On 17 Jun., 10:34, Tim Little <t...@little-possums.net> wrote: > On 2010-06-17, Peter Webb <webbfam...@DIESPAMDIEoptusnet.com.au> wrote: > > > Cantor's proof applied to computable numbers proves you cannot form > > a computable list of computable numbers. > > Cantor's construction applied directly to lists of computable numbers > shows almost nothing. Given any list of computable reals, you can > produce an antidiagonal real not on the list.
That is the first, and most important, error. A real number cannot be defined by an infinite seqeunec of digits alone. A real number and any such sequence must *necessarily* be defined by a finite definition. A list of finite definitions, however, does not allow for the construction of a diagonal.
All real numbers that can be constructed, defined, computed, recognized, specified or what ever clever set theorists may create to veil the nonsense of transfinity, ... all those real numbers in all possible (countably many) languages in all possible (countably many) finite alphabets are contained in the following list (where every line may be repliceated infinitely often - and afterwards the list is transformed to become a real sequence by Cauchy-diagonalistaion)
0 1 00 01 10 11 000 ...
> Unfortunately the > construction says nothing about whether the antidiagonal is computable > or not.
And that is completely irrelevant. > > It takes quite a bit of extra work (not part of Cantor's proof) to > deduce that there is no computable list of all computable reals.
That result is wrong. There is no computable list of only computable reals. But that is not interesting. The computable reals belong to the set of all finite definitions. This set can be cast into a list. That is important. That shows that the computable reals can be in a list all together. > > > Cantor's proof applied to Reals proves you cannot form a computable > > list of Reals. > > Incorrect, and even WTF-worthy. Cantor's proof *is* applied to reals, > and nothing in it has anything whatsoever to do with computability.
You disagree with Fraenkel (and with common sense): The diagonal is constructed. And if the diagonal can be constructed, then it has been computed. > > > The property "that you cannot form a computable list" is not the > > same as the property "is uncountable". For example, computable > > numbers have the first property but not the second.
That is correct. The notion "is uncountable" is nonsense per se.
> > Cantor's proof is about what can be expressed in a list, and not > > directly about uncountable sets (which don't even get mentioned in > > the proof). > > "Cannot be expressed in a list" is actually one of many equivalent > definitions for the term "uncountable".
Therefore this theory is inconsistent.
There is a set that cannot be expressed alone in a list, but is a subset of a countable set.
> Whether Cantor himself used > that specific word is irrelevant - the mathematical content of his > proof was to establish uncountability.
And this proof failed. Obviously it is wrong to conclude that unlistability implies a cardinal larger than aleph_0. But this is what Cantor and his followers did. Otherwise there was no hierarchy of cardinals.