Bruno Marchal wrote:

At 07:58 18/11/03 -0800, Norman Samish wrote:
Thanks for the opinions. You have convinced me that at least the empty set MUST exist, and "The whole of mathematics can, in principle, be derived from the properties of the empty set, ." (From <> .)

I don't see why the empty set MUST exist. It seems there is a confusion here between "no things", and "nothing", or if you prefer between



Besides, I don't see how the whole of math can be generated from
the empty set. You need the empty set + a mathematician (or a least
a formal machinery, or a theory).
BTW, in "infinity and the mind" Rudy Rucker gives the best (imo) popular
account of the "schema of reflexion", a powerful axiom (or theorem
according to the chosen formal set theory) for generating almost
everything from almost nothing ... (it was an important axiom in my older
"machine psychology", but I succeed to bypass it since I use the Solovay
logic G and G*...


Yeah, Rudy Rucker's book is a great introduction to set theory and mathematician's notions of infinity. After reading that book I finally understood the concept of "aleph-one", "aleph-two", and so forth. Basically an ordinal is defined as any collection of smaller ordinals, with the empty set being the minimum ordinal. So 0={}, 1={0}={{}}, 2={0,1}={{},{{}}}, 3={0,1,2}={{},{{}},{{},{{}}}}, and so forth. Since you're allowed to have sets with an infinite number of elements, you can also have infinite-sized ordinals--the smallest possible infinite ordinal is omega, which is just the set of all finite ordinals, or {0,1,2,3,4,...}. Then the next ordinal after that is omega+1, or {0,1,2,3,4,...,omega}. Both these ordinals are countable, and you can construct much higher countable ordinals like omega^2, omega^omega, omega^omega^omega^omega..., etc. Then the first ordinal with cardinality aleph-one is simply defined as "the set of all countable ordinals", which set theory says should be an allowable set, and which by the definition of ordinals must itself be an ordinal. Likewise, the set of all ordinals with cardinality less than or equal to aleph-one should also be an allowable set, so that represents the first ordinal with cardinality aleph-two, and so forth.

Personally, I'm a little suspicious of whether this is really meaningful in a "Platonic" sense, since you get a self-contradiction if you try to talk about "the set of all ordinals" (that set would itself have to be an ordinal larger than any of its members), which shows you can't just assume any collection of ordinals can be a set. So, the mere fact that no obvious contradiction has been found in assuming you can make sets like "the set of all countable ordinals" or "the set of all ordinals with cardinality less than or equal to aleph-one" doesn't completely reassure me that such objects actually "exist" in Platonia, or that questions like "is the cardinality of the continuum equal to aleph-one" have any "true" answer.

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.

Jesse Mazer

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