FOM: Hersh on the axiom of infinity etc.

Stephen G Simpson simpson at
Thu Sep 17 18:16:23 EDT 1998

Is Reuben Hersh willing to consider views contrary to his own?

Discussing a series of difficulties which, according to Hersh,
undermine "foundationalism", Hersh 12 Sep 1998 18:06:45 writes:

 > One famous difficulty is [the] axiom of infinity.  You can't do
 > modern math without it.

Are you sure?  For a contrary opinion, read my paper

    Stephen G. Simpson, Partial realizations of Hilbert's Program,
    Journal of Symbolic Logic, 53, 1988, pp. 349-363.
which is also available on-line at

 > We could dispense with infinity, and only do the finitistic part of
 > mathematics.  But we don't want to give up analysis, geometry, and
 > so on, so we accept the axiom of infinity.

Not everybody accepts it.  Hilbert rejected it.  That was the point of
Hilbert's program of finitistic reductionism.  Read my paper "Partial
realizations of Hilbert's Program".

 > It is widely thought that Godel killed Hilbert's project.

Not everybody agrees with this conventional wisdom.  Read my paper
"Partial realizations of Hilbert's Program".

 > There are also the axiom of replacement, which I would defy you to
 > claim is intuitively obvious, and the axiom of choice--need I say
 > more?

The axiom of replacement flows from the idea that a set is a
collection of objects which is limited in size, i.e. "not too big".
The axiom of replacement says intuitively that if you start with a set
S and replace each object belonging to S by another object, the
resulting collection is still a set, because it isn't any bigger than
S.  Many people think this is intuitively obvious, and they certainly
have a point.

The axiom of choice says intuitively that if you have an indexed
family of nonempty sets X_i, i in I, where I is an index set, then
there exists an indexed family of objects a_i in X_i, i in I.  This
also has a certain intuitive appeal.  In any case, the axiom of choice
is almost universally accepted; it is part of the current consensus
with regard to mathematical rigor.

-- Steve

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