| |
 |
|
|
Science Forum Index » Mathematics Forum » Questioning the defintions of set and element.
Page 1 of 5 Goto page 1, 2, 3, 4, 5 Next
|
| Author |
Message |
| Mark |
 Posted: Thu May 01, 2008 6:55 am |
|
|
|
Guest
|
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? |
|
|
| Back to top |
|
| Martin Wanvik |
| Posted: Thu May 01, 2008 6:55 am |
|
|
|
Guest
|
|
| Back to top |
|
| Arturo Magidin |
| Posted: Thu May 01, 2008 6:55 am |
|
|
|
Guest
|
In article <AijSj.861$NZ7.158@newsfe10.ams2>, Mark <user@home.com> wrote:
Quote:
"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.
Quote: "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?
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.
Thomas Jech's massive "Set Theory", for example, begins simply by
listing the axioms of set theory, never bothering to say what "set"
and "is an element of" are supposed to mean; merely "sets" are the
objects of discourse, and "is an element of" is a binary relation
among objects in our discourse. The axioms describe certain rules
whereby their uses are bound. Only later does he describe what
"intuitively" a set may be ("Intuitively, a set is a collection of all
elements that satisfy a certain given property.") But this is an
->intuitive idea<-, a hook to try to help you internalize and get some
meaning out of the axioms, ->NOT<- a definition.
Paul Halmos, in his very readable "Naive Set Theory", says in the
second paragraph of the book:
One thing that the development will not include is a definition of
sets. The situation is analogous to the familiar axiomatic approach
to elementary geometry. That approach does not offer a definitio of
points and lines; instead, it describes what it is that one can do
with those objects. The semi-axiomatic point of view adopted here
assumes that the reader has the ordinary, human, intuitive (and
frequently erroneous) understanding of what sets are; the purpose
of the exposition is to delineate some of the many things that one
can correctly do with them.
[...]
The principal concept of set theory, the one that in completely
axiomatic studies is the principal primitive (undefined) concept,
is that of "belonging". If x belongs to A (x is an element of A, x
is "contained" in A) we shall write x e A. [italics in the
original instead of quotation marks].
Page 2 of Thomas Hunberford's "Algebra" has:
In the Goedel-Bernays form of axiomatic set theory, which we shall
follow, the primitive (undefined) notions are "class",
"membership", and "equality". [...] A class A is defined to be a
set if and only if there exists a class B such that A e B.
In GB, then, the notion of "set" is defined, but only in terms of the
undefined notion of "class".
And so on and so forth.
--
======================================================================
"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 |
|
|
| Back to top |
|
| Martin Wanvik |
| Posted: Thu May 01, 2008 6:55 am |
|
|
|
Guest
|
Quote:
"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 |
|
|
| Back to top |
|
| Arturo Magidin |
| Posted: Thu May 01, 2008 6:55 am |
|
|
|
Guest
|
In article <oClSj.1209$7z4.15@newsfe13.ams2>, Mark <user@home.com> wrote:
Quote:
"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.
[...]
Quote: 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.
Quote: [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.
Quote: 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.
Quote: 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 |
|
|
| Back to top |
|
| Arturo Magidin |
| Posted: Thu May 01, 2008 6:55 am |
|
|
|
Guest
|
In article <oClSj.1209$7z4.15@newsfe13.ams2>, Mark <user@home.com> wrote:
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 |
|
|
| Back to top |
|
| Arturo Magidin |
| Posted: Thu May 01, 2008 6:55 am |
|
|
|
Guest
|
In article <oClSj.1209$7z4.15@newsfe13.ams2>, Mark <user@home.com> wrote:
Quote: I don't see how a logical theory can be based on the undefined.
Quoted from "Introduction to Logic and to the Methodology of Deductive
Sciences", by Alfred Tarski, translated by Olaf Helmer; Dover
Publications, Inc., New York, unabridged republication of the 9th
printing, 1961, of the 1946 second revised edition. ISBN
0-486-28462-X, pp 117 ff. What appears in the original in smallcaps
font I place in quotation marks.
We shall now attempt an exposition of the fundamental principles
which are to be applied in the construction of logic and
mathematics. The detailed analysis and critical evaluation of
these principles are tasks of a special discipline, called the
"methodology of deductive sciences" or the "methodology of
mathematics." For anyone who intends to study or advance some
science it is undoubtedly important to be conscious of the method
which is employed in the construction of that science; and we
shall see that, in the case of mathematics, the knowledge of that
method is of particular far-reaching importance, for lacking such
knowledge it is impossible to comprehend the nature of
mathematics.
