FOM: {n: n notin f(n)}

Dean Buckner Dean.Buckner at
Sun Sep 1 14:09:09 EDT 2002

There is little I can add to my previous arguments except to clarify some
points, and to put them in context.  I don't see any substantial objection
so far to my claim that we can put natural numbers and reals in 1-1
correspondence, without dropping Axiom of Separation.

1.  I am of course committed to a claim of the sort Richard Heck mentions

    (ER)(S)[(x)(Sx --> Px) --> (Ey)(Py & (z)(Sz <--> yRz))].

but, since I have specified that S be a singular concept, and P a predicate,
it does not follow we can substitute "x is not in f(x)" for S, and so it
does not follow, as Richard argues, that I have to reject Axiom of

2.  A logically singular concept (which can of course be numerically plural)
is one that specifies *which* bunch of things satisfy it.  For example "x is
identical with Socrates", or "x is a natural number between 1 and 10".  To
understand that x = Socrates, you have to grasp the identity of the
set of things corresponding to the concept.  A predicative or general
concept by contrast does not specify which bunch of things satisfy it, but
merely that the satisfying object has some characteristic or other.  Thus "x
is a natural
number" is predicative and so is, crucially "x is not identical with
Socrates".  x
could be *any* object whatsoever (so long as not Socrates).

_Thus, while "x is in f(x)" is a singular concept, "x is not in f(x)" is
and cannot be substituted into Richard's formula_.

The set theoretic interpretation makes it seem more complicated than it is.
All we're saying, is that we can map 1 to {1,2,3}, for example, 2 to {3,4,5}
and so on, always mapping a known identifiable object to a known
identifiable bunch of things.  But "all x that are not in f(x)" does not
such a known identifiable bunch of things.

On Harvey's points:

3.  That the set theoretical interpretation of mathematics is coherent, and
natural.  "The lingering life maintained by the old Aristotelian and
scholastic logic .. is an extraordinary fact in the history of philosophy; I
believe it can be accounted for only by supposing that the syllogism is
substantially the correct analysis of the process which passes through the
mind in reasoning".  This was written by a professor from Princeton in 1870,
on the dawn of an apocalypse for that system.  The fact that ZFC is the
currently accepted standard, i.e. is established, could equally well have
been written in 1870, of the syllogistic logic.  I.e. being established or
apparently natural or whatever is not really an argument for anything.
Similarly for historical status.  The syllogistic did after all survive
about 2,300 years.

4. " The progress of mathematics and science has necessarily lead to the
consideration of concepts that do not coincide with any that are
commonly used in ordinary informal discourse. For instance,
physicists and chemists do not convey their discoveries in terms of
"fire" and "hot" and "wet" and "dry". "

(a) Aristotelian physics survived as long as it did, because of the time to
develop a framework of enquiry, and a climate that allowed people (in a
reasonable way) to question the existing assumptions.  This was the real
intellectual achievement of the Western tradition.   (b) However, are any of
the concepts embedded in ZFC different  from any of those used informally,
ot in the logical systems that preceded it?

5.  "The kind of attitudes and objections being raised in the FOM discussion
...are merely incoherent, useless, vague, and arbitrary - at least in their
present form."

It's nice of Harvey to add the last rider.  I would say: a discussion group
like FOM is like a workshop, not a shop window.  So don't expect to see any
finished goods!

That said, what is incoherent about 1-2 above?  It consists of an idea that
we have managed to shoehorn into a set-theoretic (actually second order)
formulation, plus the ideas of "singular" and "general".  The latter are
well-established philosophical concepts, though maybe not so familiar in
mathematics.  If they are not clear, then someone speak up, please.

Dean Buckner

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