# Tagged Questions

In mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and a stronger semantics. (Def: http://en.m.wikipedia.org/wiki/Higher-order_logic)

105 views

### $\exists x ~ \forall y ~ f(x, y) \iff \forall y() ~ \exists x~ f(x, y(x))$ Name? Proof?

I was looking at potential theorems, and this one came up: $$\bigg(\exists x ~ \forall y ~:~ f(x, y) \bigg) \iff \bigg(\forall y() ~ \exists x ~:~ f(x, y(x))\bigg)$$ (where the second $y$ is ...
54 views

143 views

### Types versus kinds and sorts

In the context of logic, especially Higher‑Order‑Logic and Calculus‑of‑Construction, what is a kind and how does it relates to and differs from a type? My raw guess if that a kind is the higher level ...
249 views

### Why aren't there any first-order sentences which have the property of being true in all non-standard models of PA and false in the standard one?

I'd like to know where the following result comes from (that is, whether there is a more general result from which it follows or else how it can be proven): There is no first-order sentence which is ...
87 views

### A consistent first-order theory whose impredicative second-order variant is inconsistent

Let's assume that we have a consistent first-order theory, which was derived from a second order theory by replacing universal quantification over second order variables by axiom schemes for ...
315 views

### annihilator method confusion

I have a final in the morning and I am extremely confused on the annihilator method. I have been googling different explanations all night and I just dont get it at all. I am looking at an example: ...
109 views

### Does second-order arithmetic (Z2) prove soundness and uniform reflection for first-order arithmetic (PA)?

(1) Does full second-order arithmetic (Z2) prove soundness and uniform reflection schemas for first-order arithmetic (PA)? That is, do we have for all formulas $\phi$:  \underset \phi \forall \; ...
319 views

### There is a second-order sentence that is valid in standard semantics but not valid in Henkin semantics?

Let $\Sigma^\mathrm{ST}$ be a set of sentences that is valid in standard semantics and $\Sigma^\mathrm{Henk}$ be a set of sentences that is valid in Henkin semantics. Since ...