Articles of higher order logic

Peano arithmetic with the second-order induction axiom

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 […]

advantage of first-order logic over second-order logic

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 […]

Henkin vs. “Full” Semantics for Second-order Logic and Multi-Sorted First Order Interpretations

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?

Is there any generalization of the hyperarithmetical hierarchy using the analytical hierarchy to formulas belonging to third-order logic and above?

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 […]

Any “natural” examples of true statements in number theory not provable in 2nd order systems?

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 […]

Independence results in first-order PA and second-order PA

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 […]

what are first and second order logics?

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

are there non-standard models of arithmetic in second order arithmetic?

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 […]

Is First Order Logic (FOL) the only fundamental logic?

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 […]

What's an example of a theory that's consistent yet has no model?

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 […]