[FOM] Finite Set Theory

Dean Buckner d3uckner at btinternet.com
Wed Feb 22 12:40:46 EST 2006

```Friedman:
> I really appreciate this frank answer of no. I.e., "not as well as I
have
> explained the emptyset and singletons".

You have explained the contents of a mirage well.  I have explained that
it may be a mirage!

DB:
> I'll try.  Some words like 'shoe' apply to just one thing at a time.
> But collective nouns like 'pair' apply to more than one thing at a
time.
> The word 'one' in 'one pair' tells you that 'pair' applies once, but
to
> two things.  Thus if there is just one pair of things, there are only
> two things, no third thing.

Friedman:
>  This is incorrect. There is a third thing here, in the pair written
<x,y>.

The third thing is something you have written down.

> There is x, there is y, and there is the ordered pair <x,y>. (Or we
could do
> this with the unordered pair).  This is the way students are taught
analytic geometry
> starting in high school.

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.

>This banning suggestion is quite bad for the development of mathematics
and
science.

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

>>>
What about the set that has nothing in it? And what about the set that
has
my computer in it and nothing else? And what about the set that has my
computer in it and my house in it and nothing else? Not only are all of
these crystal clear as single objects, but clarity about these obvious
constructions is essential for the progress of mathematics and science.
>>>

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'.

>>>
I am sure that you are not proposing a ban on such constructions just
because you personally dislike them. But I am at a loss to know just why
you
want to ban them?
>>>

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.

```