On 4/14/2013 12:04 PM, WM wrote: > On 14 Apr., 16:46, dullr...@sprynet.com wrote: > >>> Please be aware, that I require mathematics, not matheology. In the >>> latter case the faithful believe that A contains only all finite >>> natural numbers and that B also contains all finite natural numbers, >> >> A "finite natural number" is the same as a natural number. > > Absolutely true, but it has turned out to be useful, sometimes, to > recall this fact.
As it has not been "useful" in this forum, its "use" must generally be found in misrepresenting mathematics to your students.
> >> So asking for a number in A not in B makes no sense, there >> are no numbers in B in the first place. B is a list of sets >> of numbers. > > Yes, B is a list or sequence of sets of numbers. > >> A is a set of numbers not in B. Answering >> your question. > > You claim that B does not contain the set of numbers A. This claim is > false. > > Proof: Consider the table T of numbers: > > 1 > 2, 1 > 3, 2, 1 > ... > n, ..., 3, 2, 1 > ... >
Yes. Everyone knows that WM likes pushing the clock of time back.
Proof by "triangular" numbers.
Why not use the sexagesimals?
> Every line is a set of numbers, every columns is a set of numbers too. > (I drop the curly brackets.) The lines are the same as my set B, the > first column is the same as my set A. > > You claim that A = |N but that no term of the sequence B is |N. > > It is easy to see that for every *finite* natural number n, there is a > term of B that has the elements 1 , 2, ..., n. >
Do you not mean for "every *given* natural number"?
> Therefore B is a majorant of the finite initial segments (FISs) of A > *for all n* - and there is nothing else in A, by definition.
There is no "*given* natural number" that is an upper bound for "*every* finite initial segment of natural numbers" from A.
> This > cannot change by unioning its elements or FISs, because the union does > not change the elements and does not add anything that has been > missing before. >
No change is necessary because the analysis is faulty.
> So your claim is false, unless you think that after all finite natural > numbers n something unexpected and infinite happens. But that is > matheology.
That would be WMytheology.
Nothing unexpected and infinite happens because what can be proven from axioms is proven from what is given as prior in the statements of the axioms.
You confuse yourself with your failure to recognize the difference between the definition of an inductive set and the use of induction as a proof method.
Having confused yourself once, you confuse yourself further by failing to understand the metamathematics that distinguishes the second-order induction axiom from the first-order inductive axiom schema.
That is actually forgivable since these matters are not what one would expect with your background. But your failure to make an effort to understand these matters in order to formulate legitimate mathematical and metamathematical arguments is not.