in TST (tommy1729 set theory)

On 01-05-2008 16:47, Mark 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.

And I don't see how can a logical theory define all of the terms that it
uses. Because in order to define a word you need words. And in order to
define these words you need other words and so on.

So, a logical theory starts with undefined objects and undefined
relations. Then, it states a list of properties that relate these these
objects and relations.

Best regards,

Jose Carlos Santos

My question was about the definition for a collection, and nothing to do
mathematic definitions, axioms, modern set theory etc.

I would say that the definitions of the objects which the
theory is based upon are not part of the theory, and therefore the theory
starts with a previously defined concept.

All elements of any set have something in common by the virtue of the
fact that they belong to that set. A good example would be collection
of crackpots posting on sci.math.

On Thu, 1 May 2008 19:26:09 +0100, "Mark" <user@home.com> wrote:

I would say that the definitions of the objects which the
theory is based upon are not part of the theory, and therefore the theory
starts with a previously defined concept.

Try Gottlob Frege, "The Foundations of Arithmetic", if you want
to go into these questions seriously (but not too technically -
it's a very readable book). But bear in mind that Frege's theory
of the foundations of mathematics ran into apparently insuperable
difficulties (in particular, Russell's Paradox, which is very easy
to state, but very hard to see a way around), leading to a century
of complicated and difficult developments.

Or, if you /don't/ want to go into these questions in a serious
philosophical way, then try to rest content with your existing
understanding of what the words "set" and "collection" both mean -
because that understanding is probably already quite good enough,
for all practical purposes, and not much worse than anyone else's!

There are a lot of different books you can go on to read, depending
on the nature, intensity, and seriousness of your puzzlement. I
don't yet know what to recommend, or indeed whether you really need
anything other than simple reassurance that there isn't some complex
technical definition of "set" and "element" which, mysteriously, you
can't find anywhere, and aren't being told about.

On a more strictly mathematical level, there are also early works
by Dedekind and Cantor, which are pretty readable, and available in
translation and in cheap paperback editions. And, of course, there
are plenty of modern textbooks - but be prepared for frustration in
your search for, er, definitive definitions!

--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril

On May 1, 7:55 am, "Mark" <u...@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.


OK, that was discussed many times, but let's repeat it anyway. A set
is undefined concept and can only be characterized by what's known as
"set membership". We say that s in S and call s "an element" of a set.
However, at least in ZF set axioms we do not distinguish elements of
set from the set. An element of set is a set on its own, so s is a
set. For instance, we can think of a set of all natural numbers N,
and say that 1 in N, but 1 is a set on its own (check Von Neumann
numerals).

So whats a collection?

There is no formal definition of collection. Informally we say that
set is a collection of unique elements, OK, but "collection" is not
defined just as "set".

In practice, however, we often have the following situation. We work
with a set, say S, and even its members s1, s2, ... are also sets, we
would like to pretend that they are atomic (if this assumption simply
our logic but doesn't screw anything). Now suppose we need to create
the 'higher-level' set, A, consisting of S1, S2, ...; often to avoid
confusion we would like to call that second level aggregation by some
name other than set; a "family" or "collection" are often used, but
once again, that's just a matter of convenience; formally A is a set
consisting of S1, S2, ..., in turn each Sn consists of something else,
and that something else is what we treat as 'atomic element' simply
because we don't care about its internal structure. E.g., when I work
with a set N, normally I don't care about set representation of 1,
2, ... (but if we do need to prove the laws governing the natural
numbers, we have to look at the internal structure of the elements of
N, and this is where Von Neumann representation comes to play).

So you may run into something like "consider the family of all compact
sets on something". Here the "family" is still a set, we just call it
a family for convenience. You'll also see the 'first level set' are
designated by uppercase, the second level - by fancy script letter in
the beginning of alphabet, and the third level by the fancy script
letter closer to the end of alphabet. If I remember correctly, Halmos
in Naive Set Theory discusses those conventions. (Normally we wouldn't
go higher than three levels - at least I hope so.)



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


No, they don't. Neither Wolfram nor Wiki use the word 'collection'.
Try to understand what they say and not to read "between the lines".

This basically gives the following definitions.

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


No, both Wolfram and Wiki give exactly the same definition: multiset
is generalization of set; if multiset we allow multiplicity of
elements; a set is multiset with multiplicity of 1.
So multiset is defined as generalization of set, this is as much as we
can say. Again, seems like you just can't comprehend neither Wolfram
nor Wiki definitions. That's *your* problem.

So whats a collection?

An informal synonym for a set.

Would this be a good definition of colletion,

Just as a set, it's undefined.

A collection is any elements which have something in common.


All elements of any set have something in common by the virtue of the
fact that they belong to that set. A good example would be collection
of crackpots posting on sci.math.

Or could someone give a better definition?

You can't define what's fundamentally is undefined. Any definition in
this situation would boil down to creating yet another undefined
synonym of something being undefined. A set is a collection is a
family is an aggregation is a set.

