I have just discovered (during homeschooling) that my children are being taught at school that infinitely many numbers exist. Where do I have to send my petitions against this devastating creationist pre-Hilbertian falsehood?
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What you've got to realize is that infinities are the end result (or limit) of a process. If the universe is a Zeno machine (if it turns out that in the limit computation is in fact unbounded), then infinities could in fact be implemented at the limit. https://en.wikipedia.org/wiki/Zeno_machine …
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So , I think mathematicians are simply exploring the space of all possible worlds. Given that there's at least one possible world (where the universe is a Zeno machine) in which infinities could actually be implemented physically, that's enough to clinch their existence.
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Where does one draw the line? A number is an abstraction and can be implemented in many ways and to many degrees. A shape, a concept representing a count, a fraction of a diameter over circumference. Is the abstraction of an infinite set less “implemented” than above? Why?
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What if we accept a Deleuzian ontology, in which the virtual and actual are both real? Perhaps infinite numbers could exist in virtual space
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Since an infinity of numbers is conceivable, and seems to logically follow from our common-sense understanding, and an infinity of numbers could conceivably be brought forth in infinite time, it could be said that this infinity of numbers exists in virtual space
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