> On 27 Jul 2019, at 23:59, John Clark <[email protected]> wrote: > > > On Sat, Jul 27, 2019 at 2:42 PM Lawrence Crowell > <[email protected] <mailto:[email protected]>> > wrote: > > > I think that mathematics in general is not as rich if you work exclusively > > in finitude. Fraenkel-Zermelo set theory even has an axiom of infinity. The > > main point is with axiomatic completeness, and mathematics with infinity is > > more complete. > > I wonder If "more complete" just means more opportunity to write stories in > the language of mathematics that have no plot holes but are nevertheless > fictional; just as a fantasy novel by JK Rowling is still fictional even if > she maintains perfect internal consistency within her story that is written > in the language of English. For example take Euclid's proof that there is no > largest Prime Number, it's a beautiful mathematical story and it has no plot > holes, but is the story true? > > Unless it turns out we were very very wrong about General Relativity and > Quantum Mechanics I don't believe the universe has the computational > resources to calculate the 10^(10^9)^(10^9) prime number, not even if the > universe is infinite in extent because it is expanding and accelerating. So > if the word has any meaning how can the 10^(10^9)^(10^9) prime number be said > to "exist”?
We should always be clear about three forms of existence, and mechanism provides a way to make that clear: 1) You have what you assume to exist in the ontology. Exemple: 0, 1, 2, … in an ontology which is sufficient and necessary, and cannot be completed. 2) Then you have the things who existence, can be derived directly in the theory, examples are the prime numbers, or the relative universal numbers, the combinators, the Turing machine, etc. 3) Then you have the phenomenologies: the things that the universal numbers will themselves postulate, either as tools or as possible ontological things added to their possible basic ontological commitment (the universal machine will debate this). The axiom of infinity is a good example of this. It simplifies the life a lot, but with mechanism, its existence is phenomenological and not part of the ontology, nor are the real numbers, or even the negative numbers. The choice of the ontology is not important, and the difference between 1) and 2) is not conceptually important. You can take the combinators as ontology (K, S, KK, KS, …) and then prove the existence of the natural numbers from there, or you can take the numbers as ontology, and prove the existence of the combinators from the numbers. That will not change anything in the “machine’s” theology, nor the machine’s physics. But it is important to distinguish 1) and 2) to fix the discourse. Mechanism needs just one universal machinery, and we get all the others from it. Now, if you assume *any* universal machinery, (and classical logic, to remain simple), it is a theorem that 10^(10^9)^(10^9) prime number exists. That existence has nothing to do with the idea that a universe exists or not, and that we can represents such number in some way or not. In this case, you don’t even need the excluded third principle. A universal machinery can always be specified by first order (non logical) axioms, which makes all the notion of existence definable in small amont of second order logic, or set theory. We have many notion of existence in arithmetic. ExP(x) ontological existence (here existence is defined in the usual way of first order logic). []ExP(x) ontological existence accessible by the machine specifying the “[]” []Ex[]P(x) ontological constructive existence (the machine can prove the existence, and find how to build the existing object) Then the same with [1], [2], [3], … [7], and even other related to the quantisation of the observable, which involved the variants of "[]<>” in front of the propositions and quantifier. Bruno > > John K Clark > > -- > 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/CAJPayv1DyckEeJmJqcnsoC-MBDgMSAXNWL5gM_Pt4xn8%2BD96hw%40mail.gmail.com > > <https://groups.google.com/d/msgid/everything-list/CAJPayv1DyckEeJmJqcnsoC-MBDgMSAXNWL5gM_Pt4xn8%2BD96hw%40mail.gmail.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/32913DEE-8DF5-4760-900B-EC32380DED7E%40ulb.ac.be.
