Re: Cantor's definition of set

On Oct 27, 11:19?pm, G. Frege <nomail@invalid> wrote:
On Sat, 27 Oct 2007 14:22:28 +0200, Jan Burse <janbu...@xxxxxxxxxxx>
wrote:



A sequence is something like this:

x1, x2, x3, ...

Thus it is a function from the natural numbers to
a certain domain. Namely the function f with:

f(i) = xi

Completely agree with you.



This way one can also model finite and uncountably
infinite, or multi dimensional "sequences" etc...

Actually, in set theory sequences are not just "modeled" this way, but
sequences ARE (certain) functions.



But a sequence is not a set.

Sorry. You are wrong here. Functions, sets, <whatever> ARE sets in set
theory. It's even worse:

"Our contemporary orthodoxy: To show that there are so-and-sos
is to prove 'So-and-sos exist' from the axioms of set theory."

(Penelope Maddy, Mathematical Existence)



A set is ignorant on duplicates.

Right. But not sequence.



But a sequence isn't.

Exactly.

Consider the (finite) sequence

(a_i) = (x x y y z z).

Here the sequence is a function a(i) (which usually is written a_i) with

a(1) = x
a(2) = x
a(3) = y
a(4) = y
a(5) = z
a(6) = z.



Maybe you should first work out what a function is.

Exactly. Hint: It's a (special) set - at least in the context of set
theory.



This thread is about sets...

Exactly.

F.

--

E-mail: info<at>simple-line<dot>de

It's my night off so you are getting more than your fair share of
replies.
Q: How is a function like a set? We can call it a set, but why?

.

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