On 7 Feb., 08:15, fom <fomJ...@nyms.net> wrote: > On 2/7/2013 12:45 AM, WM wrote:
> > For every n: (a_n1, a_n2, ..., a_nn) =/= (d_1, d_2, ..., d_n). > > For every n: (a_n1, a_n2, ..., a_nn) is terminating. > > For every n: (d_1, d_2, ..., d_n) is terminating. > > > For all n: (a_n1, a_n2, ..., a_nn) =/= (d_1, d_2, ..., d_n). > > For all n: (a_n1, a_n2, ..., a_nn) is terminating. > > For all n: (d_1, d_2, ..., d_n) is *not* terminating.
> that the irrational number, in virtue of > the property given to it by the definitions > has just as definite a reality in our minds > as the rational numbers or even the integers, > and that one does not even need to gain it > through a limiting process, but by possession > of it one becomes convinced of the practicability > and evidence of limiting processes in > general." > > Notice the word DEFINITION in Cantor's > statement.
Definition or not - all cases have to be treated similarly: "for all" either expresses a limit or not. There is not a bit of logic in arbitrary choice.
Cantor said: Die transfiniten Zahlen stehen oder fallen mit den endlichen Irrationalzahlen. (The transfinite numbers stand or fall with the finite irrational numbers.) He should have said they stand or fall with the *complete decimal representations* of irrational numbers.
> That is how LOGIC is applied > in the FOUNDATIONAL STUDY of DEMONSTRATIVE > SCIENCES.-
In the logic applied in mathemaics "for every" is tantamount to "for all". Instead of blatheríng in capitals you should apply logical thinking. Sorry if I demand an impossibility.