> On 30 Aug 2019, at 04:33, Alan Grayson <[email protected]> wrote: > > If there are infinities in mathematics, but not in physics or in nature, is > that a problem? AG
Is that an interesting problem? I guess so. Some theories in mathematics assume an axiom of infinity, like in set theory, analysis, etc. That has often led to paradoxes, but they have been “solved” by diverse means. So most such theories are considered not being problematic. We can also show that, even restricted on the arithmetical truth (which has no axiom of infinite, as all natural numbers are conceived to be finite), adding an axiom of infinity lead to stronger provability abilities. The set theory ZF proves much more than the arithmetic theories PA, even on just the numbers relations. Yet ZF, and actually all effective theories are limited on a small spectrum of the arithmetical reality. The omega-initial segment of ZF mirrors PA faithfully. In physics, the universe itself could be infinite, without having any infinite things in it, like the model of Arithmetic (all numbers are finite, and the set of all numbers is just a meta-concept, not representable directly in the theory, but still manageable (you can prove in PA that there is an infinite of prime numbers, by proving For x (prime(x) -> It exist y (y bigger than x) and prime(y)). Are there actual infinite object in the universe? I can prove that if mechanism is false, then there as such object. With Mechanism, the mind is infinite, and physics is somehow the Mind seen from itself internally. That might favours an infinite physical universe. Does our substitution level depend on Planck Constant? Open problem. With mechanism, the axiom of the infinite is inconsistent on the ontological level, but is a theorem on the phenomenological level. It shortens the proofs, and provide many tools to to handle mathematically the semantic, the notion of limit, many form of approximation, even to learn just about the natural numbers or the combinators. Mechanism provides a testable account of the mind-body relation, an account which does not assume more than elementary arithmetic, and which doesn’t involve any other ontological commitment. So let us see. The quantum structure, and time, admits a “simple" arithmetical interpretation, but space, dimension, energy remains in the shadow. Science has not yet decided between Plato's and Aristotle’s conception of reality, and not all people are aware of the hypothetical nature of those options, nor that Digital Mechanism, or even just the Church-Turing thesis, leads us back to Pythagorus and Plato. The natural numbers realised the infinities without making them into existing things, or beings. Bruno > > -- > 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/5be11cc0-b02f-48b5-a190-f282fef5a063%40googlegroups.com > > <https://groups.google.com/d/msgid/everything-list/5be11cc0-b02f-48b5-a190-f282fef5a063%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]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/AB5418D5-0ACD-4A32-9974-F270C3738296%40ulb.ac.be.
