So I’m writing an essay about Cantor’s work.
Apart from solving Zeno’s paradoxes, what other major implications came from his idea of multiple infinities?
Cantor’s work on infinite cardinalities and ordinals didn’t solve Zeno’s paradoxes, or even really have much relation to them. Historically, I suppose it was the previous work on limits and infinite summation which was considered to have solved Zeno’s paradoxes, although even that is rather an odd way of looking at it. Fundamentally, the solution to Zeno’s two most famous paradoxes is like this:
The dichotomy paradox: There is no difficulty in having infinitely many points within a finite span. Why should there be any difficulty in this? Consider the very example provided: the distance from 0 to 1 is the finite quantity 1, yet there are the infinitely many numbers 1, 1/2, 1/4, 1/8, 1/16, etc. There is no contradiction in any of this. Just look at it.
Achilles and the tortoise: The same. There is no difficulty in having infinitely many points within a finite span. There is no contradiction in it. Why would there be?
Neither of these fundamentally needs any recourse to discussion of limits or summation, though one could bring it in, as is often done. And neither of these is much illuminated by Cantor’s work on cardinality and ordinals; not all talk about the infinite has to do with it.
Zeno’s paradoces were solved long before Cantor, with the invention of calculus at the very latest. I’m not aware of any consequences outside of math (but of course, the boundary between math and philosophy is not always clear-cut).
This may or may not be along the lines of what you’re looking for, but I am reminded of a discussion I took part in, over whether it’s true that “Anything that can happen, will happen, given enough time.” The part involving Cantor’s multiple infinities really gets going on page 2 of that thread. Basically, one poster originally claimed that
[QUOTE=Keeve]
Your dart board has an infinite number of points on it. Each one is infinitely small. If you would repeatedly throw your dart for an infinite amount of time, you WILL hit every point.
[/QUOTE]
not realizing that there are multiple infinities, and so a countably infinite sequence of points/events cannot contain all possible outcomes in an uncountably infinitie sample space.
Have you seen the bit on Wikipedia about Philosophy, religion and Cantor’s mathematics?
Godel’s Incompleteness Theorem, which had implications for the rest of philosophy (though perhaps not as many as are sometimes supposed—but in any case he showed that “true” is not synonymous with “provable”), was proved using a “diagonalization” argument similar to the one(s) Cantor invented to prove that there are multiple infinities.
Interesting thread, thanks. Not exactly what I was looking for but an interesting read nonetheless.
Yeah this is more what I was looking for; only this section is very brief and I haven’t a very good grounding in intuitionism or constructivism. Actually I was hoping someone could point me toward some sort of more detailed explanation of what implications Cantor’s work had for non-set theory philosophical topics.
Thanks for your help so far.
My answer: None.
Yeah, that’s kinda what I’m thinking. Cantor’s work, like anything else mathematical, is saying something very specific, and areas of philosophy outside of logic don’t generally meet the criteria to apply it.