[FOM] Set Theory and Intuition

[Jeremy Clark]

Dean Buckner wrote:
>
> There is an argument that frequently occurs in the popular literature on
set
> theory that I ought to mention.  It goes: one intuition, a very powerful
> one, is that any set is somehow larger than any of its proper subsets.
> However there is another intuition, almost as powerful, that we can match
> every member of a infinite set to a unique member of one of its proper
> subsets (Galileo, and numbers and their squares, usually come into the
story
> here).  Thus, our ordinary intutions are inconsistent.  We can only
resolve
> the inconsistency by dropping one of them (the first).  Thus, our ordinary
> conceptual scheme leads (for all but the slow-witted or deliberately
obtuse)
> inexorably to & through the gates of Cantor's paradise.
>

I'm afraid that I lack the intuitive feeling that a set has to be somehow
larger than
a proper subset. Or at least, if 'larger' is interpreted in terms of
equinumerability.
This is because I find the counter-examples (the usual ones which you
mention above)
perfectly intuitively graspable. I think the relationship between
mathematics and
ones intuitions is a two-way one: we choose axioms and (more importantly, I
think)
definitions which chime with our intuitive feel for numbers, geometries,
etc. But then the
results we go on to prove are inevitably going to affect our intuitions: we
get 'feedback'.

I think it is a bit artificial to call some intuitions 'ordinary' or
'natural', to distinguish them
from other intuitions which we develop as a result of understanding
mathematics. That process of understanding is also, ultimately, intuitive,
isn't it?
We read the proof over and over until something clicks and we say, 'Ah, I
*see*.'

> Wrong.  Undoubtedly set theory is consistent, it doesn't follow that our
> intuitions are not.  The whole argument rests on the assumption we have
any
> intuition about what the expression "infinite set" denotes.  I have no
such
> intuition.  Given any set of whole numbers, we can find a set of prime
> number to match, agreed.  But then the set must have a number, and
therefore
> cannot be infinite, surely?  Every set of numbers is finite, and there are
> infinitely many of them.  That's my intuition, and its perfectly
consistent
> with the intuition that if any set of F's that wholly contains some set of
> G's, then there are more F's than G's.
>

Why is set theory 'undoubtedly consistent'? This is not a rhetorical
question...

Insisting that every set is necessarily finite (as you do above) seems to me
to be less
an intuitively grasped fact and more a redefinition of the word 'set' so
that sets are
necessarily finite. Do you *really* find it impossible to grasp infinite
sets (I mean
nice friendly infinite sets, like e.g. the set of primes)?

> If any set theorist want to claim he some contrary intution, that's fine.
> But then he has to drop the intuition that is more natural (my one).
>

I think you're on dangerous ground if you claim that some intuitions are
more natural than
others. This reduces mathematics to a sort of popularity contest (which, now
that I think
about it, isn't a bad analogy). There is a danger here of slipping into a
sort of
mysticism with regard to certain mathematical objects, as Brouwer was
supposed
to have done with regard to the continuum.

Jeremy Clark