Finitism and primitive recursive arithmetic

[JOSEPH SHIPMAN]

In the context of ZFC, which is a fine context for foundational discussions, it seems to me that “finitist result” ought to mean “provable without using the axiom of infinity”, which gives us, for arithmetic, the same theorems that PA does.

Did Hilbert or his collaborators ever explicitly either accept or reject the claim that Ackerman’s function was finitistic?

— JS

Sent from my iPhone

> On Dec 11, 2021, at 9:43 PM, WILLIAM TAIT <williamtait at mac.com> wrote:
> 
> Although my main Concern in the paper “Finitism” was with the conceptual question of what finitism should mean, I ventured some remarks about the historical question of what Hilbert and Bernays meant by the term.  In his dissertation, a version of which he later published, Richard showed clearly that I had overstated my case (in the case of Hilbert).
> 
>  Bill Tait
> 
> Sent from my iPad
> 
>>> On Dec 9, 2021, at 7:43 PM, Richard Zach <rzach at ucalgary.ca> wrote:
>>> 
>> 
>> Thanks for the plug, Alasdair!
>> 
>> Bill Tait has written a followup to his "Finitism" paper which, with discussion, is reprinted in his collected essays (The Provenance of Pure Reason). The original (without the added discussion) is on his website: 
>> 
>> http://home.uchicago.edu/~wwtx/finitism.pdf
>> 
>> He also has a more recent paper:
>> 
>> Tait, William. "What Hilbert and Bernays Meant by “Finitism”". Philosophy of Logic and Mathematics: Proceedings of the 41st International Ludwig Wittgenstein Symposium, edited by Gabriele M. Mras, Paul Weingartner and Bernhard Ritter, Berlin, Boston: De Gruyter, 2019, pp. 249-262. https://doi.org/10.1515/9783110657883-015
>> 
>> It's unfortunately not on his website but you might be able to look at it on Google Books:
>> 
>> https://books.google.ca/books?id=60DEDwAAQBAJ&lpg=PR1&pg=PA249#v=onepage&q&f=false
>> 
>> If I'm not mistaken, he discusses something very much like the idea you propose (the Sieg/Ravaglia argument).
>> 
>> FWIW I agree with Bill that the conceptually best analysis of what Hilbert *should have* meant by "finitism" is PRA. My "scepticism" about Tait's proposal only concerned the historical question of whether Hilbert et al. ever used methods that went beyond PRA in his proof theory. I think Bill agrees with me on the historical question.
>> 
>> -RZ
>> 
>> On 2021-12-08 10:21, Alasdair Urquhart wrote:
>>> Richard Zach has given a detailed logical and historical 
>>> analysis of this question in his 2001 Berkeley doctoral thesis. 
>>> It's available at Richard Zach's website. 
>>> 
>>> Zach expresses scepticism about Tait's proposal.  If you are interested 
>>> in this question, I strongly recommend his dissertation. 
>>> 
>>> 
>>> On Tue, 7 Dec 2021, martdowd at aol.com wrote: 
>>> 
>>>> FOM: 
>>>> 
>>>> I was just reading 
>>>>  Takeuti's Well-Ordering Proof: Finitistically Fine? 
>>>>   by Eamon Darnell and Aaron Thomas-Bolduc, 
>>>>  philsci-archive.pitt.edu/15160/1/Takeuti.pdf 
>>>> They mention Tait's thesis that PRA exhausts finitistic methods. 
>>>> 
>>>> It occurred to be that the terms of PRA can be augmented by adding a function, 
>>>> which is the universal function for primitive recursive functions.  If 
>>>> a nice extension of PRA existed allowed proving facts about this, 
>>>> wouln't this still be finitistic? 
>>>> 
>>>> One possibility might be to add some version of Hoare's axioms to Meyer- 
>>>> Ritchie loop programs. 
>>>> 
>>>> Martin Dowd
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20211211/59c310c2/attachment-0001.html>