Limits from Beyond

16 06 2006

I just began to write this paper on limits which covers them from a non common direction with the aim to have them be more thoroughly and intuitively understood. Ive still got a bit to go since I wish it to be self contained. A snippet:

A set may also contain itself as a member. So we can have D = {a,b,c,D}. Now, consider a set S={Any set which does not have itself as a member}. So we know that D is not a member of the set S since it has itself as a member. But is S itself a member of this set? Is it a member of itself? Can we have {…,S}?

If S is a member of itself then obviously something is wrong since the requirement of membership is a set must not have itself as a member to be included. But then if we say it is not a member of itself then there is still a problem. Why? Because then it would meet our requirement of a set that does not include itself as a member. Therefore it would in fact be a member of itself! This is obviously a paradox

All these though are solutions to problems offered to make the system which we are working with more consistent and free from problems. Mathematics in this sense is like a patchwork quilt. We sew, sew and if we sew in such a way as to tear our quilt we must patch it up lest the poor person who uses our quilt freeze to death. There are many implications to this paradox and the assumptions required to resolve it (assumption of axiom of choice or well ordering for hierarchy) or ability to prove no strong formal mathematical theory may ever be complete but these are outside the scope of this paper.




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