[FOM] Finite Set Theory

[Harvey Friedman]

On 2/22/06 12:40 PM, "Dean Buckner" <d3uckner at btinternet.com> wrote:

Friedman wrote:

>>  This is incorrect. There is a third thing here, in the pair written
> <x,y>.
> 
> The third thing is something you have written down.

I wrote it down the usual way people generally write it down. This is all
standard fare.

> Ergo, to demonstrate that something exists it is enough to state that it
> exists, and to say that students starting in high school are taught that
> it exists.

I just wanted to remind everyone that they were taught all of this
elementary stuff in high school, to help them recall it.

> It is healthy for the development of mathematics and science to ask
> questions about some of its fundamental assumptions from time to time.

I do not object to this, as long as it is understood that the inquirer is
not dogmatically asserting that there is a real incoherence.

If it is presented as a challenge to make things clearer than they usually
are, clearer in some novel way, then that is healthy.

> You say the idea of a set is crystal clear.  The fact it can have two
> distinct meanings (the ordinary one, meaning many things, the
> mathematical one, meaning one thing) suggests it is not entirely
> 'crystal clear'.

On the FOM list, it is reasonable to assume that we are talking about the
mathematical sense - "meaning one thing".

On a philosophy of language list, or linguistics list, or some other quite
different kind of list than f.o.m., one may wish to NOT focus on the
mathematical sense, and instead focus on other senses (of which there are
apparently many).

If you propose that some other sense of set is interesting from a
logical/mathematical point of view, and propose that it be analyzed
seriously, then that could be a good thing for the FOM list.
> > 
> Proposing a ban?  You claim in your original post is that it is very
> easy for ordinary people to understand how set theory works.  The point
> of my post is that it is more complicated than you suggest.

I'm glad that you are NOT proposing that the ordinary mathematical notion of
(finite) set be banned.

It would be interesting if you could identify a generally intelligent person
who has difficulty with this very elementary finite set theory. We could
invite them on the FOM to engage in a dialogue that would focus on just what
the difficulties are.

Harvey Friedman