I had to attend a seminar today and it was very boring so i decided to write this instead.

I was thinking about russell's paradox and i watched an explanation for it. the video also said that russell's paradox does not actually say anything significant about set theory because this paradox actually shows up in predications themselves and not only in systems of logic. for example, " "is a cat" sounds funny" is true because if you say "is a cat" enough times, because of semantic fatigue it can sound funny. we can also form predicates of predicates. for example, " "is a predicate" is a predicate" is true because "is a predicate" really is a predicate. however, " "is a cat" is a cat" is not true since "is a cat" is a predicate and not a cat. now we can consider all predicates which are true of themselves and form a predicate that is true of itself, the predicate is " "is true of itself" is true of itself".

the relation between sets and objects is that sets contain objects and the relation between predicates and the subject is that the subject is true of the predicate. that means " "is true of itself" is true of itself" is the same as "the set of all sets that contain themselves". in order to re-generate russell's paradox in predicate form, we simply say "is not true of itself" and ask "is that predicate true or not true of itself?", if it is true of itself, then it is not true of itself and if it is not true of itself then it is true of itself.

i'll try to explain my thoughts about this through the polyform framework and try to relate it to incompleteness and proofs. now this is where i conjecture something else because it seems really obvious and maybe it is true idk. but to me, proofs or truths are just cross mappings between 3 systems in a particular structure and that when this 3 system cross mapping is absent, it creates unverifiability and when self-reference comes into the picture, it creates the paradoxes.

so consider the predicate " "is a cat" is a cat", here we can say that the predicate within the predicate "is a cat" is a state of S1 and write it as S1. similarly, "a cat" is a state of S2. the word "is" represents the mappings. So all predicates can be represented by the general mapping structure S1 —> S2. but when we make a statement by mapping two things together, we are not proving it, we don't know if it is true. thus, proofs or truths are when there exists states in a system of verification say S3 and there is a cross mapping between S1, S2 and their corresponding states in S3. So a statement is true when S1 —> S2, S1 —> S3, S2 —> S3, and S3 —> S3 (here we are talking about mapping of the state and their units and not the whole system). This essentially means that a predicate S1 —> S2 is true when their corresponding statements in a system of verification S3 have coherent mappings (in systems of logic we can say that both states in the S3 system must be able to be generated using the rules of S3 while maintaining axioms). Thus the predicate " "is a cat" is a cat" is false because we have mapped "is a cat" to a cat using our system of representation but not in the underlying system of proof. and the predicate " "is a predicate" is a predicate" is true because "is a predicate" is a predicate and "a predicate" is a correct description of it. here, the system of verification is a sort of english logic hybrid i suppose.

to now analyze the structure of the paradox we can represent the entire paradox as S1 —x—> S2, S1 —> S3, —> S2 —> S3 and we are trying to figure out if S3 —> S3. in order to understand the paradox, we have to know what S2 is. and what S2 maps to is essentially a state S3 in system of proof says that S3 —x—> S3. so if S3 —x—> S3, then then S1 —> S2, but S2 says S3 —X—> S3. S2 saying that S3 —x—> S3 is essentially the same as the system of S2 referencing itself since one of the units of S2 IS S2. so in a sense, its just like incompleteness in polyform but somehow not exactly. in polyform, a statement such as "the statement g is provable" is essentially like saying S1—>S2, S1—>S3, S2—>S3 and S3 —> S3. BUT since being provable means a mapping exists between units, one of the units here is S2 so there is no truth or proof here because the mapping just infinitely recurses. so this makes me think maybe incompleteness and russell's paradox are just slightly different cases of the cross mappings of proof that i mentioned earlier. it sort of ties these two issues and describes them in quite similar structures but slightly differently. in russell's paradox, we make a statement in a system that is set theory and then we want to know if that statement is true or not true. we write "the set of all sets that do not contain themselves contains itself" and written in a more verbose way is "the set of all sets that do not contain themselves contains the set of all sets that do not contain themselves". here s1 is the set of all sets that so not contain themselves, and s2 is the set of all sets that do not contain themselves. "contains" here is analogous to "is true of" in the system of predicates. let me reframe this in set theory notation: R = {x | x is a set and x ∉ x} is russell's paradox, here s1 is x and "contains" is implied by the presence of x between the curly brackets aka R "contains" x, and s2 is x but the 2 xs serve different purposes. in this case, x is a set and an object at the same time as in the case of the previous examples i mentioned (predicates).

proofs can be obtained using other methods such as simple extension. but since representations of systems and the systems are different, when we construct a statement using the extension rules of the representational system, that doesn't necessarily guarantee that those manipulation rules preserve the axioms of the underlying logic system even if it preserves the axioms of the first system.

the self reference problem does NOT arise from s3 mapping to s3. when s3 —> s3, it means they are states which are a part of the same system and when s3 —x—> s3, they are states which are not a part of the same system. this is significant for s3 —> s3 and not for s1 —> s2 because s1 —> s2 IS the thing we want to prove, it is what we constructed. but s3 —> s3 has to hold in order to prove the truth of the statement because it is the system in which actual logical manipulation is done and not the representational layer.

suppose we have representation layer R, logical layer L and verification layer V. we can say that a statement is true when R —> L —> V. when a mapping fails between any of these layers, it is said to be false. now if the case is of self reference, paradoxes are generated when R loops back in on itself, Incompleteness when L loops on itself and undecidability when V loops on itself.

clarification: s doesn't mean system, s means state. what we want to do is prove that the states are a part of the same system. S1 maps to S2 is a statement about the states. when the mapping holds they are a part of the same system. just like how when we say s3 maps to s3, we are saying that a state in s3 maps to s3. and when we say s1 maps to s3, we are saying the states are the same even though the system is different. same with s2 maps to s3. i wrote this in a book and the s1 to s3 and s2 to s3 mappings were written vertically so it made the distinction clear which i now realize is not clear when writing everything horizontally sorry