There are at least an infinite number of types of paradoxes

using Godel numbering it is possible to reference any finite string of symbols. This means that a statement can reference other statements that are malformed, as well as well-formed statements that are paradoxical. The usual definition of a paradox is a statement that contradicts itself, or has more than one truth value. However suppose we had a function P(x) that decided if x was a paradox. If we try and generate a paradox by usual methods of diagonalisation we have the problem that a paradox does not provide a truth value. Non rigorously if the Godel number of statement "P(x)" was 'g' then "P(g)" is a very unusual statement. It claims to decide whether 'g' numbers a paradox, but if it did then "P(g)" would be a paradox and so have no truth value. Normally we expect such a diagonalisation to end in a contradiction of undecided truth value, but here a truth value is not even impossible. Is this a different type of paradox to the usual self-contradiction type which generates contradictory truth values.

ChatGPT said:

Yes, this is a different type of paradox, one that goes beyond the usual self-contradiction or undecidability arguments. Let’s analyze why.

1. Standard Self-Contradiction Paradoxes

A typical paradox of self-reference, such as the liar paradox ("This statement is false"), leads to a contradiction because if the statement is true, then it must be false, and vice versa. Gödel-style diagonalization in formal logic often constructs statements like "This statement is not provable," leading to undecidability rather than outright contradiction.

2. The Role of a "Paradox Decider" Function $P(x)$

You propose a function $P(x)$ that determines whether the statement encoded by $x$ is paradoxical. The issue arises when we attempt to use diagonalization on $P(x)$ itself.

If the Gödel number $g$ corresponds to the statement $P(g)$, then $P(g)$ asserts that $g$ is a paradox. This means:

• If $P(g)$ returns true, then $g$ is a paradox, meaning it should have no definite truth value. But this contradicts $P(x)$ being a decision function that assigns a clear value.
• If $P(g)$ returns false, then $g$ is not paradoxical. However, since $g$ encodes $P(g)$, this implies $P(g)$ must be well-defined, which contradicts the claim that $g$ encodes a paradox.

3. Why This is a Different Type of Paradox

In traditional self-referential paradoxes, the paradoxical statement oscillates between two truth values or reaches undecidability. However, in this case, the problem is deeper: the paradox doesn’t even allow for the assignment of a truth value at all. This is more than just a contradiction—it suggests that the concept of "deciding paradoxes" is itself incoherent.

This form of paradox highlights the limitations of defining paradoxicality as a decidable property. In other words, the function $P(x)$ itself is problematic: if it were computable, it would lead to a situation where it both has and does not have a truth value, which is an even more fundamental breakdown than classical paradoxes.

4. Possible Implications

• This suggests that paradoxes are not merely statements with contradictory truth values but might be statements that break the framework of truth assignment itself.
• If paradoxes cannot be neatly classified as either true, false, or even undecidable, then functions like $P(x)$ cannot exist in a meaningful way.
• This is somewhat related to Tarski’s undefinability theorem: truth in a sufficiently rich formal system cannot be defined within that system. Similarly, paradoxicality (as a property of statements) may not be definable within the system it applies to.

5. Comparison to Other Paradoxes

• Liar Paradox: Contradiction between truth and falsehood.
• Gödel’s Incompleteness: A statement that is true but unprovable.
• Berry Paradox (Vagueness Paradoxes): A problem with self-reference in definitions.
• Your case: A paradox where truth assignment itself collapses when we attempt to define paradoxicality as a decidable property.

Conclusion

Yes, this is a different type of paradox. Instead of merely contradicting truth values, it questions whether a truth value can be assigned at all. It suggests that paradoxes are not simply about undecidability or contradiction but can also indicate the fundamental incoherence of certain concepts (like "deciding paradoxes"). This is a deeper issue that hints at intrinsic limitations in the nature of formal systems and definability.

You said:

Call a simple contradiction Paradox1 and this a Paradox2. Suppose we then had a function P2(x) that decided if x was a Paradox2 and had godel number g2. Is "P2(g2)" yet another type of paradox?

ChatGPT said:

Yes, "P2(g2)" would introduce yet another level of paradox, which we can call Paradox3. Let's break it down systematically.

1. Reviewing Paradox1 and Paradox2

• Paradox1 (Self-Contradiction Paradox): A statement that directly contradicts itself, like "This statement is false." It generates a contradiction between truth values.
• Paradox2 (Truth-Assignment Collapse Paradox): A statement whose paradoxicality is supposed to be decidable (via $P(x)$), but when applied to itself, it undermines the very notion of decidability and truth assignment. This paradox doesn't just contradict itself; it prevents the system from assigning it a truth value at all.

