On 4/17/2013 2:45 AM, WM wrote: > On 16 Apr., 22:45, Virgil <vir...@ligriv.com> wrote: > >> It is not clear to me, or to anyone sensible, that the entire sequence >> of all naturals in A, which has no maximal member, is "in" any line of >> naturals that has a maximal member, and it is equally clear that every >> line does have a maximal member. > > Is there any number of A that is not in at least one line?
Responding to a statement saying "It is not clear..." with a question. How typical.
It may be "Socratic", but, since the developments of early nineteenth century mathematics had led to a need for a more rigorous approach, Aristotle's influence won the day on behalf of the advancement of science.
> For all n : (1, ..., n) of A is in line n of B.
You really need to become more consistent concerning when a number corresponds with a mark ('1') and when a number corresponds with a finite initial segment of the natural numbers *given* by the axioms.
> For all n : line n ob B is in (1, ..., n) of A >
This current mode of argumentation corresponds to the subsequence of triangular numbers
1 :=> 1 2 :=> 3 3 :=> 6 4 :=> 10
and so on
> So we have an identity. There is no actually infinite line in B, so > there is no actually infinite sequence A. (Of course A is potentially > infinite as is B.)
No. What we have it one of your particular 'thought experiments' that provided you with the directed set structure by which you came to devise your particular notion of potential infinity.
When the ancient Greeks actually considered what is involved with *calculation*, Euclid established the intrinsic directed set structure of the natural numbers with a *proof* that there is no greatest prime number.
That proof involves a successor function.
That successor function is the basis for the modern axioms for the arithmetic with natural numbers.
The systems of modern axioms are the basis of modern mathematics.