**8**

votes

**1**answer

202 views

### The Hasse Minkowski theorem in Peano arithmetic

Harvey Friedman's "Concrete Mathematical Incompleteness" at http://www.math.osu.edu/~friedman.8/pdf/0.Intro061311.pdf cites the Hasse Minkowski theorem saying quadratic forms over a number field are ...

**1**

vote

**1**answer

256 views

### Ordinal Exponentiation in Genzen's Sequent Calculus

For Genzen's sequent calculus with PA axioms, why is the proof-theoretic ordinal $\epsilon_0$? This seems to hinge on what exactly it means for the level of a cut or CJ inference figure to be higher ...

**5**

votes

**1**answer

532 views

### Axiomatization of first order logic (finitely many variables)

Standard textbooks in mathematical logic will assume an infinite supply of variables. Their axiomatization of first order logic will typically contain an axiom of the form $\forall ...

**1**

vote

**2**answers

279 views

### Embedding of consistent subset in first order logic (finitely many variables)

I am looking at FOL with no equality, no constant, no function symbol and the unique binary predicate $\in$ with variables in arbitrary sets $V$ or $W$. Specifically we define ${\bf P}(V)$ as the free ...

**12**

votes

**1**answer

699 views

### Reverse mathematics of Hilbert's Theorem 90

What is known, and what is published, on the reverse mathematics of the nest of results called Hilbert's Theorem 90?

**5**

votes

**6**answers

247 views

### Strength of Bishop style constructive mathematics vs $\mathsf{RCA}_0$

This question came out of this other MO question of mine. My question is
Is there a formal comparison between $\mathsf{RCA}_0$ and $\mathsf{BISH}$ (Bishop style constructive mathematics as used ...

**14**

votes

**1**answer

563 views

### Goodstein's theorem without transfinite induction

Goodstein's theorem is an example of a theorem that is not provable from first order arithmetic. All proofs of the theorem seem to deploy transfinite induction and I've wondered if one could prove the ...

**1**

vote

**3**answers

267 views

### Is there an recursively axiomatized system with infinitely many proofs for some propositions or a proposition [closed]

Is there any recursively axiomized system with infinitely many proofs for some propositions or a proposition? So we will have at least one proposition which is deduced from the recursively axiomatic ...

**12**

votes

**6**answers

2k views

**5**

votes

**1**answer

288 views

### Is the equivalence between a $\Sigma^0_1$ and a $\Pi^0_1$ formula defining the same recursive set provable in a sufficiently strong arithmetic ?

Let $A$ be a recursive set. $A$ is recursively enumerable, so $A$ may be defined by a $\Sigma^0_1$ formula, i.e. by $\exists \overrightarrow{a} \phi (\overrightarrow{a}, n)$, where $\phi$ contains no ...

**11**

votes

**1**answer

397 views

### Does any lower bound on proofs of FLT improve Shepherdson 1965?

In 1965 Shepherdson proved that FLT is independent of the fragment of PA that uses only open induction and signature $0,S,+\times$. Indeed $2x+1\neq 2y$ is independent of that fragment. Schmerl ...

**8**

votes

**2**answers

925 views

### When does $ZFC \vdash\ ' ZFC \vdash \varphi\ '$ imply $ZFC \vdash \varphi$?

Being a new member, I am not yet sure whether my question will be taken as a research level question (and thus, appropriate for MO). However, I have seen similar questions on MO, couple of which led ...

**12**

votes

**1**answer

276 views

### What metatheory proves $\mathsf{ACA}_0$ conservative over PA?

Simpson's book shows $\mathsf{ACA}_0$ is conservative over $\mathsf{PA}$ in the natural way by model theory using definable subsets. Of course, $\mathsf{ACA}_0$ being conservative over PA is ...

**5**

votes

**1**answer

242 views

### Interpretability and consistency strength

I have heard there is some fairly recent result showing that whenever theories $T$ and $T'$ have the same consistency strength, then each can interpret the other. I suppose it refers to first order ...

**9**

votes

**2**answers

417 views

### Reverse mathematics below RCA

I'm sure this is a fairly basic question, but I can't seem to find a solid answer:
My primary question is: Is there a reasonably nice subsystem of second-order arithmetic corresponding essentially to ...

**22**

votes

**0**answers

848 views

### Godel on recursion-theoretic hierarchies

At the end of his excellent article, "The Emergence of Descriptive Set Theory" (http://math.bu.edu/people/aki/2.pdf), Kanamori writes:
"Another mathematical eternal return: Toward the end of his ...

