My first thought upon reading this: You need to read Cantor, who showed that infinite sets can still have different cardinalities; it is entirely possible that some infinity N divided by some other infinity M is a finite quantity. You're trying to define infinity as a single thing; it's entirely possible for there to be multiple infinities.

Or, in other words, infinities are far, far messier than you're presenting them here, and your proofs are relatively naive in regard to the work that has already been done on the subject. I would suggest starting with Cantor; his is hardly the last word, but as old as it is, it is one of the best words. Kronecker, a contemporary of Cantor, is a decent read for the infinity-skeptic side of the equation, but didn't have much substantive to offer.