On 12/11/2012 12:55 AM, WM wrote: > On 10 Dez., 21:03, fom <fomJ...@nyms.net> wrote: >> On 12/10/2012 11:57 AM, WM wrote:
>> >> Yes. He did. But, Cantor's notion of a real >> number was clearly found in the completion of a >> Cauchy space. > > That is completely irrelevant for the result. > >> He found that more appealing >> than Dedekind cuts. This is evident since >> his topological result of nested non-empty >> closed sets in a complete space is closely >> related. >> >> There are ordinal numbers in set theory given >> the names of natural numbers. > > Only those which are finite. >> >> Find a different criticism of Alan's remarks >> if you must. This one is incorrect. > > So you disagree that 2 is a real number?
Since you like quoting the Grundlagen, try transcribing long detailed passages from section 9 where Cantor rejects definitions that conflate logical priority as you have been doing.
That is where he calls his construction
"... a fundamental sequence and correlate it with a number b, TO BE DEFINED THROUGH IT,..."
and goes on to say,
"Care must be taken on this cardinal point, whose significance can easily be overlooked: in the third definition (his definition of fundamental sequence, fom) of the number b, say, is not defined as the 'limit' of the terms a_v of a fundamental sequence (a_v); for this would be a logical error similar to the one discussed for the first definition, i.e. we would be presuming the existence of the limit lim(v=oo) a_v. But the situation is rather the reverse."
Cantor does not address the full nature of how real numbers are constructed. At the beginning of section 9 he writes,
"I shall not discuss the introduction of the rational numbers, because rigorous arithmetic presentations of this have often been given; among those with which I am most familiar I emphasize those of H. Grassmann and J.H.T. Mueller."
In section 11 he explains his principles of generation wherein 2*omega arises. It is based on the "repeated positing and uniting of unities" which is a far cry from his "fundamental sequences".
In section 14 he explains multiplication of his number classes but defers explanation of a form such as 2*omega as previously explained. This explanation is in section 3
"If one takes a succession, determined by a number beta, of various sets which are similar and which are similarly ordered and such that each has a count of its elements equal to alpha, then one obtains A NEW WELL-ORDERED SET whose corresponding count supplies the definition for the product beta*alpha, where beta is the multiplier and alpha the multiplicand."
So, this time let me suggest that you open a new top-level post openly apologizing to Alan.