# Absolute Infinite

The Absolute Infinite (symbol: Ω) is an extension of the idea of infinity proposed by mathematician Georg Cantor.

It can be thought of as a number that is bigger than any other conceivable or inconceivable quantity, either finite or transfinite.

Cantor linked the Absolute Infinite with God,: 175 : 556  and believed that it had various mathematical properties, including the reflection principle: every property of the Absolute Infinite is also held by some smaller object.[clarification needed]

## Cantor's view

Cantor said:

The actual infinite was distinguished by three relations: first, as it is realized in the supreme perfection, in the completely independent, extra worldly 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 order type. 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 [he appears to mean what we now call a set] 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".
...
Now I envisage the system of all [ordinal] numbers and denote it Ω.
...
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.