Thursday, March 02, 2006

hey again... so i had some trouble posting the last time, but there was something else i thought was actually the most interesting part of the link in my previous post. basically, there was a claim there that is has been proven that in fact it cannot be proven whether or not there exists an infinity between what they call alpha0 and alpha1. Thus, it is something of a choice, where of course if you choose to take either view as an axiom in your system, you get different results. Well, on the one hand, that's a bit unaesthetically pleasing... but on the other hand, it's good news regarding your intuition. even if we weren't aware of everything there was to consider about infinities, and even after i claimed that there was some good formalization of this matter, it appears there IS room for a better definition... at least better in my opinion. perhaps there's a way to think of cardinality (where equivalence is not perhaps strictly defined by bijections) where this ambiguity in the number of infinities does NOT occur. in this case, we would of course be starting with our own kind of "axioms" or decisions about what fundamentally makes the most sense to define the size of infinite sets, and we would then be able to prove, under that system, that there's no such choice of the number of infinities... a more precise number would arise, whether itself is an infinity or not, we'll have to consider.

all for now... i'm tempted to start reading some real mathematics again... to sharpen... if i come across something interesting i'll let you know :)

many many,

jls

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