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Topic: Nathan counts the powerset
Replies: 144   Last Post: Jan 26, 1999 9:26 AM

 Messages: [ Previous | Next ]
 Ulrich Weigand Posts: 72 Registered: 12/12/04
Re: Nathan counts the powerset
Posted: Dec 22, 1998 6:39 AM

>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 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, ... }

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 non-existence of sets?

--
Ulrich Weigand,
IMMD 1, Universitaet Erlangen-Nuernberg,
Martensstr. 3, D-91058 Erlangen, Phone: +49 9131 85-7688

Date Subject Author
12/20/98 Nathaniel Deeth
12/20/98 Ulrich Weigand
12/20/98 Bob Street
12/20/98 Ulrich Weigand
12/22/98 Bob Street
12/22/98 Ulrich Weigand
12/21/98 Nathaniel Deeth
12/21/98 Ulrich Weigand
12/24/98 Nathaniel Deeth
12/24/98 Dik T. Winter
12/25/98 Nathaniel Deeth
12/26/98 Dik T. Winter
12/27/98 jsavard@ecn.ab.ca
12/28/98 Math Icon
12/28/98 David C. Ullrich
12/25/98 Nathaniel Deeth
12/21/98 graham_fyffe@hotmail.com
12/22/98 Nathaniel Deeth
12/24/98 Nathaniel Deeth
12/21/98 David C. Ullrich
12/22/98 Nathaniel Deeth
12/22/98 Bob Street
12/23/98 Nathaniel Deeth
12/24/98 Bob Street
12/25/98 Nathaniel Deeth
12/25/98 Bob Street
12/26/98 Bob Street
12/24/98 Bob Street
12/24/98 Virgil Hancher
12/25/98 Nathaniel Deeth
12/27/98 jsavard@ecn.ab.ca
12/28/98 Robert Harrison
12/22/98 Bob Street
12/22/98 Ulrich Weigand
12/23/98 David C. Ullrich
12/24/98 Nathaniel Deeth
12/24/98 Bob Street
12/25/98 Nathaniel Deeth
12/24/98 David C. Ullrich
12/24/98 Bob Street
12/25/98 Nathaniel Deeth
12/25/98 Nathaniel Deeth
12/25/98 Bob Street
12/25/98 Nathaniel Deeth
12/25/98 Arturo Magidin
12/26/98 Bob Street
12/27/98 Nathaniel Deeth
12/27/98 Nathaniel Deeth
12/28/98 Arturo Magidin
12/28/98 Bob Street
12/30/98 Nathaniel Deeth
12/30/98 Bob Street
12/30/98 Matt Brubeck
12/30/98 Nathaniel Deeth
12/30/98 Bob Street
12/31/98 Nathaniel Deeth
12/31/98 Bob Street
12/31/98 modern life is rubbish
1/3/99 Nathaniel Deeth
1/1/99 standebj@SLU.EDU
1/3/99 Nathaniel Deeth
1/5/99 Bob Street
12/26/98 David C. Ullrich
12/27/98 Nathaniel Deeth
12/28/98 Math Icon
12/28/98 David C. Ullrich
12/25/98 Robert Harrison
12/25/98 Robert Harrison
12/26/98 David C. Ullrich
12/26/98 David C. Ullrich
12/26/98 David C. Ullrich
12/26/98 Bob Street
12/27/98 jsavard@ecn.ab.ca
12/28/98 Nathaniel Deeth
12/28/98 Ronald Bruck
12/28/98 feldmann4350@my-dejanews.com
12/29/98 John Savard
12/30/98 Nathaniel Deeth
12/31/98 Bob Street
12/31/98 Nathaniel Deeth
12/31/98 John Savard
12/31/98 Planar
1/2/99 Bob Street
1/3/99 Nathaniel Deeth
1/5/99 Bob Street
1/3/99 Nathaniel Deeth
1/26/99 Clothes Rod
12/31/98 Planar
1/3/99 Nathaniel Deeth
1/3/99 David C. Ullrich
1/4/99 Bob Street
1/4/99 John Savard
12/31/98 John Savard
1/3/99 Nathaniel Deeth
1/3/99 standebj@SLU.EDU
1/4/99 John Savard
12/31/98 Dave Seaman
1/3/99 Nathaniel Deeth
1/3/99 Ian Storey
1/4/99 Dave Seaman
12/31/98 John Savard
1/3/99 Nathaniel Deeth
1/4/99 John Savard
12/28/98 Bob Street
12/21/98 Nathaniel Deeth
12/21/98 Ulrich Weigand
12/22/98 Nathaniel Deeth
12/22/98 Ulrich Weigand
12/24/98 Nathaniel Deeth
12/24/98 David C. Ullrich
12/28/98 Nathaniel Deeth
12/28/98 Nathaniel Deeth
12/28/98 David C. Ullrich
12/30/98 Nathaniel Deeth
12/31/98 David C. Ullrich
1/1/99 standebj@SLU.EDU
1/3/99 Nathaniel Deeth
1/3/99 standebj@SLU.EDU
12/24/98 Ulrich Weigand
12/28/98 Nathaniel Deeth
12/20/98 David C. Ullrich
12/21/98 Kirby Cook
12/21/98 Nathaniel Deeth
12/21/98 John Starrett
12/21/98 Kirby Cook
12/21/98 John Savard
12/21/98 Jeremy Boden
12/21/98 John Savard
12/22/98 Nathaniel Deeth
12/22/98 Bob Street
12/22/98 John Savard
12/21/98 Barrie Snell
12/22/98 David C. Ullrich
12/23/98 Barrie Snell
12/22/98 Bob Street
12/22/98 zenevsky@wwa.com
12/23/98 Bob Street
12/22/98 John Starrett
12/22/98 Jeremy Boden
12/22/98 Jeremy Boden
12/23/98 Jeremy Boden
12/24/98 David C. Ullrich
12/24/98 John Starrett
12/30/98 Jeremy Boden
12/31/98 brian tivol