I think that the argument in the linked article amounts to this:
put 1,2, 3 etc in a left hand column, then 0.5, 1.00, 1.5 etc in a right hand column;
there is now a one-to-one correspondence between the items in two infinite lists;
however, it is possible to add intermediate numbers (0.75, 1.25 etc) in the right hand list but not in the left hand list;
in fact, it is possible to add an infinity of numbers between any two numbers in the right hand list;
therefore, one infinity is greater than the other - and that is just within the uni-dimensional number line.
I welcome comments from mathematicians. I am a philosopher, not a mathematician. Mathematics is important to physics which is important to hard sf where we get, e.g., Poul Anderson's rationales for FTL. For maths and physics, see here.
3 comments:
Kaor, Paul!
Again, I'm reminded of how the Sea People used mathematics in ENSIGN FLANDRY.
Sean
Kaor, Paul!
That’s not quite what’s going on, although you may understand better than your wrote. (I remember a lecture in college covering Cantor’s Proof.) You can establish one-to-one correspondence between the natural numbers (1, 2, 3 ...) and the rational numbers, including 0.5, 0.75, 0.875, 42.6, etc., but between any two rational numbers, there are an unlimited number of irrational numbers, as well as an unlimited number of other rational numbers, and therefore you cannot establish one-to-one correspondence between the natural numbers and the real numbers (rational and irrational), even though there is an infinity of natural numbers (just keep counting). It is contrary to how most people ordinarily think about numbers.
Best Regards,
Nicholas
Nicholas,
Thanks. I think I understand this.
I also love the proof that there is no highest prime.
Paul.
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