FOM: What is the standard model for PA?
Torkel Franzen
torkel at sm.luth.se
Thu Mar 26 03:33:29 EST 1998
Vladimir Sazonov says:
>Anyway, the main axioms are actually *given* by the teacher.
I don't think it would be very profitable to dwell on the essential
difference, if any, between your description above and saying that we
accept the axioms (once we see them formulated) on the basis of our
informal understanding. Your main point seems to me to be that
>Changing (or rejection)
>of some of these rules may result in a different notion of natural
>numbers with different understanding and intuition.
This points up a general problem with the sort of revisionism you
are arguing for. Your general ideas and aims are quite intelligible on
the basis of our ordinary understanding of arithmetic. You suggest (in
your paper) that this "ordinary understanding" is, on closer
inspection, contrary to basic intuition and experience, and that it
could be replaced, as stated above. However, you present these views
using a logical apparatus steeped in the tradition that you describe
as contrary to basic intuition and experience, and you yourself
assimilated arithmetical concepts just the way the rest of us did. We
don't have any actual evidence that it is even possible to assimilate
"feasible arithmetic" in childhood the way we now assimilate standard
arithmetic. This is not to claim that it is impossible, but the whole
thing is just up in the air.
This is why I don't see any real basis for an argument with you over
the intelligibility or acceptability of arithmetic as traditionally
conceived. Your ideas and inclinations about how arithmetic should be
understood, and the logical work inspired by them, aren't at all
difficult to understand or appreciate from the point of view of that
traditional conception. The view that the traditional conception can
or should be dispensed with, on the other hand, is a kind of
revisionary enthusiasm difficult to substantiate.
>What makes the powerset 2^N of natural numbers (i.e. the set of
>infinite binary strings) to be indeterminate *in contrast to*
>the powerset 2^1000={0,1}^1000 of {1,2,...1000} which should be
>determinate (according to the traditional view and *contrary* to
>my intuition)?
Answers to this question tend to boil down, one way or another, to
the statement that 2^N is not generated by a rule.
---
Torkel Franzen
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