Philosophically different. Give the axioms of the rational numbers we can construct, using ZF set theory, a set of equivalence classes of ordered pairs of integers which satisies those axioms. We can also construct a set of equivalence classes of Cauchy sequences of rational numbers which satisfy the axioms for real numbers, including algebraic and transcendental numbers.
--- Frank C. Wimberly 140 Calle Ojo Feliz, Santa Fe, NM 87505 505 670-9918 Santa Fe, NM On Sat, Jun 20, 2020, 10:45 AM Jon Zingale <[email protected]> wrote: > The isomorphism *isn't*, in some sense, enough. For instance, the rationals > can be philosophically different than the integers. Sure we can identify > them via diagonal argument, but when we want a field we don't reach for the > integers. I claim that something similar is happening here and that the > point of the article is missed when we jump to the isomorphism. Gisin would > have just talked about the rationals if he meant the rationals, instead, he > invokes Chaitin and computability on purpose. The truncation simplification > obfuscates the deeper point. He is making an ontological claim about the > universe and one that theoreticians of quantum theory may appreciate but > applied mathematicians will not. The subjectivity of an observer is forced > on us by classical logic. Here he constructs a physics over a completely > different topos and what follows is not needing to make the observer > interpretation. This point is significant enough to think about as being > *more* than just truncation, it establishes what can be meant by randomness > and the possibility that determinacy may be an illusion, even in > macroscopic > physics. > > > > -- > Sent from: http://friam.471366.n2.nabble.com/ > > - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . > FRIAM Applied Complexity Group listserv > Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam > un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com > archives: http://friam.471366.n2.nabble.com/ > FRIAM-COMIC http://friam-comic.blogspot.com/ >
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