Re: Cantor's definition of set

On Thu, 25 Oct 2007 12:12:45 -0700, John Jones <jonescardiff@xxxxxxx>
wrote:


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

Well, actually, this is indeed the case, even if no sets are involved:

No two [different] things can be identical.

The other way round: Let's consider a thing A and a thing B (which is
not necessarily different from A). If A and B are identical (A = B),
there actually is only _one_ thing.

Example: Let's consider the first man on the moon, and let's consider a
guy called Neil Armstrong (a former test pilot and astronaut). If the
first man on the moon is identical with Neil Armstrong (with other
words, if Neil Armstrong is the first man on the moon), then we are
actually talking about exactly _one_ man, namely Neil Armstrong who (in
this case) is the first man on the moon.

Now if we consider the set of all first men on the moon, then Neil
Armstrong (i.e. the first man on the moon) is an element in this set;
actually he's the only element in this set. Though the first man on the
moon is "also" an element of this set. (But since Neil Armstrong IS the
first man on the moon, etc.)


[...]

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'?

I'm sorry, didn't see any objection against the concept of 'a set of
numbers'.

Consider the following set of numbers:

{1, 2, 3}

Let's call it "M" for short. Hence:

M = {1, 2, 3}.

Now
1 e M,
and
2 e M,
and
3 e M.

With other words, the numbers 1, 2, and 3 (and only these numbers) are
elements in M. These numbers are pairwise distinct: 1 =/= 2, 1 =/= 3,
and 2 =/= 3.

Hence M contains exactly _three_ numbers:

|M| = 3.

So where's the problem with this set M?


F.

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