Re: Cantor's definition of set

On Oct 25, 8:47?pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Oct 25, 12:12 pm, John Jones <jonescard...@xxxxxxx> wrote:

Cantor said:

'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 [Anschauung] or of our thought.'
Further, except in a multiset, every element of a set must be unique-
no two members may be identical.

Aside from Cantor's work and notions, I wonder whether such
qualfications as you've made above stem from notational questions
rather than anything more substantive than the axiom of
extensionality, which stands clear enough. For example, that {0 0} =
{0}, or even a general statement of such a thing, doesn't require much
more than simply reminding as to the axiom of extensionality.

(By the way, the notion of a multiset is also easily captured in Z set
theory.)

My observation is this:
Surely, these definitions disallow a set whose individual members are
characterized by being sequenced, or heirarchical, or derive their
properties from being in a collection.

I see no such sure inference as that. Though I'm not particularly
loyal to Cantor's defintion, I don't see that having "definite,
distinct objects" in any way precludes them from also having such
properties as being in the ranges of certain sequences, etc.

For how can we have a set of numbers if, from our definition above, no
two members of a set can be identical?

What's stopping us from it?

1) If no two members are identical, then either a number is a
composite (which may allow a heirarchy or sequence)

In set theoretical terms, what's a "composite" and whatever a
"composite" is, why can't it be a definite, distinct object?

or
2) if number is not a composite then without its members exhibiting a
relationship, a set of numbers is simply a set of numerals.

A numeral is usually understood to be a certain kind of term in a
language. A set may have numerals as members. But I don't see why you
imagine that a number (no matter how you define 'number') can't be a
member of a set without being also a numeral.

My question is this: If the above objection against the concept of 'a
set of numbers' (real numbers for example) stands, then how ought we
to correctly define ' a set of numbers'?

The objection doesn't stand, so there's no problem any greater than
just defininig 'number' however you like. Once you've done that, then
'a set of numbers' is defined as easily as:

x is a set of numbers iff for all n, if n is a member of x then n is a
number.

MoeBlee

OK. But if the members of a set have nothing in common, then how can
we have a set of numbers, for numbers are derived from each other -
they have something in common.

.

Relevant Pages

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