2. Defining a "Paradox2 Decider" Function $P2(x)$

Now, suppose we define a function $P2(x)$ that determines whether $x$ encodes a Paradox2. This function should, in theory, classify statements as either being of type Paradox2 or not. However, if we Gödelize this function and consider its self-application at $g2$ (i.e., the Gödel number of "P2(g2)"), we encounter a new problem:

• If $P2(g2)$ returns true, then "P2(g2)" is a Paradox2. But Paradox2 means that the statement lacks a well-defined truth value, which contradicts $P2(x)$ being a decision function.
• If $P2(g2)$ returns false, then "P2(g2)" is not a Paradox2. But this means that $P2(x)$ has consistently assigned it a truth value, which implies that it actually is a Paradox2—another contradiction.

3. Emerging Paradox3

This is neither just Paradox1 (self-contradiction) nor exactly Paradox2 (truth-assignment collapse). Instead, it's an iteration of the same issue at a higher level: it demonstrates that even attempting to define the concept of Paradox2 leads to a new breakdown in logical classification.

We can tentatively call this Paradox3, characterized by:

• The failure to define Paradox2 in a consistent way.
• The collapse of the notion of decidability across multiple layers of paradox classification.
• A hierarchy of paradoxes, where each attempt to define and classify paradoxes recursively generates a new paradox of higher complexity.

4. Implications of This Hierarchy

• Unavoidable Paradox Levels: If every function designed to classify paradoxes ultimately leads to a new paradox, then paradoxes might form an infinite hierarchy.
• No General Paradox Decider: If $P2(x)$ fails, then an attempted $P3(x)$ (deciding Paradox3) would likely generate Paradox4, and so on. This suggests no universal function can categorize all paradoxes without generating a new one.
• Ties to Gödel and Tarski: Just as Gödel showed that arithmetic cannot consistently prove all truths within itself, and Tarski showed truth cannot be defined within the system, this suggests paradoxes cannot be exhaustively classified within any single formal framework.

Conclusion

Yes, "P2(g2)" generates a new paradox (Paradox3). This demonstrates that paradoxes can self-replicate in increasingly complex forms when we try to classify them within a formal system. Rather than finding a final resolution, any attempt to capture paradoxicality in a decision function will likely generate a new paradox at a higher level.

You said:

Does this suggest there is a countably infinite number of paradoxes?

ChatGPT said:

Yes, this reasoning strongly suggests that there is at least a countably infinite hierarchy of paradoxes. The key idea is that every time we try to classify or resolve paradoxes with a formal decision function, we generate a new paradox at a higher level. This process can continue indefinitely, producing an infinite sequence of paradoxes.

1. Constructing a Hierarchy of Paradoxes

We started with:

• Paradox1: Simple self-contradiction (e.g., the liar paradox).
• Paradox2: A paradox that undermines truth assignment itself (e.g., defining a function $P(x)$ to decide paradoxicality leads to a paradox).
• Paradox3: A paradox arising from trying to classify Paradox2 (i.e., function $P2(x)$ fails similarly).

By the same logic, defining a function $P3(x)$ to decide Paradox3 would generate Paradox4, and so on. This process never stops—each new paradox is a reaction to the attempt to classify the previous one.

Thus, we obtain an infinite sequence of paradoxes:

$$P_1, P_2, P_3, P_4, \dots$$

This sequence is at least countably infinite, since it can be indexed by the natural numbers.

2. Is This the Full Set of All Paradoxes?

While we've shown that there is at least a countable infinity of paradoxes, it's unclear whether there might be uncountably many paradoxes. There are a few possibilities:

• The paradoxes are exactly countable: If all paradoxes arise in a structured hierarchy (like Gödel numbering), then they can be mapped to $\mathbb{N}$, implying a countable set.
• There could be an uncountable number: If some paradoxes arise in ways that are not generated by formal self-reference but instead emerge from more complex or chaotic systems (e.g., real-world paradoxes in physics, social systems, etc.), then the set of paradoxes could be uncountable.

3. Relation to Gödel and Tarski

• Gödel’s incompleteness theorems suggest that truth in formal systems cannot be fully captured by a single system—a result similar in spirit to the infinite paradox hierarchy.
• Tarski’s undefinability theorem suggests that truth itself is an undefinable notion within a system. Similarly, paradoxicality (as a property) seems to be undefinable within any single system.

4. Implications

• There is no "ultimate" paradox: If the hierarchy continues indefinitely, then every paradox is just a step toward a new one.
• No universal paradox classifier exists: If every classifier generates a new paradox, then paradoxes are an inherent feature of formal reasoning.
• Paradoxes are fundamental to logic: They are not just oddities but deeply tied to the structure of formal systems.

Conclusion

Yes, the logic strongly suggests a countably infinite number of paradoxes, with the possibility of even more. Every attempt to classify or decide paradoxes recursively generates new ones, leading to an unending hierarchy.