In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> 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.
Why not. Given such a decimal sequence(along with sign and decimal point, such a number can be compared in size with any other and thus is in trichotomy with the rest of the numbers.
Since WM very often cites trichotomy as the critical test, and such sequences all pass that test,
> 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.
It does if the nth number is specified to the 2*nth place.
> > > > 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.
How about line(n) = 1/(n+1), the n+1 to allow for a line numbered 0?
What about that list is not computable?
> > > 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 cannot have a computable listing of ALL reals, but above is a computable listing of some reals, line(n) = 1/(n+1).
> > That is correct. The notion "is uncountable" is nonsense per se.
Then the notion "is countable" is equally nonsensical, but in ZFC, and most other set theories, it is quite sensible.
That WM is unable to make sense of ti is his problem, not ours. > > > > 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.
Not according to any standard meaning of "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
What is obvious to WM is often false to everyone else and what is false to WM is often obvious to everyone else. "and there is no health in him."