[FOM] Regarding Alternative Foundations

[Harvey Friedman]

In an offline correspondence I sent the following message which I
think is of general interest for FOM.

There are precise senses in which all foundations for mathematics are
equivalent - or at least comparable. So it doesn't make any difference
- for many purposes - which one you pick. It has been seen that for
most purposes - maybe not for all - the most convenient is classical
set theory through ZFC.

Let me explain a bit. Whereas there can of course be issues as to what
is meant by various subtle mathematical notions, or hugely general
notions, such as arbitrary set, the prevailing view is that there is a
common denominator where we do in fact speak the same essential
language. Namely what is commonly referred to as arithmetic sentences.
These are mathematical statements that are about the ring of integers.

And even here one can go a bit further. There are the so called purely
universal arithmetic sentences, often called the Pi01 sentences, which
asserts that for all integers n_1,...,n_k, some statement holds
involving only bounded quantification connectives, and the ring
operations. There is a huge amount of robustness here for Pi01
sentences, whereby this definition can be greatly liberalized without
admitting anything new.

In particular, for any two foundational schemes that have ever been
seriously proposed, there is a clear way of talking about the provable
Pi01 sentences, and either every such for one of the foundation is
such for the other foundation, or vice versa. This is generally true
also for all arithmetic sentences, but NOT if you admit constructive
foundations. With constructive foundations there is a bifurcation.

We can argue that the essence of mathematics is that which is easily
and directly cast as an arithmetic sentence or even as a Pi01
sentence. In which case alternative foundational schemes don't look as
important or relevant.

Harvey Friedman