Re: Questioning the defintions of set and element.

On May 1, 1:47 pm, "Mark" <u...@xxxxxxxx> wrote:
"Arturo Magidin" <magi...@xxxxxxxxxxxxxxxxx> wrote in message

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In article <oClSj.1209$7z4...@xxxxxxxxxxxxx>, Mark <u...@xxxxxxxx> wrote:

"Arturo Magidin" <magi...@xxxxxxxxxxxxxxxxx> wrote in message
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In article <AijSj.861$NZ7....@xxxxxxxxxxxxx>, Mark <u...@xxxxxxxx>
wrote:

"David C. Ullrich" <dullr...@xxxxxxxxxxx> wrote in message
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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.

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.

Perhaps you should learn how not to assume things.







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

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

Then why are you posting to a *mathematics* discussion board??? As
you just admit, your question is clearly more of a "philosophical"
nature, and is one that mathematicians avoid.

.

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