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.

"Martin Wanvik" <martinw@stud.ntnu.no> wrote in message
news:22846326.1209653540265.JavaMail.jakarta@nitrogen.mathforum.org...

"Martin Wanvik" <martinw@stud.ntnu.no> wrote in
message
news:15978560.1209647023542.JavaMail.jakarta@nitrogen.
mathforum.org...
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?
[...]

You might find the following wikipedia articles
helpful:

http://en.wikipedia.org/wiki/Naive_set_theory
http://en.wikipedia.org/wiki/Axiomatic_set_theory

-- Martin Wanvik

I've read the articles, but they do not define what a
collection is.

Then you missed my point, which probably wasn't as obvious as I though it
would be. I'll try to be clearer: Your definition above of a set isn't
really a definition at all, in the mathematical sense of the word. It is
simply an intuitive, non-formal description of what we consider a set to
be, the kind one usually employs when doing naive (or informal) set
theory. The rigorous, formal way of doing these things is referred to as
axiomatic set theory.

-- Martin Wanvik

They are definitions.
You are assuming more that what is actually there.

"Virgil" <Vir...@gmale.com> wrote in message

news:Virgil-DD2602.14422401052008@comcast.dca.giganews.com...

In article <wKpSj.1014$NZ7....@newsfe10.ams2>, "Mark" <u...@home.com
wrote:

I was basically asking whether elements in a set need to share something
in
common.

Other than membership in that set, no!

Then why would they be in a set in the first place?
Can you provide me with an example of a set whose elements have nothing in
common with each other, other than the fact that they belong to the set?

I was basically asking whether elements in a set need to share something in
common. That's all.

Then why would they be in a set in the first place?
Can you provide me with an example of a set whose elements have nothing in
common with each other, other than the fact that they belong to the set?

On May 1, 1:00 pm, "porky_pig...@my-deja.com" <porky_pig...@my-


But Mark is correct: both Wolfram and Wikipedia state [...]

It may be that those two sites corroborate one another in this
instance, and, of course, those sites are often correct on many
matters. My remark was general...In my opinion a good mathematical
vocabulary is not built by grabbing definitions ad hoc from such
sites, especially Wikipedia, which is nothing but ad hoc defintions
and formulations, written article-by-article (and edited by no single
responsible authority), without a systematic presentation among the
various articles on even one single branch of mathematics.

that the
'collection' is a common synonym for multiset. I looked at the
'multiset' definition on both Wolfram and Wiki sites but missed the
Wolfram definition of 'collection' as well as the fact that Wiki
redirects 'collection' to multiset. OK, mea culpa.

Though other people may have different experiences, it is my
impression that 'multiset' is a special notion in a way that
'collection' is not.

There ->is<- not definition of primitive terms in modern
axiomatic theories. In most set theories, "set" is not defined at all;
in a few, such as Goedel-Bernays, the definition is only one or two
levels above the undefined terms.
[...] Moreover,
an implicit or intuitive definition is usually based on a MODEL of an
axiomatic theory, and as such do not form part of the theory but
rather of a particular INTERPRETATION of the theory.

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?

Well, my experience is that "collection" is used in the same way a
"family" is used: to informally distinguish the 'second level set' (an
aggregation of sets) from the "first level sets".

Frankly, I would rather *not* have "collection" as a standard
synonym for multiset since it only confuses the issue.- Hide quoted text -

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?

On Thu, 1 May 2008 12:55:46 +0100, "Mark" <user@home.com> 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

In article <pjqj14ps39u6jacq45m2jbn9r8mrlo9bf9@4ax.com>,
Angus Rodgers <twirlip@bigfoot.com> wrote:

Isn't there a dilemma here? When a mathematician thinks about e.g.
the set of zeros of the Riemann zeta function, must we suppose that
(unlike Riemann himself) he is "really" thinking about some element
of some model of some fixed, formal axiomatic theory of sets, which
must therefore be taken as foundational for all of mathematics?

There is of course no telling how one "thinks" about something;
and the source of my (or your) intuition is irrelevant provided that
you can present an acceptable correct proof following the usual
canons, regardless of whether you came up with it by thinking about
some element in some specific model of some formal theory, or you came
up with it by thinking of functions as sheep jumping a
fence. Whatever representation goes on in your mind is irrelevant so
long as you can "map" that representation into something formal enough
to constitute a proof (that is why one tries to avoid "obvious" as
part of an argument).