On May 1, 3:42 pm, MoeBlee <jazzm...@hotmail.com> wrote:
On May 1, 12:41 pm, "Mark" <u...@home.com> wrote:

http://en.wikipedia.org/wiki/Collection_%28mathematics%29

You will immediately improve your mathematical vocabulary by ceasing
to rely on Wikipedia for it.

MoeBlee

But Mark is correct: both Wolfram and Wikipedia state 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.

And yet, once again, the fact that 'collection' is a common synonym
for multiset does not mean more than saying that 'family' is a common
synonym for the aggregation of sets. Synonym is not a definition.
Neither sets not multisets are defined. OK, the multiset is
generalization of a set, but that does not make multiset more defined
than set, ditto for 'collection' understood as a synonym of multiset.

In fact, the only time I saw the word 'collection' is as a reference
to the 'second level aggregation' of sets, just like family. But then
I never read anything on multisets, so may be this is a common
convention. *Still* does not make it a definition.

http://en.wikipedia.org/wiki/Collection_%28mathematics%29

In article <xmnSj.44$SY5.40@newsfe13.ams2>, Mark <user@home.com> wrote:

"Arturo Magidin" <magidin@math.berkeley.edu> wrote in message
news:fvcp9o$27f6$1@agate.berkeley.edu...
In article <oClSj.1209$7z4.15@newsfe13.ams2>, Mark <user@home.com
wrote:

[...]

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


Then perhaps you should learn some basic mathematical logic.

Perhaps you should learn how not to assume things.

You're right. I owe you a big apology. Just because you posted to a
mathematics group (sci.math), with a mathematical question about the
meaning of some mathematical terms of art, quoting a mathematics book
written by mathematicians, and a specific mathematician (Cantor), I
should not have assumed that you were looking for a mathematical
answer.

My deep and sincere apologies. I should have realized that it was just
a fickle twist of fate the directed your question to sci.math instead
of alt.usage.english, where it belonged.

--
======================================================================
"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

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

"Arturo Magidin" <magidin@math.berkeley.edu> wrote in message
news:fvd8jt$2cl1$1@agate.berkeley.edu...
In article <xmnSj.44$SY5.40@newsfe13.ams2>, Mark <user@home.com> wrote:

"Arturo Magidin" <magidin@math.berkeley.edu> wrote in message
news:fvcp9o$27f6$1@agate.berkeley.edu...
In article <oClSj.1209$7z4.15@newsfe13.ams2>, Mark <user@home.com
wrote:

[...]

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


Then perhaps you should learn some basic mathematical logic.

Perhaps you should learn how not to assume things.

You're right. I owe you a big apology. Just because you posted to a
mathematics group (sci.math), with a mathematical question about the
meaning of some mathematical terms of art, quoting a mathematics book
written by mathematicians, and a specific mathematician (Cantor), I
should not have assumed that you were looking for a mathematical
answer.

My deep and sincere apologies. I should have realized that it was just
a fickle twist of fate the directed your question to sci.math instead
of alt.usage.english, where it belonged.

Apology accepted. But you're wrong, it does belong here.
I was basically asking whether elements in a set need to share something in
common. That's all.
Why would you think that it would be more appropriate in alt.usage.english?

In article <wKpSj.1014$NZ7.376@newsfe10.ams2>, "Mark" <user@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!

On May 1, 5:29 pm, "Mark" <u...@home.com> wrote:
"Arturo Magidin" <magi...@math.berkeley.edu> wrote in message

news:fvd8jt$2cl1$1@agate.berkeley.edu...



In article <xmnSj.44$SY5...@newsfe13.ams2>, Mark <u...@home.com> wrote:

"Arturo Magidin" <magi...@math.berkeley.edu> wrote in message
news:fvcp9o$27f6$1@agate.berkeley.edu...
In article <oClSj.1209$7z4...@newsfe13.ams2>, Mark <u...@home.com
wrote:

[...]

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

Then perhaps you should learn some basic mathematical logic.

Perhaps you should learn how not to assume things.

You're right. I owe you a big apology. Just because you posted to a
mathematics group (sci.math), with a mathematical question about the
meaning of some mathematical terms of art, quoting a mathematics book
written by mathematicians, and a specific mathematician (Cantor), I
should not have assumed that you were looking for a mathematical
answer.

My deep and sincere apologies. I should have realized that it was just
a fickle twist of fate the directed your question to sci.math instead
of alt.usage.english, where it belonged.

--
======================================================================
"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

Apology accepted. But you're wrong, it does belong here.
I was basically asking whether elements in a set need to share something
in
common. That's all.
Why would you think that it would be more appropriate in
alt.usage.english?

By the way: the elements of a set always share something
in common: they are all elements of that set!

-- m

"Virgil" <Virgil@gmale.com> wrote in message
news:Virgil-DD2602.14422401052008@comcast.dca.giganews.com...
In article <wKpSj.1014$NZ7.376@newsfe10.ams2>, "Mark" <user@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?