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To show that the infinite measure (= infinity as a whole) is absurd, we must show that an infinite set can contain elements ALEPH or, equivalently, a set infinity can not be an infinite ALL.
Let N be the set of integers, and AB the bijection of N with one of its parts.
1) N is equipotent to any of its parts.
Eg: N is equipotent to 2N (there the same number of elements in N and 2N).
2) If N is equipotent to 2N, it does not contain more elements than 2N.
3) If N does not contain more elements than 2N, and 2N if N is a part, then N is also an infinite subset of N and N does not contain all the elements up to Aleph (no ALL can count down to the last).
4) N is not an infinite whole (it can not be distinguished from one of its parts, at least on a cardinal).
5) If N is not an infinite whole, it does not exist in overall N larger than N, and N is equipotent to P (N).
N contains only elements that it is possible to count until last countable, that is to say that it contains all the elements up to (n + 1), no more.
A set is infinite (or unfinished):
1) If and only if it contains the empty set.
2) If it is a finite set can be increased by one element and one.
3) If the sum of its parts it is equipotent.
In an infinite set, we can always count as many items as you like, but never infinite.
Infinity as a template is an absurdity, since we can not conceive of object that is infinitely filled - as it has been imagined for the Universe.
Such an object (infinitely full) excludes the existence of the empty set (which implies that we can always add an element and only a finite set).
If N contains the empty set, then N = N + 1. Gold + 1 = Aleph Aleph (it is impossible to add one item at infinity).
N can both contain the empty set and an infinite integer.
Consider the infinite as matrix is the result of a projection of the finite to the infinite - what I call "the obsession at all."
I think our concept of infinity is symptomatic of the megalomania that drives our technological civilization.
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