Re: Questioning the defintions of set and element.

On Thu, 1 May 2008 18:42:07 +0100, "Mark" <user@xxxxxxxx> wrote:


"John O'Flaherty" <quiasmox@xxxxxxxxx> wrote in message
news:nlrj14976helcnlne7fnqavsrllvme3qa8@xxxxxxxxxx
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.
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?

What do you mean by "definition"? It would seem, from the other
answers, that a definition in mathematics is a statement about
something in terms of other mathematical entities. Since no
mathematical system can be all-encompassing, for any particular system
there must be a ground floor of mathematically undefined somethings.
In ordinary language, however, a definition is a statement about
something that describes it (informally), and may try to exclude other
things. You should be able to define terms in this sense. A set is a
grouping of elements - a notional grouping based on a common property
of the elements, which may be as trivial as that they were assigned to
the same set.

--
John

By definition, I mean a statement which descibes some concept or object.
The standard meaning of the word defintion.

_If_ that's what you mean by "definition" then that's exactly why
you should be posting to alt.english.usage instead of here, as
Arturo suggested. That is _not_ the meaning of "definition" in
mathematics.

In mathematics, if D is a definition of "gizmo" then D must
determine precisely, for every possible value of x, whether
or not x is a gizmo.

"Determine" here doesn't mean it's possible to actually
calculate the answer - there may be x's for which nobody
can figure out whether or not x is a gizmo. But D has
to determine whether x is a gizmo unambiguously,
in a theoretical sense: whether or not x satisfies definition
D has to be something that's precisely defined, _not_
something that might vary from person to person depending
on how he thinks about this or that.

A statement that "describes" some concept or object
is not a definition in that sense (unless you mean the
term "describe" much more precisely than you seem
to). For example Cantor's "definition" of set that you
quoted is certainly not a definition in this sense -
what is or is not an "object of thought" is not clear,
and may certainly vary from person to person.

****************

Technicality, added in case you're puzzled about
the bit above regarding how a definition has to
specify whether x is a gazebo even if we can't
figure out whether it's satisfied or not:

First, one can prove the following from the
standard axioms of set theory:

Theorem. There exists exactly one set x with
the property that x has no elements.

Now since that's a theorem, we're allowed to
make the following definition;

Definition: The _empty set_ is the set with no elements.

Now. Define S to be the set of all even natural numbers
such that S is not the sum of two primes. Is S equal
to the empty set or not? Nobody knows. But the
definition of "empty set" _does_ specify whether or
not S is the empty set: If there exists an even natural
number which is not the sum of two primes then yes,
otherwise no.



David C. Ullrich
.

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