> On 3 Jul 2020, at 14:30, Lawrence Crowell <[email protected]> > wrote: > > It does not need that sort of intense structuring. The extension of real to > complex numbers is a case of forcing of real numbers into pairs that obey > multiplication rules for complex numbers.
That is not forcing in the theoretical meaning of the papers referred to by P. Thrift. Forcing is an advanced technic in the study of the models of set theory, and it requires much more work in mathematical logic and set theory than what I use in my contribution. To be sure, Forcing is used in other model theory, including the models of Arithmetic. (See the book by Boolos and Jeffrey, or the new editions with Burgess on this). In this case of forcing viewed as computation, it concerns a non standard notion of computations, and it might be of some use in the phenomenology of matter, as most technic in model theory does. Technically, forcing can be seen as an exercise in modal logic. For those remembering Kripke semantics, the resemblance is striking, but it took some time before this has been made explicit, like Smullyan and Fitting did in their book on (Von Neuman Bernays Gödel) set theory. They found a logic of type S4 (like the first person mechanist phenomenology which isolate S4Grz1), and amazingly, where obliged to consider a quantization ([]<>p) in their modal translation, but this might be only a coincidence. Bruno > The extension with the identification 1/kT ↔ it/ħ is the basis for quantum > criticality and the foundation of phase transitions. In this sense the > quantum and classical domains of physics are different phases of matter and > energy. > > LC > > On Friday, July 3, 2020 at 2:30:13 AM UTC-5, Philip Thrift wrote: > Whether forcing does anything for physics, I don't know, but ultimately it > could be in future proof assistants (like Lean, Coq, etc.) and programming > languages as a continuation of > monadic programming > Every function can be computable! > <http://jdh.hamkins.org/every-function-can-be-computable/> > A Haskell monad for infinite search in finite time > <http://www.google.com/url?q=http%3A%2F%2Fmath.andrej.com%2F2008%2F11%2F21%2Fa-haskell-monad-for-infinite-search-in-finite-time%2F&sa=D&sntz=1&usg=AFQjCNFuGhQ_ip4HSUdQcVCDEOkZffCNKg> > Computing in Cantor’s Paradise > <https://jeapostrophe.github.io/home/static/toronto-2012flops.pdf> > > https://codicalist.wordpress.com/2020/05/23/dialectical-programming/ > <https://codicalist.wordpress.com/2020/05/23/dialectical-programming/> > > @philipthrift > > On Thursday, July 2, 2020 at 5:02:42 PM UTC-5 Lawrence Crowell wrote: > On Thursday, July 2, 2020 at 6:48:08 AM UTC-5, Philip Thrift wrote: > https://arxiv.org/abs/2007.00418 <https://arxiv.org/abs/2007.00418> > > [Submitted on 1 Jul 2020] > Forcing as a computational process > > Joel David Hamkins > <https://arxiv.org/search/math?searchtype=author&query=Hamkins%2C+J+D>, > Russell Miller > <https://arxiv.org/search/math?searchtype=author&query=Miller%2C+R>, Kameryn > J. Williams > <https://arxiv.org/search/math?searchtype=author&query=Williams%2C+K+J> > > We investigate how set-theoretic forcing can be seen as a computational > process on the models of set theory. Given an oracle for information about a > model of set theory ⟨M,∈M⟩, we explain senses in which one may compute > M-generic filters G⊆ℙ∈M and the corresponding forcing extensions M[G]. > Specifically, from the atomic diagram one may compute G, from the Δ0-diagram > one may compute M[G] and its Δ0-diagram, and from the elementary diagram one > may compute the elementary diagram of M[G]. We also examine the information > necessary to make the process functorial, and conclude that in the general > case, no such computational process will be functorial. For any such process, > it will always be possible to have different isomorphic presentations of a > model of set theory M that lead to different non-isomorphic forcing > extensions M[G]. Indeed, there is no Borel function providing generic filters > that is functorial in this sense. > > @philipthrift > > Forcing is a bizarre idea, where one introduces some new category or objects > that extends a model. In some ways we do this with the extension of > mathematics from the reals to complex numbers. The forcing of one system into > another has some connections to the Jordan-Banach algebra as well. The > replacement of a Poisson bracket structure {X, Y} → i/ħ[X, Y], is given by > the replacement of generators that are Jordan with product rule A∘B = ½(AB + > BA) to a Lie algebraic ad_A(B) = [A, B] product. For a general quantum system > with a Lindblad type equation > > Idρ/dt = [H, ρ] + ½([Aρ, A^†] + [A, ρA^†]), > > such that the Hamiltonian is an element of the Lie group L, usually as a > weight, and the operators A and A^† are operators so that AρA^†, AA^†ρ, and > ρAA^† are operations on the density matrix that are Jordan. Depending upon > how one looks at this we can think of the quantum mechanical theory on > Fubini-Study metric on ℂP^n, say a metric based action or Lagrangian ℒ = > g_{ab}y^ay^b to ℒ = g_{ab}y^ay^b + U(x_1, x_2, … ) with a general > Euler-Lagrange equation > > F_i = ∂/∂x_i(∂ℒ/∂y_a)y^a - ∂ℒ/∂x_i. > > This is component version in Finsler geometry of a differential 1-form ω = > πdq – dη. It should be apparent this is similar to the first law of > thermodynamics. This is a “practical lesson” in what is meant by forcing. The > imaginary valued system is extended to components that are further multiplied > by i = √(-1). This is a sort of Wick rotation version of the forcing from > reals to complex numbers. > > > The Jordan-Banach algebra though is just a start. We have the skew symmetry > that corresponds to the conjugate on I = √(-1). This Lie algebraic system is > in contrast to the Jordan product and algebra. We also have in physics > fundamental properties due to bosonic and fermionic statistics. In > supersymmetry there are the Grassmann numbers that enter in to intertwine > between these. Further, on a 2-dimensional space the interchange of two > bosons or fermions is ambiguous. This is because a clockwise and > counterclockwise rotational exchange are topologically distinct. In 3 > dimensions of space the paths can be exchanged, but no in two. This means a > sign change is possible in either boson or fermion cases. The stretched > horizon of a black hole is a 2-dimensional surface and an anionic system for > quantum fields. I think this is a foundation for how there are these > dichotomies in physics, such as quantum mechanics and macroscopic physics and > different statistics. Instead of physics being founded on very large > algebraic systems I think that Wigner had something right with the idea of > small groups. The big symmetry systems come about from symmetry breaking or > the removal of a degeneracy. > > The fibration and Noether’s theorem in differential forms is λ = p_idq_i – > Hdt, where the 1-form has a horizonal p_idq_i and vertical Hdt principal > bundles. This is a Finsler geometric way of seeing Lagrangian dynamics. The > differential dλ = dp_i∧dq_i – dH∧dt, vanishes on the contact manifold. In a > strict Finsler context the horizontal and vertical bundles are annulled > separately. The q_i is the generator of p_i and dp_i = 0 means momentum is > invariant under translation and similarly energy is conserved under time > translations. If there is a Lagrange multiplier that connects the vertical > and horizontal parts of the bundle then we have > > dp_i∧dq_i – dH∧dt = (dp_i/dt)dt∧dq_i – (∂H/dq_i)dq_i∧dt = [(dp_i/dt) + > (∂H/dq_i)]dq_i∧dt > > which gives us the dynamical equation (dp_i/dt) = -∂H/dq_i. > > Hamkins et al paper is dense in set theoretic constructions. These things I > have some introductory acquaintance with. I am also more interested in the > practical transduction of such ideas. I will try to see what I can gather > from this paper. Forcing does have some algorithmic connection and the above > connects 1/kT ↔ it/ħ with macroscopic system with a temperature, or gravity > in the case of spacetime physics, to time in the case of quantum mechanics. > > 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/49bc5442-f75f-4e69-9fea-d9fc8aa420f3o%40googlegroups.com > > <https://groups.google.com/d/msgid/everything-list/49bc5442-f75f-4e69-9fea-d9fc8aa420f3o%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]. 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