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Topic: Continuum Hypothesis Solution Posted
Replies: 44   Last Post: May 1, 1999 5:30 PM

 Messages: [ Previous | Next ]
 Dave Seaman Posts: 2,446 Registered: 12/6/04
Re: Continuum Hypothesis Solution Posted
Posted: Apr 16, 1999 1:58 PM

Webster Kehr <websterkehr@sprintmail.com> wrote:

>What I prove is that any number system with a countable base, a countable
>width and a countable number of dimensions is a countable number system.
>This should be obvious, but it is not so easy to prove. I actually don't
>deal with dimensions in my complete paper, but I did deal with it in a 1993
>paper. It is the first two items that are difficult to deal with, once
>those are taken care of the dimensions thing is not so difficult.

Are you saying that uncountable sets exist, but R and P(N) just don't
happen to be among them? Can you give an example of what you consider
to be an uncountable set?

>Let me talk about axioms. I mentioned this before, but there are so many
>posts to this thread (some of them are starting to drop off now) that it
>would take a week for you to find it. Cardinality is based on the "size" of
>a set. If two sets have different cardinal numbers, then they had better
>have a different "number of elements." The very concept and definition of
>"cardinality" is one of size.

What do you mean by "number of elements"? Most people understand that
to mean the cardinality, but that is evidently not what you mean, since
your statement would then reduce to "if two sets have different
cardinal numbers, then they had better have different cardinal
numbers."

I take it you did not intentionally utter a tautology, and therefore
you need to explain what your meaning of "size" is.

Then, if your definition turns out to be inconsistent with the one the
rest of us use, why should we take any of your claims about "size"
seriously? Are you claiming Cantor was wrong about the size of sets
because he wasn't enlightened enough to choose your definition of
"size"?

>Axioms are totally subordinate to the concept of "size." Axioms in Set
>Theory were designed to provide a shortcut mechanism to determine the "size"
>of sets.

None of the axioms says anything about size. That is a defined
concept. If they did, then presumably we would not be in disagreement
about the meaning of the term.

>So were definitions involving mappings. But neither Axioms nor
>definitions involving mappings are superior to the concept of size. For
>example, if a mapping definition, such as a definition of uncountable, is
>used to prove a set is uncountable, but in fact the set can be shown to be
>countable by a better method, then the definition is false. Ditto for
>axioms.

Definitions and axioms cannot be false. Axioms can be inconsistent,
and definitions can be unconventional, but that does not make them
"false."

Cantor's proof shows that if X is any set, then there is no surjection
f: X -> P(X). The proof does not mention cardinality or countability
in any way. Despite repeated challenges, you have been unable to state
which part of the proof does not logically follow from previous
statements, or the axioms or definitions.

Let's make it simple. Do you believe there exists a set X such that a
bijection f: X <--> P(x) actually exists? Notice that my question does
not mention size or countability or cardinality at all.

>In some fields of mathematics, axioms and definitions cannot be false,
>because they are the core of the field. But the core of the field of
>Cardinality (i.e. Set Theory) is "size," not any axioms or definitions.

Nonsense. The usual set theory is called ZF, because that is the name
given to the axioms. Nobody talks about "ZFS" meaning "ZF with
Cantor's idea of size" as opposed to "ZFK" meaning "ZF with Kehr's
definition of size."

>Thus in Set Theory any definition and any axiom can eventually be proven to
>be false, or at least "weak," meaning it only applies in a limited number of
>cases, but not all cases.

If there is a case where a definition does not apply, then it is not a
definition. For example, the definition of a perfect number allows us
to conclude that cows and horses and bulldozers are not examples of
perfect numbers. The definition is universally applicable.

>By the way, the third paper on my web site proves that there are only a
>countable number of Turing Machines. It is quite a clever proof but you
>would have read the complete paper before you could even start the paper.

Amazing. I can prove that in one sentence. The Goedel numbering of
Turing machines establishes an injection from TM -> N, and therefore TM
is countable.

--
Dave Seaman dseaman@purdue.edu
Pennsylvania Supreme Court Denies Fair Trial for Mumia Abu-Jamal

Date Subject Author
4/14/99 Papus
4/15/99 Webster Kehr
4/16/99 Alan Morgan
4/18/99 Webster Kehr
4/16/99 Dave Seaman
4/16/99 Bill Taylor
4/16/99 Nathaniel Deeth
4/16/99 Jake Wildstrom
4/19/99 Sami Aario
4/16/99 Ken Cox
4/19/99 Michel Hack
4/20/99 Nathaniel Deeth
4/20/99 Ulrich Weigand
4/21/99 Nathaniel Deeth
4/21/99 Nathaniel Deeth
4/21/99 Ulrich Weigand
4/21/99 Brian David Rothbach
4/21/99 Virgil Hancher
4/22/99 Nathaniel Deeth
4/22/99 Sami Aario
4/23/99 Nathaniel Deeth
4/25/99 Sami Aario
4/22/99 Ulrich Weigand
4/23/99 Nathaniel Deeth
4/23/99 Ulrich Weigand
4/20/99 Nathaniel Deeth
4/17/99 Papus
4/19/99 Kevin Lacker
4/20/99 Bill Taylor
4/19/99 Andrew Boucher
4/19/99 Kevin Lacker
4/21/99 Andrew Boucher
4/22/99 Keith Ramsay
4/23/99 Andrew Boucher
4/24/99 Keith Ramsay
4/25/99 Andrew Boucher
4/27/99 Bill Taylor
4/27/99 David Petry
4/30/99 Keith Ramsay
5/1/99 Keith Ramsay
4/16/99 Andrew Boucher
4/16/99 Dave Seaman
4/18/99 Webster Kehr
4/19/99 Jeremy Boden
4/19/99 Sami Aario