekenstein bound S = kA/4ℓ_p^2, or quantum modified version S = >>>> kA/4ℓ_p^2 + quantum corrections. Then the quantum processing or quantum >>>> Church-Turing thesis is I think equivalent to the information processing >>>> of >>>> spacetime as black holes and maybe entire cosmologies. >>>> >>>> These are exciting developments. >>>> >>>> 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/b14b7394-7fce-434d-82a4-107b6ce84cf1%40googlegroups.com.
