On Saturday, July 4, 2020 at 5:31:24 AM UTC-5, Bruno Marchal wrote: > > > On 3 Jul 2020, at 14:30, Lawrence Crowell <[email protected] > <javascript:>> 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. >
It is and is not. Agreed forcing is a dense set theoretic technique on how to extend a model into some other model. I think it was Hamkins who wrote on how forcing was really not that mysterious and used the example of extending integers into rationals, those into reals and those into ,,, . Set theory without some connection to actual mathematics is about as useless as tits on a boar hog. LC > > 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/ >> >> @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 >>>> >>>> [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] <javascript:>. > 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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