Bruno Marchal wrote:
>
>At 07:58 18/11/03 -0800, Norman Samish wrote:
>>Gentlemen,
>>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, Ø." (From
>><http://www.hedweb.com/nihilism/nihilf01.htm>http://www.hedweb.com/nihilism/nihilf01.htm
>>.)
>
>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
>
>
>and
>
> {}
>
>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*...
>
>Bruno
>
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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Received on Wed Nov 19 2003 - 19:43:27 PST