FOM: Re: Godel set and integers (Platonism and Logicism in set theory)

[Joseph Vidal-Rosset]

>I think the issue here is simply intertranslatability of concepts.
>The validity of the statement isn't affected by the particular brand
>of set theory one uses.
>
>					--Randall

Dear Randall,

In fact there is maybe here an interesting illustration of 
indeterminacy of reference. I remember that you wrote somewhere that 
the definition of strongly cantorian sets has no precise meaning in 
ZFC (sorry, I do not remember where you wrote that and so I am not 
sure to quote exactly what you said.) What I have in mind is partly 
inspired by this remark.
I do not doubt that you can find inter-translatability of concepts 
between for example ZFC and NFU because I have learned from you that 
you can have models of the former in the latter and models of the 
latter in the former (if I have understood well the basic idea). And 
you can describe Arithmetics inside ZFC as well as inside NF(U). 
Right ? But my point is elsewhere. Strictly speaking I do not think 
that Gödel was right when he wrote:
>The concept of "integer" can be replaced by the concept of "set" 
>(and its axioms).
I see what Gödel means and what it means is right in practice. But he 
is not well inspired to talk about the concept of integer and the 
concept of set. Set thoeries like NF imply that the universe is a set 
and Specker shows that in NF such a set cannot be well ordered and is 
infinite so it cannot be replaced by what we mean usually by an 
integer. Thinking about Forster wrote about two traditions in set 
theory, I think that uses and functions of the concept of set are 
more free than than the concept of natural number.
Thus what I had in mind, when I sent this mail was that: I believe 
there is a working  concept of set  basically different (distinct) 
from the concept of integer and maybe from the concept of number 
itself. E.g. different set theories (for example ZFC and NFU) in 
foundations of mathematics, but only one standard Arithmetic. So if 
there was a contemporary Platonism encouraged by strong realistic 
view about sets (Bernays, Gödel), this philosophy of set theory could 
be demonstrably wrong (it does not mean that Platonism i
be demonstrably wrong (it does not mean that Platonism is 

demonstrably wrong as a philosophy of Math., but only that it cannot 
be based on set theory to find its strongest arguments, and that it 
is strongly relative to one set theory, i.e. ZFC.) If I am not wrong, 
that leads to make a distinction between Realism and Platonism, the 
latter being anti-empiricist, the former being compatible with 
empiricism, as Quine's philosophy shows.
Specker theorem shows also that when Bernays wrote the totality of 
integers is presupposed by Arithmetic is a Platonistic 
presupposition, this assertion is true only in a set theory where 
infinity cannot be a theorem but an axiom; it is wrong in NF where 
infinity is proved, contradicting Frege who believed that induction 
has to be based on Arithmetic: in NF the contrary is true because 
induction is at work to show that Arithmetic belongs to NF.
I don't know if Dedekind was right to think that 'numbers are free 
constructions of mind', we can doubt of that if we sympathize with 
Kronecker's philosophy of natural numbers, but it seems to me that 
set theory shows that sets are that Dedekind believed about numbers.
I believe that fits with what you write about "what is a set" in your 
nice book on NFU. Could you agree ?


Best wishes,

Jo.

PS: Could you change my email address in NF list to put this one I'm using ?