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, Ø." (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 > > _________________________________________________________________ > Share holiday photos without swamping your Inbox. Get MSN Extra Storage > now! http://join.msn.com/?PAGE=features/es