Re: Cantor's definition of set

John Jones schrieb:
But a sequence is a certain kind of set that "codes" the order.
There's no conflict in that.

In that case, the set should incorporate that code in the name of the
set. So instead of saying 'a set of numbers', I should also include in
the name of the set the application for generating numbers. But in
that case, I simply have an application.

Hey guys, have you totally lost sight. 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

This way one can also model finite and uncountably
infinite, or multi dimensional "sequences" etc...
Note: the plural, sequences, the aim is to define a
whole set of sequences:

finite sequence: The domain is a finite set.
For example the sequence:

x1, x2, x3.

Corresponds to the function:

f(1) = x1,
f(2) = x2,
f(3) = x3.

uncountably infinite sequence: The domain is a
uncountable infinite set. For example the sequence:

xs, xt, ... where r, s are the reals

Corresponds to the function:

f(s) = xs,
f(r) = xr,
etc..

multi dimensional sequence: The domain is
for example a cartesian product. For example the
sequence:

x11, x12, x13, ...
x21, x22, x23, ...
x31, x32, x33, ...
...

Corresponds to the function:

f(<i,j>) = xij


A further basis for defining a sequence might be that the
domain is totally ordered or some such. This means
that for the domain D, we also stipulate an order <= subset
DxD, which satisfies:

http://en.wikipedia.org/wiki/Total_order

But a sequence is not a set. A set is ignorant on
duplicates. But a sequence isn't. Take the two finite
sequences:

x1=1, x2=2, x3=2. And
x1=1, x2=1, x3=2.

These are different sequences. But viewed as a function
they have the same domain, namely the set {1,2}. Functions
(graphs) live already in set theory, but they need a
little apparatus to work with. What is a function? Maybe
you should first work out what a function is.

Further a sequence might be related to a total order
or some such. A set has not any order involved per se.
You don't find a set theory axiom that deals
with an order <=.

Orders come later, when we use set theory. We can define
them for example by membership inclusion. Or by any other
means. And we can abstract them, to ordinals. If you want
to discuss ordinals or functions, you should maybe start
another thread. This thread is about sets...

Bye



.

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