I am in the middle of my PhD and I am trying to reinforce my knowledge of mathematics by studying the foundations of Analysis. The first task is to get the bases of the natural numbers. So for this I chose the ZFC axioms which gives the consistence of the PA axioms. My first question […]

As I look over the post that has the similar question, I began to wonder: The only reason I found is that first-order logic can prove validity of some second-order logic formula/sentences, as some of second-order logic can be modelled by first-order logic, as provided by compactness/completeness theorem. However, according to my knowledge, as second-order […]

In this paper by Jeff Ketland, he notes: With Henkin semantics, the Completeness, Compactness and Löwenheim-Skolem Theorems all hold, because Henkin structures can be re-interpreted as many-sorted first-order structures. What is it about the full semantics for second-order logic that defies re-interpretation into a many-sorted first-order logic?

As I understand, hyperarithmetical sets are defined according to the analytical hierarchy, that is, second-order-logic formulas. There is a generalization of hyperarithmetic theory named α-recursion theory. Do this extension generalizes the hyperarithmetical sets to sets definable with any arbitrary higher-order logic? NOTE: I know that α-recursion theory extends the hyperarithmetical sets from ${\omega}_1^{CK} $ to […]

I know that there are a few theorems in number theory that are somehow known to be true, but have been shown not to be provable in first-order Peano arithmetic (PA). Have any so-called “natural” examples of true statements in number theory (not some variation of the Liar Paradox) that have been shown to be […]

There are statements $\varphi$ that are independent of first-order Peano Axioms. Are these statements also independent of second-order Peano Axioms? I’m reading Wikipedia articles around independence results and it has come evident that the second-order formulation of PA is strictly more powerful than first-order formulation. I don’t, however, see if this has any connection to […]

This question already has an answer here: First-Order Logic vs. Second-Order Logic 3 answers

non-standard models of arithmetic in second order arithmetic? Background: According to Godel’s theorem, if we have, in a given consistent system S, a non-provable wff. A, then we can extend the system to either S1 or S2 by including A or ~A as a new axiom, respectively. Both S1 and S2, according to Godel, will […]

I’m far from being an expert in the field of mathematical logic, but I’ve been reading about the academic work invested in the foundations of mathematics, both in a historical and objetive sense; and I learned that it all seems to reduce to a proper -axiomatic- formulation of set theory. It also seems that all […]

By the completeness theorem for first order logic, if a theory is consistent then it has a model. But what’s a counterexample to this : what’s an example of a logic where some theory is consistent yet has no model? More specifically, I want to know about second order logic… Since from what I understand […]

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