Absolute Infinite - Cantor's View

Cantor's View

Cantor is quoted as saying:

The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type.

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 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 of the sequence of all its preceding elements (including 0). (The sequence Ω has this property first for ω₀+1. )

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 numbers is an inconsistent, absolutely infinite multiplicity.