Re: MY LIST of the subsets of N
From: Brian Smith (brianscsmith_at_yahoo.com)
Date: 07/02/04
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Date: 2 Jul 2004 06:15:09 -0700
This looks very much like when I attempted to prove Cantor wrong. And
like my failed proof, you also forgot to include any infinite subsets.
For example, the set of all odd numbers is never generated. There
is still some use to the algorithm, though. When I learned a little
more math, I found that this algorithm is perfect for proving the
algebraic integers are countable.
david.ferguson1@cox.net (David P. Ferguson) wrote in message news:<2cf2d21.0407010607.bb3a27e@posting.google.com>...
> This takes into account Cantors diagonal argument which purports to
> show that there is no complete list of the finite Natural numbers.
> iF YOU CONSIDER THE SET OF NATURAL NUMBERS TO HAVE A DEFINITE
> CARDINALITY xO, THEN: THE SUMS REFERNCED HERE ARE (I HOPE) ALL HAVE A
> DEFINED NUMBER OF TERMS"
>
> >>first we need to know that the set of Natural numbers is "listable"
> >>A PROGRAM:
> (0) SET CLOCK:T=0
> (1) SET VARIABLE:n=0
> (2) n=n+1
> (3) AT:t=n/(n+1):PRINT:"n=",n,"TIME=",n."/",(n+1),"sec"
> (4) GO TO (2)
> (5) AT:t=1:PRINT"Elements of N+ lised. Time=1 sec.""
> THE PROGRAM WILL LIST THE ELEMENTS OF SET "N+"
>
> PROGRAM OUTPUT:
> n=1 Time=1/2 sec.
> n=2 Time=2/3 sec.
> n=3 Time=3/4 sec.
> - - -
> - - -
> Elements of N listed. Time=1 sec.
>
> The standard method of listing the elements of the set of ordered
> pairs of the set N is:
> Where (a,b): an elt of N x N = (participant a ,paticipant b): GP
> ("Greatest participant") = greater(a,b)
>
> (0) SET CLOCK:t=0
> (1) SET VARIABLE:n=0
> (2) n=n+1
> (3) At t= n/(n+1);Print "(",n,")",2n-1, SUB[List elts of NxN w/ GP =
> n]
> (3) GO TO (2)
> (4) At t=1 sec PRINT "Elements of N x N listed."
>
> PROGRAM OUTPUT:
> (1) 1 (1,1)
> (2) 3 (2,1),(2,2),(1,2)
> (3) 5 (3,1),(3,2),(3,3),(2,3),(1,3)
> (4) 7 (4,1),(4,2),(4,30.(4,4),(3,4),(2,4),(2,4)
> - - -
> - - -
> Elements of N x N listed.
>
> We use a similar program to list the subsets of N+.
>
> Given Set N+ ={1,2,3,4,...|n elt N+}
>
> A PROGRAM:
> (0) SET CLOCK:t=0
> (1) SET VARIABLE:n=0
> (2) n=n+1
> (3) At t= n/(n+1);PRINT "(",n,") ",2^(n-1),:
> (4) SUB[Print subsets with Greatest elt = n}
> (5) GO TO (2)
> (6) At t= 1 sec PRINT""SUBSETS LISTED"
>
> THE PRINOUT:
> (1) 1 {1}
> (2) 2 {2},{1,2}
> (3) 4 {3},(1,3},{2,3},{1,2,3}
> {4} 8 {4},{1,4},{2,4},{1,2,4},{3,4},{1,3,4},{2,3,4},{1,2,3,4}
> - - - -
> - - - -
> (Xo) - ...,{3,4,5,etc},{1,3,4,5,etc},(2,3,4,5,etc},{1,2,3,4,5,etc}=N+
>
> SUBSETS LISTED.
>
> THE INDEX NUMBER Ind(A) OF a subset A of set N+:
> LET SET A = {a1,a2,a3,...}: Then Ind(A) = Sum(2^(an):an elt A)
>
> EXAMPLE:
> FOR FINITE SETS:
> If A={1,2,3,4} Then Ind(A)=[2^0+2^1^2^2+2^3] = 1+2+4+8 = 15
> Set A is, therefore, the 15th elt of my list.
>
> FOR TRANSFINITER SETS:EXAMPLE
> IF A = N+ then Ind(A)=1+2+4+8+16+etc... (Xo terms)
>
> This is the sum of the numbers listed in collumn 2 of the printout.
> The set N+ itself id the last entry in the list because its index
> number is greater than the INDEX NUMBER of any proper subset of
> itself.
>
> Since "0" is not an elt of N+; I can use "0" as a place holder to
> represent elt of N+ not in a listed set.
>
> I will label the sets in the list sccording to their INDEX NUMBER:
>
> LET CANTOR APPLY HIS DIAGONAL PROCEEDURE TO MY LIST:
> S1 = 1,0,0,0,0,0,etc
> S2 = 0,2,0,0,0,0,etc
> S3 = 1,2,0,0,0,0,etc
> S4 = 0,0,3,0,0,0,etc
> S5 = 1,0,3,0,0,0,etc
> S6 = 0,2,3,0,0,0,erc
> S7 = 1,2,3,0,0,0,etc
> - - -
> - - -
> SN = 1,2,3,4,5,6,etc
>
> The diagonal of MY LIST is: Sd = 1,2,0,0,0,0,etc
> And the Compliment of diag: Sx = 0,0,3,4,5,6,etc
>
> (NOTE: The diagonal of MY LIST has ~2^Xo elements)
>
> (That there are lists with more than Xo elements is trivially
> provable)
>
> According to Cantor the set Sx is NOT in MY LIST
> But set Sx has an INDEX NUMBER.
> Ind(Sx) = 0+0+2^3-1+24-1+2^5-1+etc...
> Ind(Sx) = 0+0+4+8+16+32+etc..
>
> The last five entries in MY LIST are:
>
> INDEX NUMBER SET
> 1+3+0+8+etc... {1,2,0,4,5,6,etc}
> 0+0+4+8+etc... {0,0,3,4,5,6,etc} = Sx
> 1+0+4+8+etc... {1,0,3,4,5,6,etc}
> 0+2+4+8+etc... {0,2,3,4,5,6,etc]
> 1+1+4+8+etc... {1,2,3,4,5,6,etc} = N+
>
> So the set Sx is in My LIST afterall!
> There is a simple explanation for this:
> SUm{1+2+4+8+etc|Xo terms} = 2^(Xo-1)? is greater than than
> Sum{1+1+1+1+etc|Xo terms} = Xo.
>
> Cantor was restricted to considering lists of subsets which have the
> sane cardinality as the set in question. "N+ or N"
>
> This is because he was using the DIAGONAL of the array formed by
> the list of subsets. It should be clear to nearly everyone that the
> number of "elements" in the diagonal neccessarily equals not only the
> the number of elements in the master set but also the number of
> elements in the list!
> Cantor DID prove that every set has more subsets than subsets. In
> the process he proved that the the cardinality of the set of subsets
> of N ia greater thsn the cardinality of N (Xo). But he did not prove
> that there is no list of the subsets of N what-so-ever.
> This I respectfully submit.
> david.ferguson1@cox.net
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