### proof theory - Notion of strongness in cut rule

I've read somewhere that the cut rule in sequent calculus$$\frac{A \vdash \mathbf{C}, B \qquad A',\mathbf{C} \vdash B'}{A,A' \vdash B,B'} (\text{cut})$$states that the $\mathbf{C}$ on the right is stronger than $\mathbf{C}$ on the left.I would like to know what is this notion of strongness and how is $\mathbf{C}$ on the right stronger than $\mathbf{C}$ on the left....Read more

### np complete - Proof that SAT is NPC

I am not really sure I understand the idea behind Cook theorem (it says that SAT is a NP-complete problem).I read the proof with all its parts corresponding to the Turing machine TM solving it (TM is in a single state at any given time, only single cell is read by the head of TM in every single moment, only single symbol is on the tape etc...) + the fact that SAT is in NP (which is obvious)... but I just dont understand how does it all prove anything?The definition of NP-complete problem is, that a problem $U$ is in NPC, if all NP problems are ...Read more

### proof theory - Subscript 0 in Reverse Mathematics

What does the subscript 0 mean on terms like $\mathsf{ATR}_0$? Does it mean the same thing in $\Pi^1_k\text{-}\mathsf{CA}_0$?If I frame higher order analogues of these, should I change that subscript?...Read more

### proof theory - ERA, PRA, PA, transfinite induction and equivalences

I'm quite sure I don't understand very well the links between proof theoretical ordinals of theories, the axioms of transfinite induction and the objects a theory can prove to exist.For instance I'm considering Peano Axioms ($\mathbf{PA}$), of proof theoretic ordinal $\epsilon_0$, Primitive Recursive Arithmetic ($\mathbf{PRA}$) of proof theoretic ordinal $\omega^\omega$ and Elementary Recursive Arithmetic ($\mathbf{ERA}$), which is a fragment of $\mathbf{PRA}$.I was wondering if $\mathbf{PRA}+TI\{\alpha\in\epsilon_0\}$ (where $TI$ stands for tr...Read more

### proof theory - computational complexity of primitive recursive functions

If we have a rewrite system for primitive recursive functions, which simplifies each term according to how the function was defined, then what is the computational complexity of this calculation? That is, what is the coplexity of the normalization procedure? I have heard a claim that for a closed term calculating the value of the function requires transfinte induction up to $\epsilon_0$. Is this true and where can I find a proof of this? For example in (Schwichtenberg & Wainer 2012) there is a lemma which says that a primitive recursive fun...Read more

### proof theory - Am I counting quantifiers correctly?

I think this is right but I want to check. The theory $\mathsf{WKL}^*_0$ is conservative over EFA for $\Pi^0_2$ sentences. And the first order part of $\mathsf{WKL}^*_0$ is axiomatized by EFA plus the following formula scheme, known as the $\Sigma^0_1$ bounding principle. For every $\Sigma^0_1$ formula $\varphi(i,j)$ ( with $n$ not free):$$(\forall i<m)\, \exists j\, \varphi(i,j) \rightarrow \exists n\,(\forall i<m)(\exists j<n)\varphi(i,j).$$So this formula scheme is not $\Pi^0_2$ and I want to check that I see this correctly. I ...Read more