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Re: Nathan counts the powerset
Posted:
Dec 22, 1998 4:32 AM
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Ulrich Weigand wrote:
> Mike Deeth <mad@ashland.baysat.net> writes:
>
> > You asked, "Where are the infinite subsets?" When does a set become an
> > infinite set? (remember the INFINTIGERS) ;-)
>
> A set is finite if there is a bijection from the set to some integer.
> A set is infinite if it is not finite. Where's the problem?
>
>
In a previous post I showed a bijection between the members of the powerset of
natural numbers and the natural numbers. Several peaple commented that, although the
bijection was correct, and mapped EVERY finite subset, it missed all the infinite
subsets. ie. Which number corresponds with the set of ALL even numbers? I produce
the number "...10101010". Now they said, "if that number is infinite, its not a
natural number. If that number is finite, it doesn't map the infinite sets" There
appears to be no way out of this, catch 22, situation.
The table below shows a bijection between a Natural number and a Set with the same
number of members. None can doubt this bijection continues into the infinite. Lets
say that the last row of the table contains the infinite set of all natural numbers.
What natural number is it in bijection with? (Remember, the number must be a
Natural)
N Set-of-N
---------------------
1 {1}
2 {1,2}
3 {1,2,3}
4 {1,2,3,4}
5 {1,2,3,4,5}
. .
. .
. .
? {1,2,3, ... }
Hint: there are no infinite sets.
Nathan Deeth :-)
Age 11
> > In the past, my usage of the term "set" was not correct/consitant. Sometimes I
> > would say things like: "the set of all natural numbers" BUT - There are no
> > infinite sets. All Sets have a specific number of members.
>
> This is an extremely non-standard use of the word 'set'. As this word is
> usually used throughout mathematics, there are indeed infinite sets --
> the most prominent of them being the set of all natural numbers.
>
> If you want to insist on talking only about finite sets, I'd advise you
> to state this clearly in advance; otherwise you only create confusion.
>
> > Just as there are infinitly many finite numbers. There are infinitly many sets
> > that contain a finite number of members.
>
> True. On the other hand, there are also infinitly many sets that contain
> infinitly many members each.
>
> > Axiom 2 (succesor) is responsible for
> > both the infinity of numbers and of sets.
>
> The Peano axioms talk only about existence of natural numbers, they don't talk
> about sets at all. If you want to reason about sets in an axiomatic theory,
> you'll have to choose one that actually talks about sets -- the one that
> is mostly used would be Zermelo-Fraenkel set theory.
>
> > All the infinite subsets you talked about can be attained by applying (forever)
> > axiom 2 (succesor).
>
> The Peano axioms talk about natural numbers. If you identify natural numbers
> with finite sets, you might also say they talk about finite sets, but that's
> it. Most definitely the Peano axioms don't talk about infinite sets.
>
> > In the bijective mapping below, every specific finite set maps to a specific
> > finite binary string. There is NO FINITE LIMIT to the size of sets that can be
> > mapped.
> >
> > {2,4,6,8,10, ...} <-> ...1010101010
> > {1,2,3,4,5,6,7, ...} <-> ...1111111
> > {1,4,9,16, ...} <-> ...1000000100001001
> > {2,3,5,7,11, ...} <-> ...10001010110
> >
> > The "..." used above repressents a list of non-specified (but finite) set
> > members or a string of non-specified (but finite) binary digits. The "..."
> > doesn't repressent infinite numbers of members or digits.
>
> Then you still aren't talking about infinite sets, but maybe sequences of
> finite sets.
>
> > Infinity is are process not a value. The infinite subsets are located in *the
> > process* not in the table. When you realize this, you will realize that there
> > is only one infinity - absolute never ending infinity.
>
> Nobody is talking about a 'value infinity', whatever this is supposed to mean.
> We are talking about sets, i.e. collections of objects. If you want, you can
> use e.g. ZF set theory to reason about them, but even in naive set theory
> (i.e. not formulated axiomatically) certainly also infinite collections of
> objects, e.g. all natural numbers, all even numbers, etc. exist as sets.
>
> --
> Ulrich Weigand,
> IMMD 1, Universitaet Erlangen-Nuernberg,
> Martensstr. 3, D-91058 Erlangen, Phone: +49 9131 85-7688
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