[FOM] Cantor and the Axiom of Infiinty

[Dean Buckner]

Cantor writes (in the Grundlagen)

. however contradictory it might be to speak of a greatest number of class
(I), there is nevertheless nothing offensive in thinking of a new number
which we shall call w, and which [my emphasis] will be the expression for
the idea that the entire assemblage (I) is given in its natural, orderly
succession. (Just as v is an expression for the idea that a certain finite
number of unities is united to form a whole.)

It is accordingly permissible to think of the newly created number w as the
limit to which numbers v approach, if by it nothing else be understood than
that w is the first integer which succeeds all numbers v, that is, it is to
be regarded as greater than every one of the numbers v.

This looks like a philosophical justification of the axiom of infinity
(rather than a mere statement of it).  A question: did Cantor attempt such a
justification in many other places, or just some other places in this work
or elsewhere?

Also, is this the first formulation of the axiom?  Grateful for any help.



Dean Buckner,    : If one person can see it as a paradise
London,              : of mathematicians, why should not
ENGLAND        : another see it as a joke?

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