Re: MY LIST of the subsets of N
From: Andrzej Kolowski (akolowski_at_hotmail.com)
Date: 07/11/04
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Date: 11 Jul 2004 16:24:47 -0700
david.ferguson1@cox.net (David P. Ferguson) wrote in message news:<2cf2d21.0407081704.5cfb5b6d@posting.google.com>...
> akolowski@hotmail.com (Andrzej Kolowski) wrote in message news:<a1fa83d9.0407041306.16435a7@posting.google.com>...
> > david.ferguson1@cox.net (David P. Ferguson) wrote in message news:<2cf2d21.0407031637.61db8765@posting.google.com>...
> > > akolowski@hotmail.com (Andrzej Kolowski) wrote in message news:<a1fa83d9.0407020756.293c6e59@posting.google.com>...
> > > > 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"
> > > > >
> > > >
> > > > [snip]
> > > >
> > > >
> > > > > 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.
> > > > >
> > > >
> > > > The latter phrase is true for any other unbounded subset also.
> > > >
> > > >
> > > > > 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.
> > > > >
> > > >- - - - - - - -- -- - - - -- - - - - - - - -- - - - - -- - - - - - -
> > > --
> > > Comment 1
> > > This seems interesting to me but I wish you would explain it so I
> > > can understand it better: I note that you claim that the "range?" is
> > > the set of natural numbers: I think that this practice of presuming
> > > the range to be something is dangerous. We know that every set{even
> > > the set N) has more subsets than elements. This means that N is a
> > > proper subset of the range of the function if the domain is the set of
> > > all subsets of N. --or not?--
> > > f:Domain=> range. Right?
> >
> >
> > Every function has a domain and a range. I think now I misinterpreted
> > what you were doing, especially with respect to 0. Below I offer an
> > alternative explanation. First a side note -
> >
> > You defined N+ as follows:
> >
> > "N+ = {1, 2, 3, 4, ... | n elt N+}"
> >
> > which is of course a circular definition: the definition of N+
> > has N+ itself on the right side of the equation. Perhaps you
> > had a misprint.
> >
> > ==========================================================================
> >
> > The following is my current understanding of what you were
> > trying to do:
> >
> > You are defining an "index" for every subset A of N as follows:
> >
> > If A = {a1, a2, a3, ... } then
> >
> > Ind(A) = 2^(a1 - 1) + 2^(a2 - 1) + 2^(a3 - 1) + ...
> >
> > For finite sets, this is not an unreasonable index, and
> > in fact it is a one-to-one function from the set of finite
> > subsets of N into N itself: that is, if Ind(A) = Ind(B),
> > then A = B. You can show this by noting that there is a
> > correspondence between Ind(A) and a binary expansion: for
> > example, if A = {2, 5, 6, 8}, then you could represent your
> > index in binary as:
> >
> > Ind(A) = .01001101
> >
> > But for infinite sets, what is the meaning of Ind(A) ?
> > I believe you are thinking of it as a member of some kind of ordered
> > infinite set. You are thinking that bigger subsets are farther
> > along in the list: that is, if A is contained in B, then
> > Ind(A) < Ind(B).
> >
> > Again writing the index in reverse binary,
> >
> > Ind(N) = .111111111...
> >
> > Note that if E = {2, 3, 4, 5, 6, ...} then
> >
> > Ind(E) = .011111111...
> >
> > Note that if D = {1}, then
> >
> > Ind(D) = .100000000...
> >
> > So what your index defines is actually a function from the
> > subsets of N to the real numbers between 0 and 1. It is not
> > a one-to-one function (because Ind(E) = Ind(D), for example), but
> > it is a surjection. This is sufficient to show that the set of
> > all subsets of N has cardinality at least as large as that of the
> > set of all real numbers between 0 and 1. The binary version of
> > your index function also does have the property that if A is a subset
> > of B, then
> >
> > Ind(A) < Ind(B).
> >
> > There is clearly a partial inverse function: given a real number
> > between 0 and 1, express it in binary and find the corresponding
> > subset of N: for example,
> >
> > x = .011011001,
> >
> > then define H(x) = {2, 3, 5, 6, 9}.
