On 24 Jun., 11:04, Virgil <Vir...@home.esc> wrote:
> > Thise prodedure is abbreviated by > > (..., An, ... A2, A1, A0, L0) > > but of course there cannot appear at any stage a list without first > > line. > > But that list can be rearranged into a more standard form, for example > with the An and the members of L0 listed alternatingly, and from such a > rearrangement a non-member anti-diagonal can be constructed.
What should that be good for? The countable set under investigation contains the antidiagonal of the arragement that I prescribed.
Your arguing reminds of another argument: I can eat cherries, therefore the list is neither complete nor incomplete.
I for my person would not accept that. Perhaps set theorists have another taste. >
> > It is as simple as that. > > > It is not a big deal. Set theory shows many contradictions arising > > from the idea, that from "every n in N can be constructed" it is > > implied that "N can be constructed". > > In, for example, FOL+ZFC, there is no assumption that every n in N "can > be constructed". What is assumed is that there exists a set, along with > all its elements, having the properties that we want for N and its > elements , which we then call N.
And being ready to be used for the construction of a bijection. > > This is wrong, because for every > > > n, there is an infinite set of unconstructed elements of N. > > But we do not "construct" any of them. Though we do construct various > naming conventions for elements of N.
Bijections from M to N are constructed. Sequences (having natural indices) are constructed. Many constructivists do not believe in uncountability but in N because N can be constructed.