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 - 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

Which ordinals are proof-theoretic ordinals?

Few months ago I have posted this question on MO, but I must admit that at the time, admittedly, I had no idea on how technical proof-theoretic considerations can be. I have decided to revise this question, and I thought that asking a new question is a better option that rewriting the old one (especially given that the other one has a partial answer to old question).Given theory $T$ and recursive relation $\prec$, we say that $T$ proves $\prec$ to be well-ordered if $T$ proves that $\prec$ is a linear order and that $\forall X:((\forall n\prec ...Read more

computational complexity - Characterizing visual proofs

``Proofs without words'' is a popular column in the Mathematics magazine. Question: What would be a nice way to characterize which assertions have such visual proofs? What definitions would one need?I suspect that in order to make this question precise, one will have to define a computational model for the ``visual verifier'' and postulate the possibility of a visual proof if there is a quick verification algorithm, and the visual proof itself is short.Rev. 1:To elaborate: the intriguing and essential feature of a visual proof is that the proof...Read more

Is Kolmogorov complexity (KC) relevant for proof theory?

Note. The title was modified. Previous title was"Every theorem t has a proof no more complex than~|t|. Is this right?"The question ("Is Kolmogorov complexity (KC) relevant for proof theory?") arises because every theorem has a proof with low Kolmogorov complexity". To be more specific,Every theorem $t$ has a proof $p$ with $K(p)\leq K(t)+c\leq|t|+c'$, where $c$ and $c'$ are constants.Proof: It is assumed that there is a proof checking algorithm $C(x)$ thatoutputs TRUE if $x$ is a correct proof, FALSE otherwise.Then there exists a fixed (dependi...Read more

Understanding the nature and structure of proofs; Reverse Mathematics and Proof Theory. Prerequisites? Good introductory texts?

I'm still studying maths at undergraduate level, but intend to continue exploring topics in pure maths after I have graduated, so am thinking already about what directions I'd like to persue now, (as I like to be several steps ahead of myself!)One area which seems particularly interesting is 'reverse mathematics'. I'd be interested to learn what the prerequisites would be for understanding it. Presumably it is a part of logic, and is related in some way to proof theory? If say, I wanted to explore questions such as, 'are there a finite number o...Read more

"Strange" proofs of existence theorems

This question isn't related to any specific research. I've been thinking a bit about how existence theorems are generally proven, and I've identified three broad categories: constructive proofs, proofs involving contradiction/contrapositive, and proofs involving the axiom of choice.I'm convinced that there must be some existence theorem that can be proven without any of these techniques (and I'm fairly confident that I've probably encountered some myself in the past haha), but I can't come up with any examples at the moment. Can anyone else c...Read more

proof theory - Applicability of Deduction theorem to Primitive recursive arithmetic

Hello. I already asked the question here. The main point is that I tried to prove in Primitive recursive arithmetic (PRA) the totality of the Ackerman function, and I found, that the single thing which can prevent it - nonapplicability of the Deduction theorem to PRA. But I know, that totality of the Ackerman function is unprovable in PRA. Does it mean, that the Deduction theorem is non-applicable to PRA?People commented, that: "the main reason that PRA does not prove the Ackerman function is total is that PRA does not include enough induction ...Read more

Writing "Semi-Formal" Proofs

I am very interested in proofs. I have taken an undergraduate coursecalled "Logic and Set Theory" which I found very interesting, but ultimatelyunsatisfying. My biggest disappointment has to do with the language in whichproofs are expressed. It seems to me that we have all of the symbols necessaryto express a proof in "pure math". By which I mean, only using symbols and afew specialized words (iff, let, ...). And yet most proofs that I have seen arejust walls of English text, interpolated with mathematical symbols.When I read a complex proof,...Read more

proof theory - Doesn't cut elimination make intuitionistic logic equivalent to classical logic?

Suppose we have a proof by contradiction of $A$, meaning we've proven $(A \to \bot) \to \bot$.If we eliminate all cuts in the proof, then the last step of the proof will be an implication elimination involving $A \to \bot$ and $A$, meaning that the proof will contain proof of $A$.Equivalently, if we have a lambda calculus term of type $(A \to \bot) \to \bot$, then its $\beta$-normal form must be of the form $\lambda f : A \to \bot. fX$ where $X : A$.Doesn't this mean that any proof involving double negation elimination can be rewritten (using c...Read more