FOM: reply to Vidal-Rosset
holmes@catseye.idbsu.edu
holmes at catseye.idbsu.edu
Wed Aug 9 13:48:01 EDT 2000
Dear Joseph (cc FOM, NF lists):
You are putting me on my mettle for this reply by posting the
correspondence to all these lists :-)
On the whole, I find little to agree with in what you write. In
many places, I feel that I don't understand what you are saying;
this may be a contributing factor.
You quoted G\"odel:
>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.
and suggested yourself:
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.
My comment:
I think I stand by what I said earlier. I don't think anything of
great philosophical interest is going on here -- the issue is simply
that concepts are intertranslatable and it is to some extent optional
which ones you take as basic.
I don't understand "where the concepts of set and the concepts of
integer are precisely distinct"; the concepts "set" and "integer"
are certainly distinct.
I now pass to your second letter.
You wrote:
I remember that you wrote somewhere that
the definition of strongly cantorian sets has no precise meaning in
ZFC.
I reply:
Actually, the definition of strongly cantorian set makes perfect sense
in ZFC (or almost any set theory). A set is strongly cantorian iff
the restriction of the singleton map to that set is realized as a set.
In ZFC this is well-defined, but not useful, since _every_ set is strongly
cantorian. In NF or NFU, some sets are demonstrably not strongly cantorian,
and the question of which sets are strongly cantorian and which are not
is mathematically interesting.
You wrote:
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.
I reply:
Your statement "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." doesn't make sense to me. The fact that some sets cannot be
integers does not obstruct the coding of integers as sets. In fact,
NF supports a perfectly adequate coding of the integers as sets,
and this is helped rather than hindered by the proof of infinity.
Your point "that uses and functions of the concept of set are
more free than than the concept of natural number." is much stronger.
See my concluding remark.
You say:
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.
I reply:
This seems confused. No one (especially not G\"odel) proposes to
_identify_ the concept of set and the concept of integer (or number).
So no one is surprised that the concepts are distinct. What is
proposed is to explain (or "implement") "integer" in terms of "set".
Again, it is important to observe that the notion of "integer" seems
to be better defined than that of "set" (though it is not actually
true that there is only one theory of integers).
You wrote:
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
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.)
I reply:
This is a non sequitur; it does not seem to follow from anything you
say earlier. Moreover, I don't think it is correct. It is a
sociological fact that most Platonists (with a strong realist view
about sets and committed to a definite theory of sets) have been
committed to ZFC or something like it. But there is no philosophical
reason why a Platonist could not be committed to a different set
theory (for example, NF). There is nothing about strong set
theoretical realism that dictates the adoption of any particular set
theory. (This is not to say that I don't think there are
philosophical arguments which might lead a realist to adopt one set
theory rather than another: I think that the Platonist who believed in
NF would have more explaining to do than the one who believed in ZFC).
You wrote:
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 reply:
I have trouble understanding this. (for the information of readers,
Specker proved that the axiom of choice is false in NF; the "axiom of
infinity" for NF thus follows as a theorem). Suppose that one is a
Platonist about NF. One can then define the notion "natural number"
in terms of "set" using Frege's definition of the natural numbers (as
is usual in NF). Thus it is a consequence of one's set theory that
there is a model of arithmetic.
I don't understand what you say about Bernays. The hypothetical
NF Platonist does presuppose the totality of "integers" (as defined
in NF) in a characteristically Platonist way. He is not impeded in
this presupposition by the fact that Infinity is a theorem rather than
an axiom for him.
The totality of integers (as defined in NF) is presupposed if one
supposes that the axioms of NF are true. This has nothing to do with
the fact that Infinity is a theorem rather than an axiom. In fact, it
is certainly possible to axiomatize NF in such a way that Infinity
will be an axiom, and probably even possible to do this in a natural
way. For a Platonist, for whom mathematical theorems are _facts_,
there really should be no philosophical difference between an axiom
and a theorem _as mathematical facts_ (of course an axiom plays a
distinguished role in his derivation and organization of the facts,
but for the Platonist this is quite independent of the facts
themselves).
Your assertion that "Frege ... 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" admits the following
refutation. In fact, Frege proposed to define "number" in terms of
"set", and the definition of "number" that he used is precisely the
one that is used in NF. I fail to see that the development of
arithmetic in NF (which is essentially Fregean) can be a
counterexample to anything Frege thought. Cocchiarella has proposed
that Frege's system could be fixed in a natural way by adopting the
stratification criterion for comprehension (he proposes NFU rather
than NF).
(Readers should be aware that in describing the hypothetical Platonist
committed to NF I am not describing my own views).
You wrote:
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 ?
I reply:
I don't entirely understand what you write here, but I don't think it
has much to do with my account of "what is a set".
I conclude:
There is one useful point which you hint at but never quite make.
This is a further development of your point that the notion of "set"
is underdetermined to a much greater extent than the notion of
"integer". It can be argued that it is a dubious procedure to explain
"integer" (which we understand pretty well) in terms of "set" (which
we understand rather less well). I'm not sure I agree with this, but
it is a good arguing point.
And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the | Boise State U. (disavows all)
slow-witted and the deliberately obtuse might | holmes at math.boisestate.edu
not glimpse the wonders therein. | http://math.boisestate.edu/~holmes
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