FOM: Re: Godel set and integers (Platonism and Logicism in set theory)
Joseph Vidal-Rosset
jvrosset at club-internet.fr
Mon Aug 7 16:11:46 EDT 2000
>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 ?
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