[FOM] From theorems of infinity to axioms of infinity

[Alasdair Urquhart]

It seems fairly clear that it was the paradoxes that prompted the move
to postulate the axiom of infinity outright.  You can see this from
the example of Russell.  In the "Principles of Mathematics" he makes
the remarkable assertion:

 	That there are infinite classes is so evident that it will scarcely be
 	denied.  Since, however, it is capable of formal proof, it may be
 	as well to prove it [PoM p, 357].

He proceeds to give no less than three proofs of the axiom of infinity, 
including versions of the notorious `proofs' of Bolzano and Dedekind.

Even as late as 1904, Russell continued to cling to the view that the
axiom of infinity was a logical truth, when he engaged in a debate on the 
matter with Cassius J. Keyser.
In his reply to Keyser, Russell repeats one of his proofs of the axiom of 
infinity from the pages of the "Principles of Mathematics", concluding 
triumphantly: ``Hence, from the abstract principles of logic alone, the 
existence of infinite numbers is rigidly demonstrated.''

In 1905 and 1906, Russell had high hopes for his new substitutional 
theory, and in his reply to Poincaré published in 
September 1906, proves the axiom of infinity in his paper by 
constructing an infinite series of propositions.

In the end, though, the substitutional theory succumbed to the paradoxes,
and Russell was forced to adopt type theory against his will and logical
inclinations.  In Principia Mathematica, the Axiom of Infinity is
not postulated, but is present as an explicit antecedent in a lot
of arithmetical propositions.

Russell, unlike Zermelo, was committed to a logicist view of mathematics,
and so clung desperately to the idea that the existence of infinite
classes was a logical truth.  It was clearly the paradoxes that
forced him to give up this view.