> > But is there some mathematically meaningful definition of "almost > infinite"? If we say that m is a "nearly infinite" number, where > m < omega, but with m having some property that in general makes it > larger than "almost all" finite n?
According to von Neumann's account of ordinal numbers, "x < omega" means just the same as "x is an element of omega", and omega is just the set of natural numbers.
But there are other accounts of infinite number.
-- When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him. Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting