**4**

votes

**1**answer

169 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 ...

**12**

votes

**1**answer

529 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) ...

**7**

votes

**5**answers

835 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

275 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

101 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 ...

**9**

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

247 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

347 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

188 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

102 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

131 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

364 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

781 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

69 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 ...

**7**

votes

**0**answers

305 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

355 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

190 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 ...

**5**

votes

**3**answers

551 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

480 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

603 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

170 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

123 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

242 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 ...

**22**

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

559 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

325 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

404 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

736 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

460 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

458 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

326 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 ...

**7**

votes

**2**answers

581 views

### Ordinal Analysis of Peano Arithmetic with Restricted Induction

If we take Peano Arithmetic and restrict induction to formulas over various fragments of the arithmetic hierarchy, say to the $\Sigma^0_n$ formulas for various $n$ or some other interesting fragments, ...

**3**

votes

**3**answers

2k views

### About the proof of the proposition “there exists irrational numbers a, b such that a^b is rational”

What does the classical proof of the proposition "there exists irrational numbers a, b such that $a^b$ is rational" want to reveal? I know it has something to do with the difference between classical ...

**1**

vote

**2**answers

257 views

### Positive & Negative Arity

Hi,
You can talk about the arity of a function or an operation - something like addition could have an arity of 2, and negation usually has an arity of 1.
A paper I am reading is talking about ...

**17**

votes

**6**answers

2k views

### Writing “Semi-Formal” Proofs

I am very interested in proofs. I have taken an undergraduate course
called "Logic and Set Theory" which I found very interesting, but ultimately
unsatisfying. My biggest disappointment has to do ...

**8**

votes

**5**answers

2k views

### Why is it OK to rely on the Fundamental Theorem of Arithmetic when using Gödel numbering?

Gödel's original proof of the First Incompleteness theorem relies on Gödel numbering.
Now, the use of Gödel numbering relies on the fact that the Fundamental Theorem of Arithmetic is true and thus the ...

**11**

votes

**1**answer

874 views

### Proof theoretic ordinal

In Ordinal Analysis, Proof-theoretic Ordinal of a theory is thought as measure of a consistency strength and computational power.
Is it always the case? I. e. are there some general results about ...

**1**

vote

**2**answers

595 views

### Equational logic

I'm a beginner to this. Can anyone please point me to any resources for studying about equational logic, preferably with some example proofs to wet my feet in?
Thanks in advance!

**9**

votes

**2**answers

731 views

### Asymptotic density of provable statements in ZFC

This question is in response to one of the questions asked here. The OP wanted to know if the percentage of statements provable from ZFC tended to some value, and if so, what it was. In particular, ...

**5**

votes

**4**answers

1k views

### Zero-knowledge proof that 0 = 1

Suppose one day I came up with a proof that 0 = 1 in some formal system such as PA or ZFC that cannot prove its own consistency (unless it is inconsistent). Would it be possible to have a ...

**6**

votes

**1**answer

276 views

### Strength of Transfinite Induction on the Difference Hierarchy

I'm wondering if a particular theory of second order arithmetic has been studied or is known to be equivalent to some other theory.
Consider the formulas generated by $\Pi^1_1$ and $\Sigma^1_1$ ...

**4**

votes

**5**answers

833 views

### “Let $x \in A$”, beginning a proof of “$\forall x \in A$ …”, if A were empty [closed]

I work at a four-year teaching school, where we pride ourselves on teaching pure math, proof, and a rather obsessive carefulness of work. Recently I have been criticized for saying that "Let $x \in ...

**5**

votes

**6**answers

1k views

### Difference between turnstile and implication

Hi,
Does anyone know the difference between proving that
|- phi
------------------
|- ( psi -> phi )
and proving that
...

**4**

votes

**2**answers

348 views

### What is the depth of the “provability heirarchy”?

I am not a logician or set theorist, so hopefully this makes sense. Let $T$ be a theory which is expressive enough to make statements like "Statement $A$ has a proof in $T$"; for example, $T$ might ...

**-1**

votes

**5**answers

784 views

### finding cutting edge papers and books

Hi all,
what are the best strategies to find cutting edge papers and books on a field of mathematics?
..
Example:
2-3 years ago I had to analyze a time series. I found a paper and showed that to ...

**4**

votes

**3**answers

475 views

### Can a typing judgment admit essentially different derivations?

In any sort of type theory, there are a bunch of rules for constructing derivations of typing judgments such as $x:A,\; y:B(x) \;\vdash\; z:C(x,y)$. (I intend to include also judgments of the form ...

**3**

votes

**1**answer

523 views

### Derivability conditions for Robinson arithmetic

Two pieces of hearsay I have encountered about Robinson's Q:
Q fails to satisfy the Löb derivability conditions;
Pudlák criticised the Löb derivability conditions and suggested rival, weaker ...

**23**

votes

**6**answers

3k views

### Why can't proofs have infinitely many steps?

I recently saw the proof of the finite axiom of choice from the ZF axioms. The basic idea of the proof is as follows (I'll cover the case where we're choosing from three sets, but the general idea is ...