> >
> > The function H is ambiguous for numbers which terminate in
> > all 1's because there are two ways to write such numbers. To remove
> > the ambiguity, always take the expansion terminating in 0's: for
> > example, if
> >
> > x = .01001000111111111..., then also
> >
> > x = .01001001000000000...,
> >
> > so define H(x) = {2, 5, 8}.
> >
> > With this definition, H is a well-defined function and it
> > is one-to-one. Moreover,
> >
> > Ind(H(x)) = x,
> >
> > although it is not true that H(Ind(A)) = A.
> >
> >
> > I think this is a clearer way of describing what you were
> > actually doing: defining a function from the subsets of N
> > onto the real numbers between 0 and 1. This is the sense in
> > which you were creating a "list". What most mathematicians
> > would mean by "list" is a sequence of items indexed by the
> > positive integers. Clearly your "list" is "longer" than
> > the positive integers.
> >
> > None of this is new. You have only rephrased very well known
> > results in your own language.
>
> >
> > Andrzej
> This is sure some juicy feedback,Andrzej! You are a product of your
> experience and education, just as I am. I have been working under the
> delusion that we can consider not only 0.00011100110000000*
> as an ACTual number but also the number0.00011101111111111*
> We might be able to proceed with some not impossible working
> assumptions.
> We <might> consider ther to be a MASTER <greatest> number much in
> the sense that the ordinal number "omega" or its cardinal assciate Xo
> are greater than any pf their predecessors. We might than be able to
> express the cardinality ]
> of a set A = {1,4,6,25,39} to be [1/1+4/4+6/6+25/25+39/39] and
> similarly for the so called infinite sets.
> If the set is infinite then it's cardinality is an infinite set
> of summands. In like manner we can consider each and every
> binary expression to have a unique value or identity. No more any of
> this stuff about numbers having an ambiguious representation.
> We have then the ability to say that A is a proper subset of
> B.(lease use the notation A < B.) implies that Card(A) is < Card(B).
> and so on. All this is somewhat classified and I can't go into full
> detail now since my ideas are still in formative stage. I am going to
> give you comment all the attention I am sure it deserves. I am
> impressed so far and I hope to have a better reply soon>
> Thank you Andrzej!
> David P. Ferguson 7/8/04
> >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>
Thanx for the comments. I must note again that you are not
using the standard definition of cardinality. You have defined
a function g from P(N) to [0, 1] which is one-to-one on all but
a countable set of subsets, where P(N) is the set of all subsets
of N. Moreover, it has the nice property that if A is a proper subset
of B (i.e., A < B), then g(A) < g(B). Thus you have a function
which is one-to-one except on the set of complements of finite sets,
and which preserves (partial) order.
But it is wrong to assume that g(A) can be identified as the
cardinality of A. That is simply not the standard definition
and not equivalent to it. You can invent your own terminology
of course, but it is not a good idea to re-invent standard
terminology. It just leads to confusion.
Andrzej
> > > Ferguson 7/3/04
> > > - - - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - -- -
> > > - --
> > > > You thus appear to be defining a function f(S) whose domain includes
> > > > any subset S of N, and which takes values in N + {0}.
> > > >
> > > > What you need is that this function is 1-1.
> > > >
> > > > It isn't. Any unbounded subset S of N is such that f(S) = 0.
> > > > - - - - - - - - - -- - - - - - - - - - - - - - - - -- - - - - -- -- - -
> > > I wish someone would explain the "Any unbounded.... f(s) = 0." to
> > > me.
> > > Ferguson 7/3/04
> > > - - - - - -- - - - - -- - -- - - - - - -- - - - - -- - - -- - - -
> > > - - -
> > > >
> > > > [snip]
> > > >
> > > > > Cantor DID prove that every set has more subsets than subsets.
> > > >
> > > > Doubtful!
> > > >
> > > > > 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.
> > > >
> > > >
> > > > He did, if by "list" you mean a function from the subsets of N
> > > > into N itself which is one-to-one. If this is not what you mean
> > > > by "list", then either (1) you need to define exactly what you mean,
> > > > or (2) a trivial construction is possible.
> > > >
> > > >
> > > > Andrzej
> > > >
> > > > > This I respectfully submit.
> > > > > david.ferguson1@cox.net
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