FOM: Cantor'sTheorem & Paradoxes & Continuum Hypothesis

Robert Tragesser rtragesser at
Sun Feb 11 08:06:33 EST 2001

Neil Tennant wrote (I quote in full because
it's brief):
> > Isn't it fair to say that that proof of
> > Cantor's Theorem is a capital example of
> > the sort of purely logical, nonconstructive
> > proof which motivated Brouwer's churlish
> > observations about the logical?
>No, it would be quite unfair. The proof of Cantor's Theorem is

It would be helpful to have a few more words from
Neil Tennant saying what he has in mind by 'constructive'?
I suspect that he is thinking in terms of
whether or not logical principles or arguments
routinely thought of as nonconstructive were
deployed.  I have another issue in mind,
a more informal notion of "constructive",
the notion one might try to analyze by means of specifying
admissible logical principles.
It could be that there is no simple way of posing
the question I have in mind.
[1] The issue I am wrestling with does not have to do
with what logical principles are drawn upon in the proof.
It does not have to do with whether or not the proof
essnetially draws upon logical principles acceptable
to a constructivist. I take it that Neil is asserting
that the proof is constructive because it draws on
logical principles acceptable to what he thinks (and
likely others think) of as being canonically 'a
[2] What is at issue for me is appreciating Brouwer's
view toward formal logic and language in mathematical thought,-
that they have practical value but are in principle
(if not always in practice) eliminable in favor of
purely "intuitive mathematics."
It is arguably the case that in order to _appreciate
Brouwer's view_, really _Brouwer's view_ and not some
revsionary reconstruction of it by a later reason,
one must study the thinking and reasoning which gave
Brouwer offense, studying it as it were psychologically
or phenomenologically.  Here is what I am looking for:
mathematical proofs, or parts of mathematical proofs,
in which logic and language play an essential role,
in which they minimally or not at all draw upon
what might be quite liberally regarded as "intuitive
mathematics"(recalling that early Brouwer - in his
dissertation, say - had a sense of intuitive mathematics
which he did not perceive to be in conflict with the
law of the excluded middle).
Now, from a psychological cum phenomenological point of
view, in RAA proofs in mathematics, in agreeing to them,
one first agrees to them by attending to their logic,
but might nevertheless feel that the mathematics is
obscure.  A good example is most students' first
encounter with the standard RAA proof of the
irrationality of the square root of 2 coupled with
the exercise of ginning up proofs on that model
of the irrationality of the square roots of 3, 5, . . .
Those that succeed can nevertheless be brought to
confess to feeling that there is some perplexity,
some mystery. The discomforts can typically be
brought out into the open with the question:
"How can you tell when the k-th root of a natural
number is irrational?"
By having them inspect the proof techniques they
used on the square root of 3, 5, 7, . . . and by
having them try it out on the square root of 4,
9, . . . they can be led to the _mathematically
illuminating lemma_, that
"if a natural number does not have a k-th root
among the natural
numbers, it does not have a k-th root among the
rational numbers at all."
This is an example of a transition from a
proof that is compelling at first by virtue
of attention to its logic, but where
there is a sense of mathematical obscurity, to
a proof which is mathematically far more
illuminating and where the logic is in
considerably lower profile. . .the mathematics
is more intuitive (present) and not hidden
beneathe the logic.
Now the proof of Cantor's Theorem really
seems to be very much more intractably
dependent on tricks of logic and language
that actually block the way to intutivie
mathematics, to mathematical insight,
in the way that the RAA proof of the
irrationality of the square root of 2 did not.
Hence my urge to say that the logical
tricks (not to say illigitimate) exploited
in the proof of Cantor's Theorem,
variations of which drive the Russell
Paradox, perhaps count as paradigmatic
examples of the uses of logic and
language that worried Brouwer.
  Agreeing to this, one can then ask
what there is about them that blocks
or fouls the way to the intuitive
mathematics (= to the mathematics).

[3] My idea of a constructive proof is
that it ought to yield some
determining information about the
property at issue, that in the
case of Cantor's Theorem a constructive
proof should yield rather more
information about the cardinality
pf the power set of a set, given
the cardinality of the set, than
that it's greater.
I can appreciate how the lemma
within the proof of Cantor's Theorem,
might be regarded as "constructive"
in flavor -- the proof shows, given any
set of subsets S of a given set M
(where S in not in general the
set of all subsets of S), where
M can enumerate [elements put
into 1-1 correspondence with
those of S)
how to frame a subset L of
M demonstrably not in S
(and where the construction
can be carried through by
the seat of one's pants
in the finite case),
but it is framed so
indirectly, by means of
logical artifice, that
one can not make it
definite (in the infinite
But in the full theorem
the property or entity
at issue is the cardinality
of the power set, and
I don't see anything
like a construction
of this yielded by the
proof (even allowing
the loose sense of "construction"
by which the proof of the
lemma might be said to
provide a construction.

robert tragesser

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