Re: The Difference between a Set and an Element


Nam D. Nguyen wrote:
Chris Menzel wrote:
On Wed, 10 Jan 2007 18:01:50 GMT, Nam Nguyen <namducnguyen@xxxxxxx> said:
Chris Menzel wrote:
On Wed, 10 Jan 2007 16:05:05 GMT, Nam Nguyen <namducnguyen@xxxxxxx>
said:
Frederick Williams wrote:
george wrote:
JohnCreighton_@xxxxxxxxxxx wrote:
Frederick Williams wrote:
.... What is the physical difference between me and a set
containing me?
The difference is that you exist and the set containing you does
not. This allegation that sets can contain concrete objects is
misleading. Sets are abstract.
May not a set theory with urelemente have a person among its
urelemente?
I think you meant "... an abstraction of a person among its
urelemente?" Mathematics is abstract. Period.
Well, granted, it makes life much easier just to stipulate that things
are thus and so, PERIOD, because that's just what your gut tells you,
but there is nothing whatever about set theory or mathematics generally
that would prevent honest to God flesh and blood persons from serving as
legitimate urelements.
You are right: I just forgot the basic math that the natural numbers
are made of hydrogen atoms and this is why they're so light that they
float around in the mind!

An excellent argument! Nearly as cogent as "<random claim> PERIOD"! I
bow to you, sir; clearly, you have reduced my argument to absurdity.


<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?
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. 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". Or is that just a random and unprovoked reaction
of yours?

There is nothing ins Shoenfield's remarks that rules out ZU. The
concept of *set* can be as he says fundamental mathematical concepts
should be: that doesn't stop the members of sets being as concrete as
you like.

.