Re: Questioning the defintions of set and element.


"Mariano Suárez-Alvarez" <mariano.suarezalvarez@xxxxxxxxx> wrote in message
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On May 2, 2:04 pm, "Mark" <u...@xxxxxxxx> wrote:
"David C. Ullrich" <dullr...@xxxxxxxxxxx> wrote in
messagenews:ps3m145ejb89843g7l6be39undqv7rleni@xxxxxxxxxx



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

"John O'Flaherty" <quias...@xxxxxxxxx> wrote in message
news:nlrj14976helcnlne7fnqavsrllvme3qa8@xxxxxxxxxx
On Thu, 1 May 2008 12:55:46 +0100, "Mark" <u...@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

I don't care about the definitions, why can't you see that?

You started this whole thread with a

[...] Or could someone give a better definition?

You should study your masters: the `good'
sci.math trolls tend to be quite more consistent!

-- m


I started this thread with,

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

Sorry, but i've been posting on newsgroups for years, and if anyone is
trolling, it is you guys.
You just keep trying to over complicate things.
I asked something simple, you didn't get it, and then insult me.
On the other hand, I ask a 9 year old, who gets the questions, but cannot
answer them due to lack of knowledge.




.

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