**18**

votes

**2**answers

880 views

### Deep theorems and long proofs

I ran across this discussion by Daniel Shanks,
"Is the quadratic reciprocity law a deep theorem?."
Solved and Unsolved Problems in Number Theory. Vol. 297. AMS, 2001. p.64ff.
which made me ...

**5**

votes

**1**answer

243 views

### the choice of representing formulas and Gödel's second incompleteness theorem

In Rautenberg's book (A Concise Introduction to Mathematical Logic, Universitext, Springer 2006), Gödel's second incompleteness theorem is stated:
Theorem 3.2 (Second incompleteness theorem). PA ...

**11**

votes

**1**answer

590 views

### Proof-Theoretic Ordinal of ZFC or Consistent ZFC Extensions?

Let the proof theoretic ordinal $\alpha$ of a theory $T$ be the least recursive ordinal such that $T$ does not prove that $\alpha$ is well-founded. This ordinal is intended to quantify in some sense ...

**9**

votes

**2**answers

255 views

### Cut elimination algorithms

Gentzen's Hauptsatz in first order logic includes an algorithm taking any proof in the sequent calculus with cut rule, and delivering a proof without cut rule (and with the subformula property). So ...

**3**

votes

**0**answers

240 views

### The substitution theorem in first order logic (finitely many variables)

We consider the language ${\cal L}=\{\in\}$ with an arbitrary set of variables $V$. Specifically we define ${\bf P}(V)$ as the free algebra of type $\{\bot,\to\}\cup\{\forall x:x\in V\}$ (with the ...

**2**

votes

**1**answer

564 views

### Hilbert style axiomatic proof or sequent Calculus?

I am puzzling with the question which of the two proof systems (Hilbert style axiomatic proofs or substructural Sequent Calculi) is the most discriminatory?
With discriminatory I mean is which proof ...

**8**

votes

**1**answer

205 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

268 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

573 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

288 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

726 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

249 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 ...

**16**

votes

**1**answer

667 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

272 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

318 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

400 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

968 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

300 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

269 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

426 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 ...

**24**

votes

**0**answers

891 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

282 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

210 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

696 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

2k 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

326 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

118 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 ...

**12**

votes

**0**answers

288 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

590 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

237 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

123 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

138 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

522 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

940 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

323 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

441 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

268 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

700 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

621 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

1k 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

192 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

135 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

246 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

584 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

346 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

433 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 ...