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Let Me Try Once More to Explain Miss Chan's Proof to Mathematics

I believe Mathematics fundamentally hasn't understood Miss Chan's proof, so let me try restating it once more. Miss Chan's proof clearly doesn't involve any numerical issues at all. Please look at her proof -- where does it involve numbers? Quoted as follows:

"Because the leftist hypothesis states that 'all linguistically expressible human thoughts can be stored in a binary computer,' these thoughts constitute a set A, where thoughts are denoted by an, with n representing natural numbers. Since the universe will perish, and the time interval for humans to produce a thought cannot be infinitely short (otherwise it would violate the principles of quantum mechanics), we can know that the elements in set A are finite, though the quantity can be very large. Then we construct a thought as follows: U (the expression of this thought differs from thought an), where U represents logical conjunction, and an represents all elements of set A. This thought is obviously not in set A, and it cannot be simplified into any thought in A, because this thought speaks of expression, so any tricks of logical simplification are useless. Since set A is finite, the expression of this thought is also finite, and this thought is obviously a human thought. The contradiction can only come from the leftist hypothesis that 'all linguistically expressible human thoughts can be stored in a binary computer' -- meaning this hypothesis is absurd."

Miss Chan's proof: one, does not involve numbers; two, does not involve the finite-infinite distinction. Yet Mathematics insists on claiming she does this -- clearly Mathematics needs to read Miss Chan's proof more carefully. Since Mathematics tends to alter things when quoting others, please read the above proof carefully, and everyone else should look too. What Mathematics calls Miss Chan's "so-called proof" is entirely his own fabrication.

Miss Chan's proof -- here I give a popular explanation, which is what Miss Chan truly means:

The key difficulty of Miss Chan's proof lies in this sentence: "Then we construct a thought as follows: U (the expression of this thought differs from thought an), where U represents logical conjunction, and an represents all elements of set A. This thought is obviously not in set A, and it cannot be simplified into any thought in A, because this thought speaks of expression, so any tricks of logical simplification are useless."

Let me explain why the thought Miss Chan constructed must be a new thought. For simplicity, let's use an example where set A contains only four propositions: B, B-, True, False, where B- represents the logical negation of B, B can be any proposition, and True and False represent the logical values 1 and 0. Following Miss Chan's construction, we obtain a new proposition:

"The expression of this thought differs from thought B AND the expression of this thought differs from thought B- AND the expression of this thought differs from thought True AND the expression of this thought differs from thought False"

The proposition in quotation marks is obviously different from the original four propositions B, B-, True, False. One cannot perform logical operations on the new proposition to produce a simplification like (B+B-+True+False)- = False, because the proposition here is defined in terms of "expression." This is also what Miss Chan means by her statement.

What if the original four propositions were: "the expression of this thought differs from thought a1," "the expression of this thought differs from thought a2," "the expression of this thought differs from thought a3," "the expression of this thought differs from thought a4"? The situation is still unchanged, because the corresponding new proposition would be:

"The expression of this thought differs from the thought 'the expression of this thought differs from thought a1' AND the expression of this thought differs from the thought 'the expression of this thought differs from thought a2' AND the expression of this thought differs from the thought 'the expression of this thought differs from thought a3' AND the expression of this thought differs from the thought 'the expression of this thought differs from thought a4'"

The proposition in quotation marks is still different from the original four. For any N propositions, the situation is analogous. Whether N is finite, infinite, or even transfinite, the essence of the proof doesn't change, because Miss Chan's entire proof does not depend on finite or infinite -- because she uses the diagonal method.

Some people just don't understand. They think the diagonal method must have an actual diagonal, asking all day: how do you find the diagonal in 00, 11, 01, 10? Or: with finite digits but infinite numbers, how do you find the diagonal? They don't realize "diagonal method" is just a colloquial name and doesn't require finding any diagonal. If the thing being proved has nothing to do with numbers, can the diagonal method not be used? That would be the greatest joke ever!

The essence of the diagonal method lies in constructing something new that differs from everything previously posited. Such construction methods can be infinitely many; whether there is an actual diagonal is irrelevant. For example, in Miss Chan's construction, "expression" can be replaced with "word count," "color," "phase," "height" -- there are infinite possibilities, as long as the result differs from all original propositions.

The diagonal method is generally used to counter propositions claiming "all" such-and-such. The method is to gather what the opponent claims as "all" into a set, then construct an element differing from all elements in the set, thereby using proof by contradiction to show the claimed "all" is wrong. Someone might say: can't I just add your constructed element back into the set? Once someone raises this question, it only shows they haven't even studied middle school math. It's like questioning mathematical induction: if assuming it holds for arbitrary N allows us to derive N+1, then what about N+2? To such people, I can only borrow Miss Chan's catchphrase: go back and study!