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Article 1223 of comp.ai.philosophy:
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>From: silber@orfeo.Eng.Sun.COM (Eric Silber)
Newsgroups: sci.philosophy.tech,comp.ai.philosophy
Subject: Twenty fingers and toes: the *completed* totality of 'primitive integers'
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Date: 6 Nov 91 19:49:20 GMT
References: <1991Oct22.041210.5931@watserv1.waterloo.edu> <JMC.91Nov5212441@SAIL.Stanford.EDU> <1991Nov6.122415.5160@husc3.harvard.edu>
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In article <1991Nov6.122415.5160@husc3.harvard.edu> zeleny@brauer.harvard.edu (Mikhail Zeleny) writes:
...(c. 50 lines deleted )
> Once
>we realize that the second-order PA, the weakest theory capable of
>describing the set of all integers up to isomorphism, in the full-strength
>induction postulate makes an explicit reference to all properties of the
>integers, and ipso facto an implicit one to all subsets thereof, it becomes
                                                                  ^^^^^^^^^^
>evident that our reference to each integer indeed depends on an implicit
 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
>reference to the *completed* totality of all integers (recall that the
 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
>Axiom of Infinity is required for the logicist proof of the postulate that
>each integer has at most one predecessor).  In other words, once your
>analogy is made to work, it starts working against your position.

  ?????
  Primitive/'naive' uses of "integers" exist independent of "logicist proof".
  Such uses are not irrational; they employ restrained, finite pattern matching
  techniques which do establish integers without reference to anything transfinite.
  ?????


