FOM: G"odel set and integers (Platonism and Logicism in set theory)

[Joseph Vidal-Rosset]

Dear Randall (cc. NF list, FOM list),

In his famous paper "Is Mathematics Syntax of Language?" Gödel wrote:

>In general the concepts and axioms occurring in the section of 
>mathematics considered need not all occur among (or be derivable 
>from) those sufficient for a consistency proof. There exist certain 
>possibilities of replacing some of them by others.

This sentence has a footnote that I quote:

>E.g. the non-constructive concept of "there is" (referring to 
>integers) can be replaced by "intuitionistic number-theoretical 
>proof, or by "computable function of finite type", or by 
>"constructive accessible ordinal" [...]. Also the general concept of 
>"set" can be replaced by that of "ordinal number" in conjunction 
>with  that of "recursive function of ordinal numbers". The concept 
>of "integer" can be replaced by the concept of "set" (and its 
>axioms). Finally it is not impossible that the non-constructive 
>concepts of mathematics can be replaced by constructive ones, 
>provided "constructivity" is taken in a sufficiently wide sense. (in 
>Kurt Gödel, Unpublished Philosophical Essays, F.A. 
>Rodriguez-Consuegra ed., Birkaüser Verlag, Berlin, 1995, p. 180 & 
>199)

I just wonder if you could accept the sentence underlined from NF(U) 
point of view. Does the distinction between cantorian sets and non 
cantorian sets complicates the situation ? I would be glad if you 
could tell me how your conception of what is a set fits with Gödel's.

What is at issue here seems to be the role of concepts of set and 
integer in different Philosophical positions in Mathematics. (Sets in 
Zermelian tradition and sets in logicist (Russelian, Quinean) 
tradition : it seems to me that if the latter succeeds in offering a 
set theory - NF(U)? - where the concepts of set and the concepts of 
integer are precisely distinct, contemporary Platonism based on set 
theory could appeared as strongly relative (to ZF set theory). But my 
philosophical claim is still unclear at my eyes, and I am maybe on a 
wrong way.

I would be happy also to learn the point of view on this question 
from others mathematicians working in set-theory. That could help me 
and that is why I take the freedom to cc my mail to NF listeners sand 
FOM listeners.

Best regards,

Jo.

------
Joseph Vidal-Rosset
page web: http://www.u-bourgogne.fr/PHILO/joseph.vidal-rosset