> Does anyone know, are there versions of philosophy-of-mathematics that > would > allow no distinctions in infinities beyond countable and uncountable? I > know > intuitionism is more restrictive about infinities than traditional > mathematics, but it's way *too* restrictive for my tastes, I wouldn't want > to throw out the law of the excluded middle.
I suggest that given any set X that is free from self contradiction, we can always form the power set P(X) of X (ie the set of all subsets of X). Cantor proved there is no onto mapping from X to P(X). Therefore P(X) must have a higher cardinality than X. This shows that there must be an infinite number of different infinities. Note however that Cantor used a "proof by contradiction" - so as you suggest, you would have to throw out the law of the excluded middle in order to allow no distinctions in infinities beyond countable and uncountable.
I guess as a mathematical Platonist my main objection is to mathematical objects that cannot be defined in any finite way. For example, pi may have an infinite number of digits, but we can define the notion of a Turing machine and then come up with a finite description of a program that will eventually output every single one of those digits. And it is possible to go further than just the computable reals--Turing defined the notion of an "oracle machine" which, in addition to the operations of a Turing machine, can also decide whether any given Turing machine halts in a finite time. It may not be possible to actually construct an oracle machine in our universe, but the notion seems perfectly well-defined, and as a Platonist I would think a question like "what is the nth digit of the oracle machine program #m" must have a single true answer. And then it is possible to define the notion of a 2nd-level-oracle machine that can decide if any 1st-level-oracle machine program halts in a finite time, an omega-level-oracle machine that can decide if any finite-level-oracle machine halts in a finite time, etc. For any number that can be specified in terms of a program for a hypothetical machine which itself has a finite well-defined description, I am willing to believe that number "exists" in the Platonic sense.
But just based on the fact that the number of descriptions of any kind in the English language must be countable, there cannot be more than a countable infinity of numbers with finite descriptions of any kind. Therefore the reals would have to include all kinds of numbers that have no finite description at all. I am not sure I believe such things exist, and for a similar reason I am not sure I believe that every member of the hypothetical "power set of the integers" exists either. Am I necessarily denying the use of proof by contradiction by doing this? Can you explain how proof by contradiction would force one to accept the existence of "indescribable" numbers/sets? I think the universe of mathematical objects with a "finite description" in the general sense I describe above is a lot larger than the universe of mathematical objects which an intuitionist would accept, although I'm not sure about that.
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