[FOM] Re: Eisworth on Axiom of Infinity

[A.P. Hazen]

  Eisworth asks how much of the  power/utility of the standard Axiom of
Infinity in ZF comes from the fact that it postulates, not just an infinite
set, but (in effect) the set of natural numbers.
  My guess is that, in the presence of the Powerset Axiom, not much.
George Boolos* pointed out that Russell,  in "Principia Mathematica,"
showed that a model of the series of naturals under successor (= a
Dedekindian "simple infinite system") could be proven to exist two types up
from any infinite set.
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* Ref: G. Boolos, "The advantages of honest toil over theft," orig. pub. in
A. George, ed., "Mathematics and  Mind" (Ox. U. P. 1994), pp. 27-44; repr.
in Boolos, "Logic, Logic and Logic" (Harvard 1998), pp. 255-274.
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Allen Hazen
Philosophy Department
University of Melbourne