Re: The Difference between a Set and an Element

On Wed, 10 Jan 2007 20:03:55 GMT, Nam D. Nguyen <namducnguyen@xxxxxxx>
said:
...
<quote>
Logic is the study of reasoning; and mathematical logic is the type
of reasoning done by mathematicians. To discover the proper
approach to mathematical logic, we must therefore *examine the
methods of the mathematician*.

The conspicuous feature of mathematics, as opposed to other
sciences [including say physics, chemistry, or biology, ...] is the
use of proofs instead of [physical] observation. A physicist may
prove physical laws from other physical laws; but he usually
regards agreement with observation as the ultimate test of a
physical law. A mathematician may, on occasions, use observation;
for example, he may measure the angles of many triangles and
conclude that the sum of the angles is always 180 degree. However,
he will accept this as a law of mathematics only when it has been
proved.

Nevertheless, it is clearly impossible to prove all mathematical
laws. The first laws which one accepts cannot be proved, since
there are no earlier laws from which they can be proved. Hence we
have certain first laws, called _axioms_, which we accept without
proof; the remaining laws, called _theorems_, are proved from the
axioms.

For what reasons do we accept the axioms? We might try to *use
observation* here; but this is not very practical and *is hardly in
the spirit of mathematics*. We therefore attempt to select as
axioms certain laws which we feel are evident from the nature of
the *concepts* involved.

We thus have a reduction of large number of laws to a small number
of axioms. A rather similar reduction takes place with
mathematical *concepts*. [...] We therefore have certain
*concepts*, called *basic concepts*, which are left undefined; the
remaining *concepts*, called *derived concepts*, are defined in
term of these. [...] We have a criterion for basic *concepts*
similar to that for axioms: they should be so simple and clear that
we can understand them without a precise definition. [...]

Hence we may suppose that all the *concepts* which appear in the
axioms are basic *concepts*
[...]
</quote>
Shoenfield, Mathematical Logic, Chapter 1, "The Nature of Mathematical logic".

You wouldn't disagree with me that mathematical concepts are abstract,
right?

Well, I'm not sure what concepts are exactly, but if there are such
things I would certainly agree they are not concrete.

My somewhat-naive-understanding-of-the-subject assertion that
"Mathematics is abstract. Period." simply asserts what I believe as a
fact of mathematical reasoning; and, from Shoenfield's passage above,
it doesn't appear to be a "random" idea, as you seem to have alluded
to.

As far as I can see, what Shoenfield is arguing has no obvious relevance
to your claim that sets cannot contain concrete objects as members. And
the claim itself is just so, well, curious. Isn't it rather obvious
that, if there are such things as sets at all, that there are sets of
concrete things? Why should the nature of the elements prevent there
being a set of them? Granted, these are not "pure" mathematical
objects, but so what? Isn't there the set of US senators? How about
the set of Volkswagen Jettas? On what basis do you claim that the set
{1,2} exists but not the set {Barak Obama, Hillary Clinton}?

So, Dr., instead of making me feel humbled - being bowed to - perhaps
you could explain why concepts which are *abstract* can be *concrete
physical entities*, such as "flesh and blood persons".

I don't follow you. You seem to think that I've claimed that flesh and
blood persons are concepts. Why do you think that? All I've said is
that there is nothing about set theory that is incompatible with
concrete things like persons being members of sets. And I haven't said
anything about concepts at all.

.

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