Re: Cantor Diagonal Proof

Lawrence D'Oliveiro <ldo@nz.invalid> writes:
[...]

Let’s start again, with the assumption that we have a list mapping all the
reals 1:1 to the positive integers. So given any real, we can assign it a
position N ∈ ℤ ≥ 1.
>
So now we apply the Cantor construction, to try to come up with a number
not in the list. But a consequence of the starting assumption is that the
number being constructed must be somewhere in the list, and therefore the
Cantor construction must map to some positive integer Nₙ.
>
So the question is: what is digit Nₙ of this number?
>
The answer is, it must be different from digit Nₙ of itself!
>
So you see, the assumption that you *can* perform the Cantor construction
on a list of the reals leads to a contradiction. Therefore the
construction cannot be performed. QED.

Right.

What we have here is duelling assumptions: either the list can be
constructed, or (according to the Cantor construction) it cannot. There is
no “self-evident” reason to say one argument is valid while the other is
not.

No.  Cantor's proof starts by *assuming* that the list can be
constructed, and demonstrates (proves) that that assumption
leads to a contradiction.  There are no "duelling assumptions".
There is an assumption (the list can be constructed) that we adopt
temporarily for the sake of the proof, and a *proof* that the list
cannot be constructed.

If the word "assumption" is troublesome, we can rephrase Cantor's
proof as "*If* the list can be constructed, *then* we reach a
contradiction, therefore the list cannot be constructed".  That's a
valid and true statement.

Assume there is a full grown elephant in my living room.  We can
conclude from this assumption that I can see, hear, and touch the
elephant.  In fact, I cannot.  Therefore the initial assumption
is false.  Again, there are no "duelling assumptions"; there is
a tentative assumption and a proof that the assumpton is false.
(There is no elephant in my living room.)  My elephant proof isn't
quite mathematically rigorous, but it demonstrates the principle.

Therefore I suggest that the Cantor construction is similarly an axiom,
that has to be added before you can construct the reals. Without it, the ℝ
you construct consists solely of computable numbers.

Do you reject the validity of proof by reductio ad absurdum?

(Amusingly, as I was writing this, I saw that today's SMBC comic
is about Cantor.  <https://www.smbc-comics.com/comic/cant>.)

--
Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com
void Void(void) { Void(); } /* The recursive call of the void */