Let Me Explain on Miss Chan's Behalf Why the New Thought She Defined Must Be a New Thought!

Miss Chan proved that Mathematics' claim that "all human thoughts can be stored in a binary computer" is wrong. The definitions and proofs within are easily understood by anyone who has professionally studied mathematics. Since there are too many humanities and engineering students here who haven't received professional training and seem to have difficulty understanding, I'll take the trouble to help enlighten them on Miss Chan's behalf.

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 "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," 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 these 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 logical simplification process like (B+B-+True+False)- = False, because the proposition here is defined in terms of "expression." This is also what Miss Chan means by "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."

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 propositions. 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 whether things are 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 that "diagonal method" is just a colloquial name and doesn't require actually 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, one can replace "expression" with "word count of the expression," "color," "phase," "height" -- there are infinite possibilities, as long as the constructed result is guaranteed to differ from all original propositions. The possibilities for such constructions are truly infinite.

The diagonal method is generally used to counter propositions that claim "all" such-and-such. The method is to gather what the opponent claims as "all" into a set, then construct an element that differs from all elements in the set, thereby using proof by contradiction to show that 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 mathematics. It's like someone questioning mathematical induction: if assuming the proposition holds for arbitrary N allows us to derive it holds for N+1, and therefore the proposition holds for all natural numbers, then what about N+2? To such people, I can only borrow Miss Chan's catchphrase: go back and study!