
Re: Nathan counts the powerset
Posted:
Dec 22, 1998 6:39 AM


Mike Deeth <mad@ashland.baysat.net> writes:
>Ulrich Weigand wrote:
>> 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".
This is not a number; its an string of digits. Now finite strings of digits are usually interpreted as representations of natural numbers. But this does not work for infinite strings of digits. If you don't agree, please tell us to which natural number the string given above corresponds ...
>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.
That's a somewhat strange way to state it; every natural number is finite (you might say that's the definition of 'finite'), and again by definition no infinite set is bijective to any finite set.
>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 SetofN > > 1 {1} > 2 {1,2} > 3 {1,2,3} > 4 {1,2,3,4} > 5 {1,2,3,4,5} > . . > . . > . . > ? {1,2,3, ... }
There is no last row. That would contain the 'last' natural number, but according to the Peano axioms which you are so fond of, every natural number has a successor, hence there is no last natural number.
Your table contains in the left column all natural numbers, and in the right column a bunch of finite sets. No single infinite set occurs.
>Hint: there are no infinite sets.
Oh, really :) Might I ask according to which axiomatic foundation you decide about existence or nonexistence of sets?
 Ulrich Weigand, IMMD 1, Universitaet ErlangenNuernberg, Martensstr. 3, D91058 Erlangen, Phone: +49 9131 857688