**4**

votes

**1**answer

264 views

### 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 ...

**4**

votes

**1**answer

201 views

### Notation for upperbound power sets.

There is a standard notation $\mathrm{ZF}[n]$ for Zermelo Fraenkel set theory with the power set axiom restricted to saying the set of natural numbers has $n$ successive power sets ...

**14**

votes

**1**answer

650 views

### Does Taranovsky's system of ordinal notations make sense?

Dmytro Taranovsky has a Web page on which he claims to define a system of ordinal notations strong enough to provide an ordinal analysis of full second-order arithmetic. I think (perhaps unjustly) ...

**8**

votes

**5**answers

1k views

### 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 ...

**2**

votes

**1**answer

303 views

### Sequent calculus: is there a complete linear reasoning (i.e., no trees)?

In Gentzen's sequent calculus, a formal proof is described by a tree, with each node representing the sequent obtained from the child(ren) by applying a given inference rule.
If no inference rule has ...

**2**

votes

**1**answer

113 views

### Role of statistical estimation in formal proof

Consider the following scenario: There is some mathematical constant $c$ that you want to compute. You don't have a formal proof for any particular value of $c$, but you have some sound statistical ...

**10**

votes

**6**answers

1k views

### Non-constructive proofs vs. efficient algorithms

My question concerns what is meant by "nonconstructive", and whether it has ever been defined in terms of computational complexity.
The wikipedia article on constructive proof begins, "a constructive ...

**11**

votes

**0**answers

277 views

### How to measure the strength of Zermelo over bounded Zermelo?

Bounded Zermelo is Zermelo set theory with only bounded separation. It has the same strength as simple type theory or MacLane set theory or ETCS. It is a finitely axiomatized fragment of Zermelo, so ...

**2**

votes

**3**answers

521 views

### Existential instantiation in Hilbert-style deduction systems

In some deduction systems there is a rule* that given $\exists x (\phi(x))$, we can infer $\phi(y)$, where $y$ is a fresh variable (i.e., one we haven't yet mentioned in this context). Call this rule ...

**5**

votes

**1**answer

224 views

### Arithmetic strength of Peano + the Howard ordinal

Consider the theory $\mathrm{PA}+\mathrm{BHO}$ consisting of first-order Peano arithmetic ($\mathrm{PA}$) enriched by an axiom scheme which allows well-founded induction up to any ordinal less than [a ...

**3**

votes

**0**answers

119 views

### Why the choice of pairing function in Subsystems of Second Order Arithmetic?

Simpson's book uses a pairing function $\langle i,j\rangle = (i+j)^2+j$. Is that choice of function simply unimportant, or does it have expository advantages over the Cantor pairing, or does it have ...

**3**

votes

**1**answer

136 views

### Does $\Pi^1_{\infty}$ comprehension imply ATR$_0$?

$\Pi^1_{\infty}\text{-}\mathsf{CA}_0$ proves existence of models of ATR$_0$. But I think it does not imply ATR$_0$, because Axiom Beta is a kind of replacement axiom. Is that right?

**5**

votes

**2**answers

481 views

### 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 ...

**8**

votes

**4**answers

892 views

### How many well orderings of $\aleph_0$ are there?

What is known about the set of well orderings of $\aleph_0$ in set theory without choice? I do not mean the set of countable well-order types, but the set of all subsets of $\aleph_0$ which (relative ...

**2**

votes

**0**answers

71 views

### Seeking name for an order raising operator in Higher Order Arithmetic.

Any class $X$ of order $j$ in HOA is in bijection with the order $j+1$ class built up from singletons $\{x\}$ of natural numbers $x$ just the way that $X$ is built up from the numbers $x$. And of ...

**8**

votes

**0**answers

319 views

### Can second order arithmetic make $\aleph_1^L$ countable?

Simpson's book Subsystems of Second Order Arithmetic shows $Z_2$ can interpret some fragments of ZF strong enough to give good theories of constructible sets and formalize statements like "there is a ...

**4**

votes

**2**answers

413 views

### When are provability predicates provably equivalent?

Fix notation
Suppose that $Prf_1(m, n)$ is the numerical relation that holds when $m$ numbers a $T$-proof of the sentence numbered $n$, according to scheme 1 for numbering wffs and sequences of wffs. ...

