FOM: {n: n notin f(n)}
Harvey Friedman
friedman at math.ohio-state.edu
Tue Sep 3 15:48:09 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
>Separation.
You are setting up some restriction that is not standard in the
accepted foundation for mathematics. You need to show that this way
of looking at things provides a coherent alternative foundation for
mathematics. I don't see anything so far that indicates how such
ideas work for even the foundation of very elementary mathematics.
Cantor's theorem about no one-one correspondence is intended as a
theorem in set theory, and therefore appears unimpeachable. You must
be talking about some unusual interpretation of his theorem. It is
not clear what this unusual interpretation is. Even more so, it is
not clear whether this unusual interpretation, if it exists, is part
of any interesting reinterpretation of mathematics.
>The set theoretic interpretation
This is the usual interpretation of a theorem of set theory such as
Cantor's - i.e., the set theoretic interpretation. You may wish to
suggest a new interpretation of set theory. If so, what?
>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
>specify
>such a known identifiable bunch of things.
This has nothing to do with the existence of a function from N onto
S(N). (That is another form of Cantor's theorem, that there is no
function from N onto S(N)). It appears that you are not talking about
set theory. The obvious question is: if you are not talking about set
theory, then what are you talking about?
Here are some other theorems of Cantor (and earlier!). What do your
unusual ideas have to say about them?
There is no function from the empty set onto S(emptyset).
There is no function from {0} onto S({0}).
There is no function from {0,1} onto S({0,1}).
There is no function from {0,1,2,...,2^1000000} onto S({0,1,2,...,2^1000000}).
There is no function from {0,1,2,...,2^1000000} onto {0,1,2,...,1 + 2^1000000}.
The image of any sequence of real numbers has Lebesque measure zero.
The image of any sequence of real numbers is disjoint from at least
one of its translates by a real number.
Given an image of a sequence of real numbers that forms a subfield of
the field of real numbers, there is another image of a sequence of
real numbers that forms a subfield of the field of real numbers,
where the intersection of the two images is exactly the rational
numbers.
>
>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.
This argument is fallacious. You can take any development in science
that achieves something far beyond earlier developments, whose
success comes partly from overhauling earlier attempts, and make your
same remark.
The syllogistic approach, with its restrictions to universal monadic
reasoning, was obviously grossly inadequate for the foundations of
even the mathematics of the time it was invented. I.e., reasoning
with inequalities are binary relations not covered by syllogisms.
Survival is not the issue. It did not even remotely form a foundation
for very elementary mathematics.
The current foundation for mathematics is obviously grossly adequate
for the foundations of current mathematics. Of course, I spend a
great deal of time trying to punch holes in this, but everything that
I do suggests that the current foundation needs to be augmented with
additional ideas, and not criticized for having wrong ideas.
>
>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?
Yes. E.g., the cumulative hierarchy has a ranking idea, or at least a
smallness idea, that was not recognized properly before Russell's
Paradox. E.g., not recognized by Frege. I've heard some people claim
that Cantor was sensitive to 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!
I'm waiting to see a plan for an alternative foundation for
mathematics, in light of your criticism of the present one.
>
>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.
>
Since you seem to be talking about foundations of mathematics
(focusing on Cantor's theorem), you can start by showing how ideas
like "singular" and "general" connect with (foundations of)
mathematics.
For example, you could propose some new interesting formal systems
and argue that, in some appropriate sense, they form a foundation for
at least some elementary mathematics, and also are insufficient to
prove things like Cantor's theorem.
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