[Falanxi Dacai Shifu], Why Are You Making the Same Error as in the Proof of All Finite Natural Numbers Again?

[Falanxi Dacai Shifu] posted another thread today, making the same error I already pointed out in his proof regarding all finite natural numbers. Let me first quote his post:

[Falanxi Dacai Shifu] posted on 2006-08-13 17:56:30
"Of course, since the Chan lady's logical errors are too numerous, I can still make concessions. This time inspired by forum user Xishan Yue.
Let's say the thought proposed by the Chan lady is a human thought. Premise: human thoughts are finite.
Then human thoughts can form a set A. Then following the Chan lady's construction, there exists a human thought not belonging to A. Therefore, the premise is wrong. Conclusion: human thoughts are infinite."
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Has everyone noticed this sentence: "Then human thoughts can form a set A. Then following the Chan lady's construction, there exists a human thought not belonging to A." But this is merely this chef's taken-for-granted assumption, because following Miss Chan's construction, she cannot prove that there exists a human thought not belonging to A. This is because set A consists of human thoughts, and the diagonal method applied to it also produces human thoughts. Human thoughts cannot construct something that isn't a human thought -- just as applying the diagonal method to the set of real numbers cannot find any number that doesn't belong to the reals.

But the current set A is not human thoughts in general, but rather all computer-recorded human thoughts. That is, the constructing agent here is a human, and the things being constructed are computer-recorded human thoughts. The constructor and the constructed do not exhibit the completely identical situation described above. That the constructor and the constructed cannot be completely identical is a basic construction principle of the diagonal method -- this is common knowledge, but this chef apparently completely doesn't understand. Therefore, please everyone pay special attention to this distinction. In this situation, Miss Chan's construction can indeed find a thought that belongs to human thoughts but does not belong to the supposedly exhaustive set of computer-recorded human thoughts -- just as applying the diagonal method to the rational numbers can construct a number that belongs to the reals but not to the rationals.

Once again: the diagonal method is a construction method. Whether the construction leads to a contradiction has nothing to do with the premises assumed in the proof by contradiction. As for why human thoughts cannot lead to a contradiction while "all computer-recorded human thoughts" can -- this ultimately comes down to the fact that human thoughts are a concept analogous to "the set of all sets," while "all computer-recorded human thoughts" is not such a concept. And this is determined by the fundamental difference in modes between the human brain and the computer.

Because this chef made the same kind of error as in his attempt to prove propositions about all finite natural numbers, he cannot prove that human thoughts are infinite. Miss Chan said long ago that human thoughts are finite, but computer-recordable human thoughts are also finite. Regardless of how large the collection of computer-recorded human thoughts may be, one can always find at least one thought that is not among them. So this has absolutely nothing to do with whether human thoughts are infinite.

Of course, people like this chef may again take for granted: since human thoughts are finite, why can't they be recorded by a computer? This kind of thinking is what only someone without mathematical training could produce. Just like n and n+1 -- even when n is infinite, these two expressions still differ, unless n ranges from negative infinity to positive infinity. When n is finite, the two sets they represent are even more different. For example, if n's maximum value is 9, then n represents natural numbers 1 through 9, while n+1 represents 2 through 10. n cannot represent n+1. Of course, this example is unrelated to Miss Chan's proof, but I hope it helps those who think "finite means anything goes" understand that finite doesn't mean you can do whatever you want either!