In article <firstname.lastname@example.org>, Webster Kehr <email@example.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 firstname.lastname@example.org Pennsylvania Supreme Court Denies Fair Trial for Mumia Abu-Jamal