[FOM] Regarding Alternative Foundations (Harvey Friedman)

Fri Dec 13 19:22:08 EST 2019

Sorry, Harvey, I didn't read your message carefully enough. I agree with 
your point that any two "foundational schemes that have ever been 
seriously proposed" are Pi01 comparable.  Indeed the examples I gave 
corroborate it.

Paul

> ----------------------------------------------------------------------
>
> Message: 1
> Date: Fri, 13 Dec 2019 05:01:14 -0500
> From: Harvey Friedman <hmflogic at gmail.com>
> To: Foundations of Mathematics <fom at cs.nyu.edu>
> Subject: Re: [FOM] Regarding Alternative Foundations (Harvey Friedman)
> Message-ID:
> 	<CACWi-GXgiXZQNM0npA_a4sjKKkVEuumm8PeQDXrmjfgRQ3E7Tg at mail.gmail.com>
> Content-Type: text/plain; charset="UTF-8"
>
> On Fri, Dec 13, 2019 at 4:41 AM Paul Blain Levy <p.b.levy at cs.bham.ac.uk> wrote:
>> 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.
> I never said that. I agree that this "claim" which I never made, is false.
>
> Harvey Friedman
>
>
> ------------------------------
>
> Message: 2
> Date: Fri, 13 Dec 2019 10:07:14 +0000
> From: JOSE MANUEL FERREIROS DOMINGUEZ <josef at us.es>
> To: "fom at cs.nyu.edu" <fom at cs.nyu.edu>
> Subject: Re: [FOM] axiomatizations of PA (Dennis E. Hamilton)
> Message-ID:
> 	<DB8PR01MB57223F83925B132CAE706026B1540 at DB8PR01MB5722.eurprd01.prod.exchangelabs.com>
> 	
> Content-Type: text/plain; charset="iso-8859-1"
>
> Sorry to reply so late to this thread. Let me recommend one piece of mine on this topic, which came out in the book Landmark Writings in Western Mathematics:
> https://www.researchgate.net/publication/289141959_Richard_Dedekind_1888_and_Giuseppe_Peano_1889_booklets_on_the_Foundations_of_Arithmetic
> [https://www.researchgate.net/images/template/default_publication_preview_large.png]<https://www.researchgate.net/publication/289141959_Richard_Dedekind_1888_and_Giuseppe_Peano_1889_booklets_on_the_Foundations_of_Arithmetic>
> Richard Dedekind (1888) and Giuseppe Peano (1889), booklets on the Foundations of Arithmetic | Request PDF<https://www.researchgate.net/publication/289141959_Richard_Dedekind_1888_and_Giuseppe_Peano_1889_booklets_on_the_Foundations_of_Arithmetic>
> Request PDF | Richard Dedekind (1888) and Giuseppe Peano (1889), booklets on the Foundations of Arithmetic | This chapter discusses the contributions of Richard Dedekind and Giuseppe Peano to the foundations of arithmetic. Although Dedekind presented... | Find, read and cite all the research you need on ResearchGate
> www.researchgate.net
> And also a PhD thesis that contains quite a lot of material, written by Fernand Doridot, for those who read French:
> Aux origines de l'arithm?tique formelle : d?finitions du nombre naturel entre Frege et Quine, postures philosophiques et d?terminations cognitives<https://www.theses.fr/2003NANT3033>
> Best, Jose
>
>
> --------------
>
> Message: 1
> Date: Sun, 8 Dec 2019 08:26:00 -0800
> From: "Dennis E. Hamilton" <dennis.hamilton at acm.org>
> To: "'Foundations of Mathematics'" <fom at cs.nyu.edu>
> Subject: Re: [FOM] axiomatizations of PA
> Message-ID: <000001d5ade4$2e23bb20$8a6b3160$@acm.org>
>
> It is gratifying to see that Cohen's book is now available in a Dover reprint
> of the typescript (Kindle edition not recommended by reviewers though).  I
> look forward to arrival of my inexpensive print copy.
>
>
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> https://cs.nyu.edu/mailman/listinfo/fom
>
>
> End of FOM Digest, Vol 204, Issue 7
> ***********************************
> -------------- next part --------------
> An HTML attachment was scrubbed...
> URL: </pipermail/fom/attachments/20191213/433afdaf/attachment.html>
>
> ------------------------------
>
> Message: 3
> Date: Fri, 13 Dec 2019 07:56:58 -0500
> From: Joe Shipman <joeshipman at aol.com>
> To: Foundations of Mathematics <fom at cs.nyu.edu>
> Subject: Re: [FOM] Regarding Alternative Foundations (Harvey Friedman)
> Message-ID: <02165702-4563-4B9E-A8FF-C0A67178C7A8 at aol.com>
> Content-Type: text/plain; charset=utf-8
>
> You didn?t read carefully. He said that all such schemes were comparable for Pi 01 sentences not that they were the same. One can be stronger than another, but it is always the case that an inclusion relationship exists in at least one direction.
>
> ? JS
>
> Sent from my iPhone
>
>> On Dec 13, 2019, at 4:40 AM, Paul Blain Levy <P.B.Levy at cs.bham.ac.uk> wrote:
>>
>> ?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
>>>
>> _______________________________________________
>> FOM mailing list
>> FOM at cs.nyu.edu
>> https://cs.nyu.edu/mailman/listinfo/fom
>
>
> ------------------------------
>
> _______________________________________________
> FOM mailing list
> FOM at cs.nyu.edu
> https://cs.nyu.edu/mailman/listinfo/fom
>
>
> End of FOM Digest, Vol 204, Issue 10
> ************************************