[FOM] Regarding Alternative Foundations (Harvey Friedman)

[Paul Blain Levy]

Dear Harvey,

I'm puzzled by your claim that every "foundational scheme that has ever 
been seriously proposed" has the same Pi01 strength as ZFC.  Surely this 
isn't correct.  For example, my system TOPS

https://arxiv.org/abs/1905.02718

is Pi01 weaker than ZFC.  On the other hand, Kelley-Morse class theory 
is Pi01 stronger than ZFC.  And there are many other examples of 
theories with different Pi01 strength.  [Under suitable consistency 
assumptions.]

You have every right to criticize these theories, of course, just as I 
have the right to criticize ZFC.  But all of them were "seriously 
proposed" as "foundational schemes".

Paul

> ------------------------------
>
> Message: 2
> Date: Mon, 9 Dec 2019 05:53:30 -0500
> From: Harvey Friedman <hmflogic at gmail.com>
> To: Foundations of Mathematics <fom at cs.nyu.edu>
> Subject: [FOM] Regarding Alternative Foundations
> Message-ID:
> 	<CACWi-GXg4dAqZ_f1tLML-WTzpXMaYrGQeEwxfTj7FKY=N8GXHQ at mail.gmail.com>
> Content-Type: text/plain; charset="UTF-8"
>
> 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
>