Most mathematicians are not actually doing mathematics within a
formalized axiomatic set theory (just think how many hundreds of pages
it took Russell and Whitehead to prove that 1+1=2). Instead, they work
within a more "naive" set theory (or even other foundational model,
such as categories, or arithmetic, or even the theory of real
numbers). For some people, specific models are a good way to get a
handle of the objects they are dealing with; for others, formalism is
the best way of thinking about it. I really don't see a "dilemma".

Or
is he (as I tend to presuppose) thinking, rather, of a collection of
elements of a set R[i], where R is a model of some theory of the real
number system? In the former case, isn't the picture of mathematics
as a whole somewhat unrealistic?

Whose picture? Which of their pictures?

And in the latter case, isn't there
a real question of explaining what a "collection" of elements of a
"set" constructed from a "model" is?

If you are thinking exclusively about the Riemann zeta function in the
context of real numbers and complex numbers, then do you really need
an explanation of what the generic notion of "collection", "set" and
"model" are? No. You just need an explanation of what "set of complex
numbers" is, of what "function" is within that context, etc. There is
no need to be informed of all the technical details of axiomatic set
theory, nor to have a model for axiomatic set theory as a whole in
order to do calculus, either; an informal understanding is sufficient,
provided you avoid the error of using the informal understanding as
part of your argument to establish something.

After all, isn't a "model" (of
/any/ theory - whether of undefined entities called "sets", undefined
entities called "real numbers", or another kind of undefined entities)
itself a kind of "collection" or "set"?

Depends on how you found your theory. You can found your theory on
categories instead of sets. [...]

Also, isn't it true that even
though axiomatic theories of sets play a vital role in mathematics,

I would call it a pervasive rather than a vital role. It is possible
to do mathematics without any axiomatic set theory on hand, though
much of what you will read assumes at least some background in the
basic notions of such a theory.

Still, I can't
deny that there seems to me, at least, to be a real problem here.

There may be a philosophical issue; which is why you still have
vigorous philosophical debates about the nature of mathematics. But
then, there is a philosophical issue about what "mind" is, and yet
that does not seem to create a problem in terms of not allowing us to
think!

In article <oClSj.1209$7z4.15@newsfe13.ams2>, Mark <user@home.com> wrote:

"Arturo Magidin" <magidin@math.berkeley.edu> wrote in message
news:fvck0e$25pn$1@agate.berkeley.edu...
In article <AijSj.861$NZ7.158@newsfe10.ams2>, Mark <user@home.com
wrote:

"David C. Ullrich" <dullrich@sprynet.com> wrote in message
news:k0fj14lbq30cmomiiaoc1f513b35bu53jr@4ax.com...
On Thu, 1 May 2008 12:55:46 +0100, "Mark" <user@home.com> 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.

No; a definition, in mathematics, is a FORMAL statement. Here, you are
presented with an informal introduction to the idea. It is not a
definition, in the sense of a mathematical definition. You are
committing the fallacy of equivocation by saying "In other words, a
definition." There ->is<- not definition of primitive terms in modern
axiomatic theories. In most set theories, "set" is not defined at all;
in a few, such as Goedel-Bernays, the definition is only one or two
levels above the undefined terms.

[...]

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.

He noted that in modern theories primitive terms are undefined. This
happens to be what Hilbert noted and what happened at the time.


[snip N/A stuff]

I don't see how a logical theory can be based on the undefined.


Then perhaps you should learn some basic mathematical logic.

Are you trying to tell me that *you* cannnot explain to someone else what
an
element or a set is?

No. I am INFORMING you of the verifiable fact that modern axiomatic
theories are based on primitive terms, and that primitive terms are
NOT defined within the theory. If you cannot handle that level of
abstraction, then I suggest you take your own inadequacies and get as
far away from mathematical logic as you can.

If you can, then surely you must agree that you have defined them.

No. Defining something in mathematics is NOT the same as giving an
intuitive or informal explanation of something to someone. Moreover,
an implicit or intuitive definition is usually based on a MODEL of an
axiomatic theory, and as such do not form part of the theory but
rather of a particular INTERPRETATION of the theory. Again: these are
some of the basic (but subtle) notions of modern mathematics and
logic. If you are unfamiliar with them, then you ought to familiarize
yourself with them before continuing to equivocate.

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================

Arturo Magidin
magidin-at-member-ams-org

In article <oClSj.1209$7z4.15@newsfe13.ams2>, Mark <user@home.com> wrote:

[snip N/A stuff]

Did you bother to read them? They were not "Not applicable". They are
direct quotes that show that the notion of "set" is simply NOT DEFINED
in modern mathematics. Perhaps you should re-read them, instead of
dismissing them out of hand.


--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================

Arturo Magidin
magidin-at-member-ams-org

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