The principles with which we shall get acquainted serve the
purpose of securing for the knowledge acquired in logic and
mathematics the highest possible degree of clarity and
certainty. From this point of view a method of procedure would be
ideal, if it permitted us to explain the meaning of every
expression occurring in this science and to justify each of its
assertions. It is easy to see that this ideal can never be
realized. In fact, when one tries to explain the meaning of an
expression, one uses, of necessity, other expressions; and in
order to explain, in turn, the meaning of these expressions,
without entering into a vicious circle, one has to resort to
further expression again, and so on. We thus have the beggining of
a process which can never be brought to an end, a process which,
figuratively speaking, may be characterized as an "infinite
regree" - a regressus in infinitum. The situation is quite
analogous as far as the justification of the asserted statements
of the science is concerned; for, in order to establish the
validity of a statement, it i snecessary to refer back to other
statements, and (if no vicious circle is to occur) this leads
again to an infinite regress.
By way of a compromise between that unattainable ideal and the
realizable possibilities, certain principles concerning the
construction of mathematical disciplines have emerged that may be
described as follows.
When we set out to construct a given discipline, we distinguish,
first of all, a certain small group of expressions of this
discipline that seem to us to be immediately understandable; the
expressions of this group we call "PRIMITIVE TERMS" or "UNDEFINED
TERMS," and we employ them without explaining their meaning. At
the samtime, we adopt the principle: not to employ any of the
other expressions of the discipline under consideration, unless
its meaning has first been determined with the help of primitive
terms and of such expressions of the discipline whose meanings
have been explained previously. The sentence which determines the
meaning of a term in this way is called a "DEFINITION", and the
expressions themselves whose meanings have thereby been determined
are accordingly known as "DEFINED TERMS."
We proceed similarly with respect to the asserted statements of
the discipline under consideration. Some of these statements which
to us have the appearance of evidence are chosen as the so-called
"PRIMITIVE STATEMENTS" or "AXIOMS" (also often refered to as
"postulates", but we shall not use the latter term in this
technical meaning here); we accept them as true without in any way
establishing their validity. On the other hand, we agree to accept
any other statement as true only if we have succeeded in
establishing its validity, and to use, while doing so, nothing but
axioms, definitions, and such statements of the discipline the
validity of which has been established previously. As is well
known, statements established in this way are called "proved
statements" or "theorems", and the process of establishing them is
called a "proof". More generally, if whithin logidc or mathematics
we establish one statement on the basis of others, we refer to the
process as a "derivation" or "deduction", and the statement
established in this way is said to be "derived" or "deduced" from
the other statement or to be their "consequence".
Contemporary mathematical logic is one of those disciplines which
are constructed in accordance with the principles just stated[.]
[...]
The method of constructing a discipline in strict accordance with
the principles laid down above is known as the "deductive method";
and the disciplines constructed in this manner are called
"deductive theories" [footnote omitted]. The view has become more
and more common that the deductive method is the only essential
feature by means of which the mathematical disciplines can be
distinguished from all other sciences; not only is every
mathematical discipline a deductive theory, but also, conversely,
every deductive theory is a mathematical discipline (according to
this view deductive logic is also to be counted among the
mathematical disciplines). We will not enter here into a
discussion of the reasons in favor of this view, but merely remark
that it is possible to put forward ponderable arguments in its
support.
Note in particular:
"we distinguish [...] a certain small group of expressions of this
discipline[;] the expressions of this group we call "PRIMITIVE
TERMS" or "UNDEFINED TERMS," and ->we employ them without
explaining their meaning<-."
And
"Some of these statements [...] are chosen as the so-called
"PRIMITIVE STATEMENTS" or "AXIOMS" [...]; ->we accept them as true
without in any way establishing their validity.<-"
Primitive terms are "employed without explaining their meaning", and
axioms are "accepted as true without in any way establishing their
validity".
--
======================================================================
"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 |
|
|
| Back to top |
|
| Arturo Magidin |
| Posted: Thu May 01, 2008 7:13 am |
|
|
|
Guest
|
In article <pjqj14ps39u6jacq45m2jbn9r8mrlo9bf9@4ax.com>,
Angus Rodgers <twirlip@bigfoot.com> wrote:
Quote: On Thu, 1 May 2008 15:57:45 +0000 (UTC),
magidin@math.berkeley.edu (Arturo Magidin) wrote:
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.
As I seem to be firmly stuck in "fools rush in" mode, this afternoon:
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".
Quote: 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?
Quote: 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.
Quote: 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.
But you have to have some kind of "primitive" notion, lest you devolve
into infinite regress.
Quote: 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.
Quote: A question like this was the very first thing I ever posted to Usenet,
back in 1992, and I'm still confused about it, in spite of having
worked through (and forgotten!) at least one textbook on mathematical
logic since then. So I mostly prefer to work at: (a) NOT thinking
about it; and (b) getting on with doing maths when I can (with the
intention of worrying about the foundations later).
