Talk:Yablo's paradox
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Potential solutions of the paradox
[edit]First, note that each statement talks about several other statements. Thus, the statements in expanded form:
S1: ¬S2 ∧ ¬S3 ∧ ¬S4 ∧ ¬S5...
S2: ¬S3 ∧ ¬S4 ∧ ¬S5 ∧ ¬S6...
S3: ¬S4 ∧ ¬S5 ∧ ¬S6 ∧ ¬S7...
S4: ¬S5 ∧ ¬S6 ∧ ¬S7 ∧ ¬S8...
...
And so on.
Suppose S1 is true. Then what it says is true, i.e. all of the following statements are false. Statement S2 is also false. However, since all statements after S2 are false, we obtain that the statement S2 is true - a contradiction! The same thing happens when you try to consider any of the statements true.
But then suppose that S1 is false. If S1 is false, then the expression ¬S2 ∧ ¬S3 ∧ ¬S4 ∧ ¬S5... is also false. Remember De Morgan's laws:
¬(¬S2 ∧ ¬S3 ∧ ¬S4 ∧ ¬S5...) = S2 ∨ S3 ∨ S4 ∨ S5...
It turns out that in order for S1 to be false, it's necessary that at least one of the statements S2, S3, S4, etc. was true. But if we assume that S2 is true, then we return to the paradox for S2 and S3, so we assume that S2 is also false. We do the same with S3, S4, S5, S6... Stop! It looks as if we're "take away" the paradoxical construction to infinity! Indeed, for Sn to be false, only one statement Sn+k for any natural k > 0 must be true. Therefore, we can increase k of Sn+k that is true to whatever values we want.
In the liar's paradox, when you try to assign a truth value to the statement S: ¬S, something like an oscillation occurs between the two states (to better understand what I'm saying, imagine an inverter looped on itself). Imagine also a Yablo-like dynamic system - in it the paradoxical construction caused by assigning the truth to a statement will "run away" towards infinity. However, to avoid paradox, we need a "static", "balanced" situation without any oscillations between states or runaways to infinity. (As an example, there are 2 statements that assert the falsity of each other. If one of them is considered true, and the other is false, paradoxes do not arise).
One way to avoid paradox is "infinitely far" true statement. However, this approach raises questions about the essence of the concept of infinity, so we will put it aside for now. Let us pay attention to the fact that in order to prove that the paradox is preserved for any n, it is necessary to prove 2 statements:
1: For n = 1, there is a paradox (base case);
2: If there is a paradox for n, there is also a paradox for n + 1 (induction step).
Thus, if we can construct an induction from n to n + 1, then we can rigorously prove that for any n there is a paradox. Then the last option remains - all the statements in the list are false, but if they are all false, then what S1 asserts is true, and therefore S1 is true - we are back to where we started.
Wait a minute! If we prove induction, then after going through the entire natural series, we will return to where we started! We seem to be making a jump from infinity to the beginning! But what if induction provides hidden self-reference of statements? If so, then the paradox is resolved.
To summarize, we have four options for solving the paradox:
1: There is a self-reference in our set of statements by induction;
2: There is a self-reference, but not by induction;
3: There's no self-reference, but we can "take away" the true statement to infinity (this option raises questions about the nature of infinity);
4: There's no self-reference, and the paradox remains.