**4**

votes

**1**answer

247 views

### History of provably total functions of a theory

Provably total functions of an arithmetical theory is one of the tools used in proof theoretic analysis of theories.
I am looking for early history of its development. In particular,
Where was ...

**6**

votes

**3**answers

657 views

### 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 ...

**1**

vote

**3**answers

580 views

### Applicability of Deduction theorem to Primitive recursive arithmetic [closed]

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 ...

**1**

vote

**1**answer

987 views

### How to explain, that logic is the correct way of describing any system, process, etc? [closed]

Logic is the philosophical study of valid reasoning. Mathematical logic is an extension of symbolic logic (which is extension of formal logic) into other areas, in particular to the study of model ...

**0**

votes

**0**answers

182 views

### subformula property (anchored proofs)

Hello,
I would like to ask for some explanation on some property of propositional sequent calculus.
The sequent calculus that I use here follows that of Stephen Cook, in "Logical Foundations of ...

**0**

votes

**0**answers

132 views

### Implications of complex solutions of Matiyasevich / Chaitin diophantine polynomials.

This is a shot in the dark: In twf:202, an isomorphism $T\cong T^{7}$ between binary trees $T$ and seven tuples of binary trees T^{7} is mentioned. The argument for this isomorphism starts with the ...

**4**

votes

**2**answers

244 views

### Is there any literature about inner-replacement rule?

Hello all,
If you have a theorem $\vdash \alpha \rightarrow \beta$ and a theorem $\vdash \gamma$ then if $\alpha$ is a sub-expression of $\gamma$, and this sub-expression has an even number of ...

**24**

votes

**15**answers

4k views

### What's a magical theorem in logic?

Some theorems are magical: their hypotheses are easy to meet, and when invoked (as lemmas) in the midst of an otherwise routine proof, they deliver the desired conclusion more or less ...

**3**

votes

**1**answer

578 views

### Proof system with same complexity as “informal mathematics”?

The Completeness Theorem in first-order logic states that any mathematical validity is derivable from axioms. Hence, any informal mathematical proof (which is rigorous) can be translated into a formal ...

**1**

vote

**2**answers

334 views

### Would intuitionistic refutation method imply permutation of premisses?

Dear All
In the classical refutation method, one searches for a proof of $\Gamma, \lnot A \vdash \bot$ instead of $\Gamma \vdash A$. The method works, i.e. is complete and correct, since it is for ...

**1**

vote

**1**answer

427 views

### Is forward chaining also a form of focusing?

Dear All
Lets restrict ourselfs to logical theories which consist only of formulas $P_1 \supset \quad ... \quad P_n \supset Q$, i.e. propositional horn clauses expressed with implication. Lets only ...

**2**

votes

**1**answer

740 views

### How establish conversion of cut-free proof into uniform proof?

Dear All
Gentzen (*) claimed that through cut-elimination, he can normalize proofs. It is well known that cut-eliminated proofs might still contain some unnecessary noise. I am trying to show that ...

**3**

votes

**1**answer

520 views

### Feferman's extensional and intensional applications of the method of arithmetization

At the very beginning of Feferman's Arithmetization of metamathematics in a general setting it can be read:
The method of arithmetization, as developed by Gödel[10], exploits the possibility of ...

**10**

votes

**2**answers

2k views

### Consequences of technically proving anything in Coq (on at least Linux) exploiting a bug? [closed]

Technically, it is possible to prove anything in Coq proof assistant [1] (on at least Linux) due to a programming feature (or bug). This seems tractable when validating large proofs. Human analysis ...

**10**

votes

**1**answer

476 views

### Left-bracketed Ackermann function also not primitive recursive?

The original Ackermann function $\varphi\colon \mathbb{N}\times\mathbb{N}\times\mathbb{N}_0\to \mathbb{N}$ as defined in [1] was invented to prove that there is a function that is recursive but not ...

**7**

votes

**1**answer

333 views

### Looking for papers and articles on the Tarskian Möglichkeit

Some background: Łukasiewicz many-valued logics were intended as modal logics, and Łukasiewicz gave an extensional definition of the modal operator: $\Diamond A =_{def} \neg A \to A$ (which he ...

**18**

votes

**8**answers

3k views

### Proofs of Gödel's theorem

I am interested in different contexts in which Gödel's incompleteness theorems arise. Besides traditional Gödelian proof via arithmetization and formalization of liar paradox it may also be obtained ...