Re: infinity


Randy Poe wrote:
> zuhair wrote:
> > Randy wrote:
> >
> > You are trying at the same time to say ordering doesn't matter
> > to set identity ("they're the same set, so they should have the
> > same cardinality") and ordering does matter ("your function didn't
> > produce my set").
> > ----------------------------------------------
> >
> > Order, set, and Cardinality: these three terms I am sure bears a lot of
> > confusion, not among members in this group only but even among those
> > who invented them.
> >
> > Anyhow, about my question , in order for thing to be clear , lets say
> > that their is Order BEFORE set generation and Order AFTER set
> > generation.
>
> No there is not, for three reasons.
>
> 1. Sets are not "generated", they are defined by defining their
> elements.
>
> 2. Once the elements are defined, the set is precisely the collection
> of everything which meets that definition. Nothing more, nothing
> less. There is no "before" and "after" the non-existent "generation".
>
> 3. There is no order in a set definition.

Why? i can define set by series. so I say for example I want the set in
which its' members are the same as the that so and so series.

For example let me take the following series (which is order generated
of coarse) S = 1,3,5,7,9,11 Now this is a series it is not a set.

But I can define set P as the collection of the terms present in S.(
see their is not order here)

so P = {1,5,3,7,9,11}

This set P is called the field of the serial relation in S.

of coarse P is not a series( review Introduction to mathematical
philosophy by Bertrand Russell) , it is a set for it is the same for
all
arrangments, the only thing which defines P is it's members, the order
is not important in P.

I agree with you their is no order in a set definition , you see I
didn't define P above by order , I defined it by being identical to
terms in series
S without respect to their order in that series.

That's what I meant by Order befor set definition ( since you object to
the term generated).

Tell me is their something wrong about P being a set . If so then alot
can be said of how Cantor regarded the field of the series of
(Omega-0)s as set that can have Cardinality Aleph-1 . and were the
series itslef had Ordinal number Omega-1. ( Review Introduction to
mathematical philosophy, and principles of mathematics , both by
Russell.)

Regards
Zuhair

>
> > Order befor set generation is involved in the set definition itself,
>
> See #3.
>
> > ie it is part of the propositional function
>
> No, membership definitions do not imply an order. See #2.
>
>
> > responsible for collecting members in the set, but not after collecting
> > them.
>
> No, everything either is or is not in the set. Nobody has to
> "collect" anything. See #1. Do you think there are some loose
> primes out there that might escape collection, so they're
> currently classified as composite?
>
> > For example when you say the set of the first five naturals starting
> > from 0.
>
> And the set of people named John? And the set of sci.math
> participants? And the set of insects? And the set of stars
> in the Milky Way galaxy?
>
> See #2.
>
> ? Here order was responsible for picking up the members into the
> > set
>
> No, the members don't need to be "picked up". See #1.
>
> > but after the set was formed rearrangment of its members( order
> > after set generation) will not change the set identity.
>
> No, there is no orer, and no "before" and "after". See #2 and #3.
>
> > but if I say for example the set of the first 10 naturals, see here I
> > changed the order defining the set and so I will have another set.
>
> No, there is no "order defining the set".
>
> > Now back to my question
>
> Your premises are completely misguided on every point. Your
> questions arising from those premises are therefore meaningless
> and based on false assumptions.
>
> Whatever your question is about, if it is based on the above
> it is not a question about sets, has no meaning in set theory
> and can not be addressed in the context of set theory.
>
> - Randy

.

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