Re: Cantor Diagonal Proof

On Wed, 2025-04-09 at 18:52 +0100, Richard Heathfield wrote:

On 09/04/2025 18:30, wij wrote:

On Wed, 2025-04-09 at 18:10 +0100, Richard Heathfield wrote:

On 09/04/2025 17:45, wij wrote:
 
<snip>
 

Stick to the problem. Such puzzle won't prove the "0.999... problem".

 
You know what? You're right. 1/3 is irrational, 1 is infinite, 3
is unnatural, and Achilles never did catch the tortoise. I don't
know what I was thinking, claiming that rationals are rational!
Clearly the arithmetic I learned at school was deeply flawed, and
I was naif to think it could be trusted. How happy I am, now that
I have learned the truth from some faceless bloke on the
Internet. Well done you, eh? 1/3 is irrational; who knew?
 

 
1/3 is rational "by definition".

 
I know.
 

0.333... (repeating decimal) is irrational.

 
If you say so, that's good enough for me. I'll let my old school
know. They'll be appalled to hear they've been teaching it wrong
all these years.
 

Achilles puzzle gives us two models. One can catch the tortoise, one
cannot. Since we believe Achilles can, so it make? people to believe
0.99.... will finish (false from the other model. Also the term
'repeating' implies (define) never terminate)

 
You make it sound like Wij's Ant is busily calculating all the
digits. Look up "pattern" when you get a second and give the ant
a rest.
 
Achilles's race gives us /one/ model with /one/ solution that we
can reach two different ways. We can use the equations of motion
or we can use the sum of an infinite series. Both ways give the
same answer - the precise moment when Achilles overtakes the
tortoise and proves that infinite series can converge despite
your objection.

"lim(x->c) f(x)=L" means the limit of f approaching c is L, not f(c)=L 'eventually'.
f at c is not defined (handled) in limit.

lim 0.333...=1/3    ... The *limit* is 1/3, not 0.333...=1/3
0.3+0.33+0.333+...  ... The sequence converges to 1/3
Σ(n=1,∞) 3/10^n     ... The sum converges to 1/3 (or you can use lim)

lim Σ(n=1,∞) 3/10^n = 1/3  ... use lim
Σ(n=1,∞) 3/10^n ≒ 1/3      ... use '≒' ('=' can be confusing)
 
'∞' is (or can be) merely a symbol like the unit of weight/length,...., then,
everything is simple and clear.

Symbol '∞'::= A never-ending counting loop procedure with the following
              algebraic rules:
         1. ∀n∈ℕ, n<∞
         2. The multiplicative inverse of ∞ is 1/∞, and the additive inverse of
            ∞ is -∞

         Note: 'never-ending counting loop procedure' is to be defined, but, as
               long as algebraic principles are followed, '∞' is computationally
               safe.

ℝ contains infinity naturally (method of exhaustion is everywhere). You don't
need all sort of NSA, Cantor's cardinality theory to handle infinity, because
they are at most (if consistent) subset of ℝ.

But hey! Maybe the equations of motion are wrong too. Why stop at
maths when you can tear down physics at the same time?

Yes, the world is a strange thing, every logical is possible.