Re: Is the set N of natural numbers well defined?

apoorv wrote:
>I do not think I have said that the n'th digit of the diagonal
>depends on any other digit in the diagonal.

Possibly not. You said "step by step", which I interpreted to mean
each step depending upon the previous step.


>Let us consider a sequence in N.Each member of the sequence is the
>image of a map f from N to N.

I'd say (and my textbooks agree) that a sequence in a set A is a map
from N to A. Do you have an alternative definition of the term
"sequence"? In particular, a sequence in N is a map from N to N.


>Each partial sequence of the first n members is the image f(In) of
>the initial interval In.

That looks confusing to me. The symbol f previously referred to a
function from N to N, now you're using the same symbol to represent a
function from the set of initial segments of N to finite sequences in
N.


> The real number,which corresponds to the sequence itself is the
>image f(N) of N.

And now you're using f to refer to a function from {N} (not N itself)
to the set of sequences in N. Or extending the second reference. Or
something: it's a little unclear and a lot unnecessary.

Certainly all these functions can be defined in terms of the original
f, but they aren't the same thing. None of the subsequent ones are
used in any proofs of diagonalization I've seen.


>To be able to speak of the entire seq. as an entity,we need to
>consider the domain of f,not as N, but as the set of initial
>intervals, extended to include N. Otherwise,we can speak only of
>individual members of the sequence.

To be able to speak of the entire sequence as an entity, we need only
refer to f. It *is* the sequence.

We need not consider any functions from subintervals of N, or other
such extraneous baggage.


- Tim
.

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