> On 2 Oct 2018, at 09:53, Philip Thrift <[email protected]> wrote: > > > > On Tuesday, October 2, 2018 at 2:20:10 AM UTC-5, Bruno Marchal wrote: > >> On 1 Oct 2018, at 14:20, [email protected] <javascript:> wrote: >> >> >> >> On Monday, October 1, 2018 at 11:47:47 AM UTC, Bruno Marchal wrote: >> >>> On 30 Sep 2018, at 16:30, Philip Thrift <[email protected] <>> wrote: >>> >>> >>> >>> On Sunday, September 30, 2018 at 4:50:01 AM UTC-5, Bruno Marchal wrote: >>> [Re:] forcing theory in set theories with classes. >>> >>> >>> Bruno >>> >>> >>> >>> Do you follow the work of Joel David Hamkins (forcing applied to >>> set-theoretic "multiverse", etc.) >>> >>> (I have a basic idea of a type-theoretic parallel to this.) >>> >>> The set-theoretic multiverse >>> >>> https://arxiv.org/abs/1108.4223 <https://arxiv.org/abs/1108.4223> >>> >>> Joel David Hamkins >>> @JDHamkins >>> Professor of Logic, University of Oxford, and Sir Peter Strawson Fellow in >>> Philosophy, University College Oxford. Formerly of New York. >>> http://jdh.hamkins.org <http://jdh.hamkins.org/> >>> >> >> The math is interesting, and could be of some use, but it is a priori far >> too much Aristotelian to be coherent with the mechanist hypothesis. That >> should follow “easily” from the result described in most of my papers on >> this subject. The author does not seem aware of the mind-body problem, which >> put extreme constraints on what the physical reality can come from. Even >> Peano arithmetic, although integral part of the notion of observer, is too >> much rich for the ontology, where not only the axiom of infinity is too >> strong, >> >> Since you want to banish the concept of infinity from mathematics, how would >> you define, say, the limit of an "infinite" series? How would you even >> discuss this series in the context of finite mathematics? AG > > > Good question. > > The answer is not simple technically. The point is that using only the theory > Q (Robinson Arithmetic) or SK (the combinators), I can define the universal > (Turing, Church) machine, and the concept of infinity will be a tool used by > them in their mathematics. > > I do not ban anything from mathematics, nor from physics. I ban only infinity > from the ontological terms. I ban only infinity in the metaphysics/theology. > (Even God is not ontological, like in Proclus or Plotinus theology). > > Have you understand the post on Church’s thesis. You might tell me as this > will help me to see how to proceed to make you grasp all this. > > Bruno > > > > > What do you think of bounded arithmetic and other "finitist" approaches? > > https://en.wikipedia.org/wiki/Bounded_arithmetic > see bibliography: http://jeanpaulvanbendegem.be/home/papers/strict-finitism/
I wrote a paper on this, in a book in honour to Jean-paul Vanbendegem. But its approach is more than finitist, and a bit less than ultra-finitism. It does not fit the study of the “theology” of the machine, and is thus useless for deriving physics. That does not mean it is not interesting pragmatically, on the contrary, it is well fitted with the goal to make usable programs. I do think that mathematically, it is also a restriction of Post creativity (Turing universality in set theoretical terms) to sub creativity. There is no possible universal machine there. > > Computable real analysis (one can teach computable calculus instead of > "conventional" calculus) is essentially finitist: > https://en.wikipedia.org/wiki/Computable_analysis > > One can formulate the Axiom of Infinity [ > https://en.wikipedia.org/wiki/Axiom_of_infinity ] in a type of bounded set > theory (Jan Mycielski [ https://en.wikipedia.org/wiki/Jan_Mycielski ], > described in > https://books.google.com/books/about/Understanding_the_Infinite.html?id=GvGqRYifGpMC > ]. What results is an "ontology" of bigger and bigger finite sets of numbers > with gaps in them. Yes, and that is interesting. But not so much for the mind-body problem, where we cannot bound anything, except by omega. The weaker theory known from which my approach can work, is the delta_0-induction based on Q + the axioms for the exponential, known as Delta_0Exp. That is Q: 1) 0 ≠ s(x) 2) x ≠ y -> s(x) ≠ s(y) 3) x ≠ 0 -> Ey(x = s(y)) 4) x+0 = x 5) x+s(y) = s(x+y) 6) x*0=0 7) x*s(y)=(x*y)+x + 8) x^0 = 1 9) x^s(y) = x * (x^y) + the scheme of induction axioms: P(0) & [For all n (P(n) -> P(s(n)))] ->. For all n P(n), with P restricted to the delta_0 (= sigma_0 = pi_0 = recursive, decidable, …) formula. That is the weaker Löbian machine known today. In between Q and Delta_0Exp, you have all the bounded arithmetics. An excellent book on this is (without the many accent for the names): Hajek, P. & Pudlak P., 1993, Metamathematics of First-Order Arithmetic, Springer-Verlag. But no need of this for the mind body problem, which needs at least Delta_0Exp (Löbianity), for the observer. Of course I use the fact that Q can mimic Delta_0Exp. But Q does not believe what Delta_0Exp is saying, and the theology is for Delta_0Exp and all its consistent extensions, like PA, ZF, and you, and me … I need the sigma_1 completeness. It is not for practical computational application, but only for guessing what is fundamental to assume, to understand where the appearances come from. It might have application in the foundations of physics, though, and is the best way to figure out the structure of the afterlife or parallel life, etc. Bruno > > > - pt > > -- > 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 post to this group, send email to [email protected] > <mailto:[email protected]>. > Visit this group at https://groups.google.com/group/everything-list > <https://groups.google.com/group/everything-list>. > For more options, visit https://groups.google.com/d/optout > <https://groups.google.com/d/optout>. -- 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 post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
