On Friday, August 30, 2019 at 10:10:22 AM UTC-5, Bruno Marchal wrote: > > > On 30 Aug 2019, at 04:33, Alan Grayson <[email protected] <javascript:>> > 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 > > > Also, some may think that because theoretical physics (field theories) is expressed in a language that includes a mathematically continuous (real number) background R^4 of spacetime and the methods of multidimensional calculus (tensor calculus, etc.), that because *mathematically infinite-divisibility* is present and *infinitary definitions* (like "limit") are present, that these "infinities" of the mathematics are real in the actual world.
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