//* Once again they want to lower a question with a piranha attack, putting negative scoring without giving reasons in the comments. Do it, it is clear that this is an illustrated Twitter, where only repetition of what is already known is allowed, with no room for innovation or creative questioning.
"Self reference is a semantic notion. Don't warry." But if it has been obtained using self-application? In that case it is a syntactic problem, and perhaps demonstrable in the PR system.
The formula satisfied by all formulas that are not self-applicable, and only by them, is self-applicable if and only if it is not self-applicable.
Because, if it is self-applicable, it is inconsistent with its definition, because it should be satisfied only by formulas that are not self-applicable.
And if it is not self-applicable then it should satisfy itself, so it would be self-applicable.
It is a syntactic paradox, therefore it should not be expressible in Peano-Russell.
But in principle, it would seem that it can be constructed:
NoAut(x): (y) ~Subst(x, y, num(y), #x)
¿Your feedback?
Note 1: The definition of self-applicability can be formalized as follows:
Autap(x) <==> <==> (Ey) Subst(x, y, num(y), #x) && Proof(x)
But to be closer to Gödel's formalization, the Fixed Point theorem, or Diagonalization lemma, can be applied (Rudolf Carnap, 1934)
For every formula Z(x) with a single free variable, there exists a closed formula C, such that
|-- C <==> Z(#C)
Using it for self-application, C: Autap(#Autap(x)), Z: Proof(x):
|-- Autap(#Autap(x)) <==> <==> Proof(#{Autap(#Autap(x)))})
The meaning is "this sentence is self-applicable."
Note 2: There are three concepts involved in this topic:
- Satisfaction: a subject satisfies its predicate, expressed in a formula, when the conditions expressed in the formula are met. It is a purely syntactic notion, and not relational, that is, it does not depend on its relationship with other formulas.
- Demonstrability: a complete formula is demonstrable if there is a formal deduction from the logical axioms and those of the system, which has the formula in question as its last component. It is a syntactic notion as well, but it is relational; it depends not only on the conditions it expresses but also on its relationship with the axioms and the deduction rules used.
- Truth: a statement (complete formula) is considered true if the formal system has a model, and it can be shown metamathematically that that statement corresponds to a true statement in the model. It is a semantic notion, much more complex and controversial than the previous ones.
In summary, to know if a subject satisfies its predicate, it is enough to follow the rules defined in the predicate, applying them to the subject. To know if it is demonstrable, we must find a deduction from the axioms. In the case of propositional logic there is at least one universal method (truth tables), but in first-order logic, although it is demonstrated (Gödel 1930, completeness theorem) that every logically valid statement is provable, there is no method to find that demonstration. In the case of systems with their own axioms, it is necessary to find the demonstration of the internal deduction of the system, the model, and the correspondence with some sentence of the same without the help of any general method.
In this particular case, to know if a formula is self-applicable, you have to find some argument that satisfies it. To know if a sentence is self-applied, it is enough to examine whether the argument is the Gödel number of the predicate.