Re: Questioning the defintions of set and element.


"Arturo Magidin" <magidin@xxxxxxxxxxxxxxxxx> wrote in message
news:fvck0e$25pn$1@xxxxxxxxxxxxxxxxxxxxx
In article <AijSj.861$NZ7.158@xxxxxxxxxxxxx>, Mark <user@xxxxxxxx> wrote:

"David C. Ullrich" <dullrich@xxxxxxxxxxx> wrote in message
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On Thu, 1 May 2008 12:55:46 +0100, "Mark" <user@xxxxxxxx> wrote:

Hi, most definitions of element and set I have come across, say
something
like,

An element is any object of our perception or of our thought.

You found this definition where, exactly?

In formal set theory the notions of "is a set"
and "is an element of" are _undefined_.

A set is a collection of unique elements.

So whats a collection?
Wolfram says it's a multiset.
Wiki says it's a multiset.

So whats a multiset?
Wolfram says it's a set-like object.
Wiki says it's a generalization of a set.

This basically gives the following definitions.

A multiset is a collection of elements
A set is a multiset of unique elements.

So whats a collection?
Would this be a good definition of colletion,
A collection is any elements which have something in common.

Or could someone give a better definition?

From, "Discovering Modern Set Theory by Winfried Just, Martin Weese,
American Mathematical Society"

This is not presenting a "definition" in the sense of a mathematical
definition; rather, it is presenting an informal idea that is what
they will be attempting to model formally.

In other words, a definition.


"By a "set" we mean any collection M into a whole of definite, distinct
objects m (which are called the "elements" of M) of our perception or of
our
thought." - Cantor

You do know it's been well over 100 years since then, and that
everyone uses formalizations of set theory that were created well
after Cantor, right?

Yes.


As David Ullrich notes, after the advent of Hilbert and
metamathematics, it is now understood that the basic notions of an
axiomatic theory, the "primitive notions" are ->undefined<-. The
axioms and rules describe what we can do with them, but those
primitive notions do not have a definition.

He didn't note anything about Hilbert or metamathematics.

[snip N/A stuff]

I don't see how a logical theory can be based on the undefined.
Are you trying to tell me that *you* cannnot explain to someone else what an
element or a set is?
If you can, then surely you must agree that you have defined them.


.

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