[Falanxi Dacai Shifu], Your Construction of the Example of All Finite Natural Numbers Is Invalid.

[Falanxi Dacai Shifu] is quite a thoughtful person. He constructed an example of all finite natural numbers, then used his understanding of Miss Chan's diagonal method to prove that not all natural numbers are finite, thereby trying to show Miss Chan made an error. But his proof is incorrect. Let me first quote his text:

"All natural numbers are finite. Do you accept this conclusion?
[Falanxi Dacai Shifu] posted on 2006-08-10 11:49:10

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But the Chan lady can prove it wrong.
Proof by contradiction: Assume the thesis holds -- all finite natural numbers can form a set A.
Construct a number: a1+a2+a3+a4...
This number is certainly not equal to any number in A, and is also a natural number. Contradiction. Therefore not all natural numbers are finite.
See where the problem is?
If you see it, you'll understand why my books are infinitely many."

But his proof is incorrect. Let me explain point by point:

Proof by contradiction: Assume the thesis holds -- all finite natural numbers can form a set A.
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The number of elements in this set A is infinite. The proof is simple: if the number of elements in set A were finite, then the largest element N could be found, but N+1 is also a finite natural number, which contradicts all finite natural numbers forming set A. So set A has infinitely many elements.

Construct a number: a1+a2+a3+a4...
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Since the set above is infinite, this number must be in A.

This number is certainly not equal to any number in A, and is also a natural number. Contradiction.
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So you cannot reach this conclusion.

Miss Chan's proof is fundamentally different from this. She first uses quantum mechanics and the finiteness of the universe to prove set A is finite. Then the proposition she constructs also satisfies the finiteness requirement but is not in set A. So understanding Miss Chan's proof does require some effort.

[Falanxi Dacai Shifu], your books can't be infinite

Following the rules, let me first quote the key part of [Falanxi Dacai Shifu]'s post in full:

"List all possible and impossible human utterances.
If using English, first the one-letter ones:
a
b
c
d
e
f
g
h
i
j
k
i
l
m
n
o
p
q
r
s
t
u
v
w
x
y
z
The first volume has these 26 books. The second volume has 26*26 books.
aa
ab
...
zz
And so on. My books total 26+26^2+26^3... this many. It can be proven that this length is countably infinite."

Please note, this chef has again made a taken-for-granted error: "It can be proven that this length is countably infinite." Anyone who's studied math knows that for the series 26+26^2+26^3... to be infinite, the sequence must be infinite, meaning this chef assumes infinitely many books. Can humanity produce infinitely many books? Obviously not. Proof:

Because the universe is finite, and the speed at which humanity produces books cannot be infinitely fast -- the reasoning here is based on the principles of quantum mechanics -- if the universe's lifespan is A and the minimum interval for humanity to produce a book is B, then the number of books produced by humanity is less than A/B (since humanity didn't produce any books during at least the past several billion years). A/B is only a finite number, so humanity cannot produce infinitely many books.

Taking the integer part of A/B, call it N. According to this chef's formula: 26+26^2+26^3...+26^N -- the length is also a finite number. In other words, his claim that "it can be proven that this length is countably infinite" is wrong, and therefore his argument is invalid!