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
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
>>>
>>>
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