[Taigangmen Simenzhu], Let Me Tell You What the Diagonal Method Is!

I came on today and immediately saw Taigangmen Simenzhu posting about the diagonal method, but his post shows he doesn't understand what the diagonal method is at all. He treats the diagonal method as a logical method like proof by contradiction, which is wrong. The diagonal method is merely a construction method. The diagonal method alone proves nothing -- it must be combined with logical methods to be useful. This is very basic mathematical common knowledge.

Let me use an example to tell everyone what the diagonal method is, and that the diagonal method is valid for any set, because it is merely a construction method. Suppose set A is any set -- its elements can be finite, infinite, or transfinite -- with elements denoted by an. The diagonal method offers infinitely many construction possibilities. For example, construct an element a such that (a differs from a1) AND (a differs from a2) AND..., going through all elements of set A. Here "differs from" can be replaced with "has expression differing from," "has expression length differing from," etc. -- there are infinitely many possibilities. This kind of construction method is called the diagonal method. It has a special case involving numbers, which is essentially equivalent to the proposition an = some number. The method is the same.

Like any construction method, the diagonal method by itself is useless -- it proves nothing. It must be combined with a logical method, such as proof by contradiction. Whether the diagonal method plus proof by contradiction is effective depends on the logical premise of the proof by contradiction. It's not the case that proof by contradiction always requires the diagonal method, or that the diagonal method always requires proof by contradiction. These two have no necessary connection. So the diagonal method is useful for any set A, but this usefulness is limited to the scope of construction. That is, for any set A, we can apply the diagonal method, but not every proposition about set A can be proven through the diagonal method plus some logical method. The diagonal method is not some magic pill that certain people love -- one that claims to cure all diseases. This point must be made clear.

Those who can't grasp this fail to distinguish between set A and propositions about set A -- two different concepts. However, among propositions about set A, there is one type involving universal judgment -- "all" such-and-such. Such propositions often employ the diagonal method's construction and basically add proof by contradiction. This is a phenomenon, nothing more. As for specific propositions, they require specific analysis. Is there really anything to doubt here? Therefore, please be sure to distinguish between construction and proof. In Miss Chan's case, the diagonal construction is only one indispensable step in the proof, not the whole thing. It also utilizes the basic mode of computer recording. This mode determines that computer records in the real universe can only be finite. This too is an indispensable step in Miss Chan's proof -- without it, her proof wouldn't hold. Anyone with some mathematical and physics background can see this clearly!

Let me also add: human thought is not the same as propositions. Propositions can be right or wrong; thoughts don't necessarily have right or wrong. This is also one of the most important characteristics of human thought. Of course, this is unrelated to Miss Chan's proof, so I won't elaborate.