FOM: Tennant and Sazonov on arithmetic
John Mayberry
J.P.Mayberry at bristol.ac.uk
Sun Oct 11 12:10:34 EDT 1998
Neil Tennant has made an important point in his reply to
Kanovei on counting. The fact that there is possibly an ambiguity in
speaking of "the set of Cardinals now in Rome" or "the set of US
citizens as of 10:00 AM (Eastern Standard Time), October 11, 1998" does
not mean that we cannot assign exact cardinalities to finite sets. The
inexactness here is in the specification of the sets. If we succeed in
specifying - exactly - a finite set, then surely it has a definite
cardinality. But he is on shakier ground when he goes on to claim
that he could, in effect, generate the natural numbers "in thought" by
a sequence of definitions
#x(not x=x) = 0
#x(x = 0) = 1
#x(x=0 or x=1) = 2
. . .
Of course it's not clear what things like #x(x = 0) are. But
leaving that problem aside, what do the dots of ellipsis mean? This,
surely, is the difficulty to which Vladimir Sazonov is calling our
attention. The problem is to give it a clear mathematical formulation.
But don't our naïve intuitions about the natural numbers and finite
iteration give rise to the same kind of barrier to thinking about
*that* problem that our naïve intuitions about simultaneity and time
give rise to when we try to understand Einstein? I think that's all
that Sazonov meant when he drew the parallel.
What DO we mean by the "unique up to isomorphism standard
model" of natural number arithmetic? What do we mean when we say that
Pi-1 propositions in natural number arithmetic have determinate truth
values? Of course we can give Dedekind's answers to these questions if
we are prepared to make the necessary set-theoretical assumptions (the
existence of transfinite sets with power sets). But what if we are not
willing to make them? It is pretty clear that naïve appeals to the
intuitive notion of finite iteration will get us nowhere (Dedekind is
quite eloquent when making this point in his letter to Kefferstein).
But most of us also think it unlikely that satisfactory answers are to
be found by investigating our *actual* capacities for counting and
calculating. (On this matter Tennant's arguments are very much to the
point.) So where do we go from here?
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John Mayberry
J.P.Mayberry at bristol.ac.uk
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