Re: Cantor Diagonal Proof

On 04/04/2025 23:49, Lawrence D'Oliveiro wrote:

On Fri, 4 Apr 2025 13:47:29 +0200, Julio Di Egidio wrote:
 

The anti-diagonal is *as computable as the list is*: and the argument in
fact proves there can be no such list, computable or otherwise...

 No it doesn’t. The cardinality of the computable numbers is ℵ₀, same as
that of the integers. And the integers can in fact be arranged in a list,
therefore so can the computable numbers. QED.

You should try and prove it formally, you'll see that it's you here
messing up definitions, even what a deductive system is, i.e. what
it means to prove things formally.
Meanwhile few more exercises for the reader:
1) Eventually you are denying that universal quantifiers make sense.
But, unless you embrace ultra-finitism, proving a fact about all
natural numbers does not need proving it for each and every one of
them singularly: even in the most strictly constructive setting,
(mathematical) induction is the thing...
2) You are indeed making mistakes in the detail: the keyword is
not *list*, not *total*, and not even *computable*, the keyword
is *complete*: there is no complete list of computable reals either.
And now define "complete" formally...
3) Look up *extensionality* for the next level...
No criticism meant, been there done that: but insistence is futile. :)
Julio