> On 7 Mar 2020, at 18:33, Lawrence Crowell <[email protected]> > wrote: > > 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.
Up to an abuse of language, ZF is mainly PA + the axiom of infinity. The set V_omega is basically the arithmetical reality, seen embedded in ZF set theory. ZF and ZFC proves the same arithmetical proposition, and much more than PA. ZFC + higher infinities proves even more. And there are simple combinatorial problem, notably on the table of Laver, which today requires super-higher cardinal to be proven, notably the cardinals of Laver. They might play a role in the derivation of space from arithmetic. They are related to the theory if braids and knots! Bruno > > 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 > <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 > <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] > <mailto:[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 > > <https://groups.google.com/d/msgid/everything-list/b14b7394-7fce-434d-82a4-107b6ce84cf1%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/83DA9C8A-E182-4C55-BDA5-A9084BE7B8E9%40ulb.ac.be.
