The Wonder of 1
To start, let's prove an important [Theorem].
[Theorem 1] The largest natural number is 1.
[Proof] Let x be the largest natural number, and assume x≠1.
In this case, since x>1,
Multiplying both sides of this equation by x gives x^2>x ... (1).
However, (1) contradicts the fact that x is the largest natural number.
From the above, by proof by contradiction, x=1.
In other words, the largest natural number is 1.//
By the way, some might say, "Isn't 2 larger than 1?" when I say this.
But wait. "2>1" is incorrect. Let's prove the following [Theorem].
[Theorem 2] "1 = 2" is true.
[Proof] Assume 1 ≠ 2.
In this case, multiplying both sides by x gives x ≠ 2x.
Substituting x = 0 into this equation gives 0 ≠ 0.
This is a contradiction.
From the above, by proof by contradiction, 1 = 2 holds.//
How about that? It was proven perfectly, wasn't it? Are you not convinced? But I did prove it.
By the way, both of the proofs above use proof by contradiction. Is proof by contradiction itself wrong? Indeed, proof by contradiction feels a bit suspicious. I understand that feeling. I felt that way too.
But proof by contradiction is indeed a correct logical method. If you write proof by contradiction as a logical formula, it becomes:
$${(\neg P\Rightarrow(Q\land \neg Q))\Rightarrow P}$$
"¬" represents negation. Please call it "not". "⇒" should be called "implies". The (Q∧¬Q) part of this formula means a contradiction. In other words, this formula means,
If you negate P and a contradiction arises from it, then P must be true.
So, it is exactly proof by contradiction. And this logical formula is a tautology (a logical formula that is always true). You can confirm this with simple operations. I will omit the explanation here, but please be assured that proof by contradiction is not wrong.
But [Theorem 1] and [Theorem 2] are still strange, aren't they? That means the [Proof] must be wrong somewhere.
Actually, in [Proof 1], two things are assumed: that "there is a largest natural number" and that "that number is 1". Since it resulted in a contradiction, the assumption must be wrong, but it doesn't necessarily mean that "that number is 1" is wrong; it could be that the other assumption, namely "there is a largest natural number", is wrong.
In [Proof 2], I started by "multiplying by x" and then "substituting 0 for x", but that is the same as "multiplying by 0", and that part is suspicious. My apologies.
◇ ◇ ◇
~ 1, 2, 3, and infinity ~
▷ The wonder of 1
▷ Rough calculation of 2 to the power of n
▷ The magic of "3"
▷ The dubiousness of infinity