Re: MY LIST of the subsets of N

From: David P. Ferguson (david.ferguson1_at_cox.net)
Date: 07/02/04 

Date: 2 Jul 2004 14:52:15 -0700

I,david.ferguson1@cox.net , do not perfectly understand what you mean
by:
 "For example, the set of all odd numbers is never generated."

Allowing for this;The set of all odd (natural) numbers is in my list.
 The set O = {1,3,5,7,9,11,...} has location: 2^0+2^2+2^4+2^6+etc..:

 [There is as set O* of odd "numbers" which is not in the list but
that set is not a proper subset of N.]
 O* = {2*n-1|n an elt of N}.
 O* has cardinality Xo [O* has Xo elements] while O has Xo/2 elements:
SEE my:
  Newsgroups: sci.math "For any sets A, B: IfA is a proper subset of
B..."

  For an explanation of why I can claim that Card(O)<Xo.

 
brianscsmith@yahoo.com (Brian Smith) wrote in message news:<12f59340.0407020515.77b3be22@posting.google.com>...
> 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

Relevant Pages

  • Re: MY LIST of the subsets of N
    ... This looks very much like when I attempted to prove Cantor wrong. ... > PROGRAM OUTPUT: ... > Cantor was restricted to considering lists of subsets which have the ... > the process he proved that the the cardinality of the set of subsets ...
    (sci.math)
  • Re: Of course Cantors diagonlisation is ok was Re: Wheres respect?....
    ... Cantor's proof shows that all lists of real numbers are ... A construction which proves there is a ... Let's remember that the Cantor construction is to construct ... That number has a rule for the n-th digit. ...
    (sci.math)
  • Re: MY LIST of the subsets of N
    ... > LET CANTOR APPLY HIS DIAGONAL PROCEEDURE TO MY LIST: ... > Cantor was restricted to considering lists of subsets which have the ... > the process he proved that the the cardinality of the set of subsets ... The main problem with any algorithm to enumerate a set is ...
    (sci.math)
  • Re: MY LIST of the subsets of N
    ... > PROGRAM OUTPUT: ... > LET CANTOR APPLY HIS DIAGONAL PROCEEDURE TO MYLIST: ... > Cantor was restricted to considering lists of subsets which have the ... > the process he proved that the the cardinality of the set of subsets ...
    (sci.math)
  • Re: Collatz .. Nice to see this problem still tickling the minds of the same fellas...
    ... Rules 1 and 2 equate to the even and odd rule with the ... I call such a list a Sequence Vector. ... So after you discard every one of those enumerated lists ...
    (sci.math)
Loading