Jesse said...

> 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.

- David



> -----Original Message-----
> From: Jesse Mazer [mailto:[EMAIL PROTECTED]
> Sent: Thursday, 20 November 2003 7:31 AM
> To: [EMAIL PROTECTED]
> Subject: Re: Why is there something rather than nothing?
> 
> 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, ."  (From
>
>><http://www.hedweb.com/nihilism/nihilf01.htm>http://www.hedweb.com/nih
il
> ism/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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