Indeed; most people in fact never worry about the foundations, later
or ever.
Quote: 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!
--
======================================================================
"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 |
|
|
| Back to top |
|
| T.H. Ray |
| Posted: Thu May 01, 2008 7:43 am |
|
|
|
Guest
|
Magidin
Quote: 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.
User
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.
thr
Would it help you to understand what is meant
by leaving primitive concepts undefined, by going back to
what Hilbert said about geometry(in Grundlagen),"points,
lines and planes" can be replaced with "chairs,
tables and beer mugs"?
We have to be able to say what is true of things in
relation, without defining things out of existence.
The "is-ness" that you demand of a set is not the
important property of the concept of set. Realize that
in the Peano-Dedekind axioms, "number," "zero," and
"successor" are also undefined. Do you expect that
you have to know what a "number" "is" in order to do
arithmetic?
Tom |
|
|
| Back to top |
|
| David C. Ullrich |
| Posted: Thu May 01, 2008 7:49 am |
|
|
|
Guest
|
On Thu, 1 May 2008 12:55:46 +0100, "Mark" <user@home.com> wrote:
Quote: 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_.
Quote: 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?
David C. Ullrich |
|
|
| Back to top |
|
| Mark |
| Posted: Thu May 01, 2008 8:09 am |
|
|
|
Guest
|
"David C. Ullrich" <dullrich@sprynet.com> wrote in message
news:k0fj14lbq30cmomiiaoc1f513b35bu53jr@4ax.com...
Quote: 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?
David C. Ullrich
From, "Discovering Modern Set Theory by Winfried Just, Martin Weese,
American Mathematical Society"
http://books.google.co.uk/books?id=x74azoKzb_MC&pg=PA1&lpg=PA1&dq=%22our+perception+or+of+our+thought%22&source=web&ots=EEHVpDqYX5&sig=Tg7he0U1spehvNQf89utd6oefhA&hl=en
"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 |
|
|
| Back to top |
|
| Mark |
| Posted: Thu May 01, 2008 8:19 am |
|
|
|
Guest
|
"Martin Wanvik" <martinw@stud.ntnu.no> wrote in message
news:15978560.1209647023542.JavaMail.jakarta@nitrogen.mathforum.org...
I've read the articles, but they do not define what a collection is. The
wiki article for collection redirects to multiset. |
|
|
| Back to top |
|
| T.H. Ray |
| Posted: Thu May 01, 2008 8:42 am |
|
|
|
Guest
|
Quote: 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
user
I disagree. 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.
thr
Provide an example of a theory which is based on
the definitions of objects that are not part of
the theory.
Tom |
|
|
| Back to top |
|
| porky_pig_jr@my-deja.com |
| Posted: Thu May 01, 2008 8:54 am |
|
|
|
Guest
|
On May 1, 7:55 am, "Mark" <u...@home.com> wrote:
Quote: 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).
Quote: 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.)
Quote: 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".
Quote: 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.
Quote: So whats a collection?
An informal synonym for a set.
Quote: Would this be a good definition of colletion,
Just as a set, it's undefined.
Quote: 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.
Quote: 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. |
|
|
| Back to top |
|
| MoeBlee |
| Posted: Thu May 01, 2008 9:35 am |
|
|
|
Guest
|
On May 1, 4:55 am, "Mark" <u...@home.com> wrote:
Quote: Or could someone give a better definition?
Ordinarily, in Z set theories, we don't find definitions of 'element',
'class', or 'set', but it is not precluded that we define 1-place
predicates, 'is an element', 'is a class', 'is a set', and other
predicates in the following manner (though, of course the 2-place
predicate 'e', read as 'is an element of' (aka 'is a member of')
remains primitive):
x is an element <-> Ey xey
x is a class <-> (x=0 v Ey yex)
x is a set <-> (x is a class & Ey xey)
x is a proper class <-> (x is a class & ~x is a set)
x is an urelement <-> ~x is a class
x is a brace <->
ESf(x=<S f> & f is a function & dom(f)=S &
Ay(y e dom(f) -> (y is a cardinal & ~y=0))
x is a multi-set <->
ESf(x=<S f> & f is a function & dom(f)=S &
Ay(y e dom(f) -> (y is a natural number & ~y=0))
Note: 'brace' is a word I made up, to generalize the notion of a multi-
set.
MoeBlee |
|
|
| Back to top |
|
| |
Page 1 of 5 Goto page 1, 2, 3, 4, 5 Next
All times are GMT - 5 Hours
The time now is Tue May 13, 2008 7:38 am
|
|
