Absolutely infinite
The Absolute Infinite is mathematician Georg Cantor's concept of an "infinity" that transcends the transfinite numbers. Cantor linked the Absolute Infinite with God. He held that the Absolute Infinite had various mathematical properties, including the reflection principle which says that every property of the Absolute Infinite is also held by some smaller object.
Cantor said:
The actual infinite was distinguished by three relations: first, as it is realized in the supreme perfection, in the completely independent, extrawordly existence, in Deo, where I call it absolute infinite or simply absolute; second to the extent that it is represented in the dependent, creatural world; third as it can be conceived in abstracto in thought as a mathematical magnitude, number or ordertype. In the latter two relations, where it obviously reveals itself as limited and capable for further proliferation and hence familiar to the finite, I call it Transfinitum and strongly contrast it with the absolute.
Cantor also mentioned the idea in his letters to Richard Dedekind (text in square brackets not present in original):
A multiplicity is called well-ordered if it fulfills the condition that every sub-multiplicity has a first element; such a multiplicity I call for short a "sequence".
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Now I envisage the system of all [ordinal] numbers and denote it Ω.
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The system Ω in its natural ordering according to magnitude is a "sequence".
Now let us adjoin 0 as an additional element to this sequence, and place it, obviously, in the first position; then we obtain a sequence Ω′:
0, 1, 2, 3, ... ω0, ω0+1, ..., γ, ...
of which one can readily convince oneself that every number γ occurring in it is the type [i.e., order-type] of the sequence of all its preceding elements (including 0). (The sequence Ω has this property first for ω0+1. [ω0+1 should be ω0.])
Now Ω′ (and therefore also Ω) cannot be a consistent multiplicity. For if Ω′ were consistent, then as a well-ordered set, a number δ would correspond to it which would be greater than all numbers of the system Ω; the number δ, however, also belongs to the system Ω, because it comprises all numbers. Thus δ would be greater than δ, which is a contradiction. Therefore:
The system Ω of all [ordinal] numbers is an inconsistent, absolutely infinite multiplicity.
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