Even Adding the Assumption of Finite Encoding Length Cannot Save the Leftist!

2006/8/7 19:28:00

After the leftist's trick of denying their own words was exposed, they deployed another common maneuver: temporarily adding an assumption, thinking this would save them. Unfortunately, it only makes them look worse! Let me first quote the proof process from this ID's reply to the leftist:

Since we are using proof by contradiction, all argumentation proceeds from the leftist's assumption that "all human thought can be stored in a binary computer," and then derives a contradiction, thereby proving the absurdity of this assumption. Because they are stored in a computer, these binary numbers can be lined up in a queue. Whether this queue is finite or not does not affect the argument—though in reality, the number is finite. Second, since "all human thought can be stored in a binary computer," every human thought is encoded as a corresponding binary number. These numbers form a set A. In other words, all binary numbers are divided into two classes: those belonging to set A and those not belonging to set A. We call numbers in set A "A-numbers" and those not in set A "B-numbers." Now let us examine those binary numbers not belonging to set A—the B-numbers. These numbers, under this encoding, do not correspond to any human thought; otherwise, by the definition above, they would belong to set A. Good. Now apply the diagonal method to set A. The binary number corresponding to its diagonal is certainly not in set A—that is, this binary number must be a B-number. But this binary number corresponds to the following thought: "its N-th digit differs from the N-th digit of the N-th binary number when set A is lined up." That is to say, this binary number should be an A-number. From the analysis above, we have found a binary number that is both an A-number and a B-number—meaning it both belongs to and does not belong to set A. This contradicts the definition of a set. Why does this contradiction arise? Because the initial assumption—"all human thought can be stored in a binary computer"—is false. By proof by contradiction, we know: "not all human thought can be stored in a binary computer."

Now, the leftist adds an assumption: the number of digits in the binary encoding of human thought must be finite. Being finite, let us say it is M—that is, all human thought can only be encoded with M-digit binary numbers. Thus, the number of binary numbers that can encode human thought is at most 2 to the M-th power. Note that here M represents an arbitrary natural number. The leftist claims that beyond a certain length, it no longer counts as a single book and cannot be encoded. Fine—we take that limit plus 1 as M. This process is always feasible.

Now consider this thought: U(this thought's expression differs from thought a_n), where U represents the logical conjunction and a_n ranges over all elements of set A—in other words, the negation of the conjunction of these 2^M human thoughts. Expressed more logically: consider the proposition that is the negation of the conjunction of all 2^M propositions. Obviously, this too is a thought, and this thought is not among the 2^M human thoughts. Since we have assumed that all human thought is already encoded within these 2^M binary numbers, this thought—which is not among the 2^M human thoughts—is also not among the 2^M binary encodings that supposedly contain all human thought. This contradicts the assumption that all human thought can be encoded.

Of course, someone may question whether the negation of the conjunction of these propositions is a new proposition. Here we are dealing with thoughts, and thoughts cannot be combined by "collecting like terms." Otherwise, the proposition "wrong" would be the same as the proposition "Fermat's theorem does not hold." By that logic, all propositions could be reduced to two: "correct" and "wrong." This kind of collecting like terms for thoughts would inevitably lead to this absurd conclusion. No propositions can be merged by collecting like terms. Even the merger of proposition A and its negation into "true" or "false" requires premises.

Moreover, the prerequisite for all propositions to be mergeable implies that every proposition is decidable—which is obviously not the case! Godel's Incompleteness Theorem guarantees the existence of undecidable propositions. How, then, do you merge these undecidable propositions? For instance, take the above "negation of the conjunction of all 2^M human thoughts," simplify it to "the negation of the conjunction of all undecidable propositions among the 2^M human thoughts, excluding the negation of any of those propositions." Since the propositions are all undecidable, and we have excluded the negation of each, there is no possibility of merging this proposition. It is of course not among the original "2^M human thoughts"—so have we not found yet another thought that cannot be encoded? In fact, such thoughts are numerous. One can even prove that under the assumption of finite encoding length, the human thoughts that cannot be encoded vastly outnumber those that can. This proof is not difficult—anyone with basic mathematical knowledge can derive it!

From the above, it is clear that even the leftist's ad hoc addition of the finite encoding length assumption cannot save them!