**4**

votes

**1**answer

188 views

### Gödel's speed up theorem and Matiyasevich polynomials

Unless I am sadly mistaken, there should exist a polynomial $ P\in\mathbb Z[X_1,X_2,\dots, X_n]$ coding for the speed-up theorem (for, say, ZFC), i.e. having the following properties :
1) P has an ...

**1**

vote

**1**answer

216 views

### notable inductive proofs relating to fractals

what are notable/ prominent inductive proofs relating to fractals?
the motivation for this question is:
fractals are very difficult mathematical objects to work with, and many problems/questions ...

**4**

votes

**0**answers

89 views

### $n$th order arithmetic with predicates for orders

Two papers I have looked at lately axiomatize $n$-th order arithmetic in a single sorted language with predicates $Z_1,\dots,Z_n$ and axioms like $\forall x(Z_1(x)\vee\dots\vee Z_n(x))$ to say ...

**7**

votes

**1**answer

330 views

### Does the totality of Ackermann's function prove the consistency of $\Sigma_1$-induction?

It is well known that Ackermann's function is not primitive recursive. Therefore, the theories of primitive recursive arithmetic (PRA) and of $\Sigma_1$-induction ($I\Sigma_1$) cannot prove the ...

**24**

votes

**2**answers

549 views

### Why is there no connection between fast-growing functions and complex analysis

I found myself wondering the other day whether the fast-growing functions from natural to naturals that are studied by people like proof theorists are the restriction to the naturals of analytic ...

**2**

votes

**1**answer

284 views

### what are the proof-theoretic ordinals of second-order arithmetic and ZFC? [duplicate]

are they still smaller than omega-1-CK?what are the notations of them?

**12**

votes

**1**answer

254 views

### Nontrivial upper bounds on proof-theoretic ordinals of strong theories: do we have any?

Motivated by Consistency of Analysis (second order arithmetic) and Proof-Theoretic Ordinal of ZFC or Consistent ZFC Extensions?, I have the following question:
Are there any examples of strong ...

**9**

votes

**3**answers

933 views

### Consistency of Analysis (second order arithmetic)

Is there a proof of the consistency of Analysis (second order arithmetic), which is similar to Gentzen's proof of the consistency of arithmetic?
Update:
Which (different) methods can be used to ...

**10**

votes

**4**answers

1k views

### Are there non-diagonal proofs for Cantor's continuum and Godel's incompletness theorems?

There is a formal definition for the notion of a formal proof.
Question 1. Is there any formal definition for the notion of a diagonal formal proof?
Consider the following theorems both proved by ...

**10**

votes

**1**answer

301 views

### Reverse mathematics of meromorphic functions on Riemann surfaces

Various sources touch briefly on the reverse mathematics of measure theory and complex analysis. But I have found none on the uniformization theorem for Riemann surfaces or the existence of non-...

**7**

votes

**0**answers

199 views

### When is a reduction not a reduction?

Every mathematician understands the concept of reducing a complicated problem to a simpler problem. "Without loss of generality, we may assume…" However, I've noticed that some kinds of "...

**1**

vote

**1**answer

136 views

### A question about consistent fragments of formalized mathematical theories with Natural Deduction

Ref to : Sara Negri & Jan von Plato, Structural Proof Theory (2001).
In Ch.6 : Structural Proof Analysis of Axiomatic Theories [page 126-on], they
give a method of adding axioms to sequent ...

**4**

votes

**0**answers

104 views

### Stabilization of recursive approximation in $PA^-+I\Sigma_1^0$

Over any model M of $PA^-+I\Sigma_1^0$. Suppose $A\in [T]$ where $T$ is a $\Delta_2^0$-tree and $A$ is one isolated path. Further, $A$ is regular, i.e. $\forall n A\upharpoonright n$ has a code in $M$....

**2**

votes

**3**answers

369 views

### Show that Z2 is not conservative over PA

It is well-known that $\mathsf{ACA}_0$ is a conservative extension of PA. I assume this theorem gets a lot of attention because $\mathsf{Z}_2$ is not conservative over PA. Thus there ought to be first-...

**5**

votes

**1**answer

338 views

### Is Kolmogorov complexity (KC) relevant for proof theory? [closed]

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?") ...

**10**

votes

**1**answer

407 views

### What is the proof-theoretic ordinal of PA + Con(PA), PA + Con(PA + Con(PA)) etc., and why?

I seem to remember having read that the proof-theoretic ordinal (sup of ordinals the theory can prove well-ordered) of $\mathsf{PA} + \mathsf{Con}(\mathsf{PA})$ is the same as that of $\mathsf{PA}$, ...

**2**

votes

**1**answer

264 views

### Elementary proof of bounds on factor polynomials

The question Getting a bound on the coefficients of the factor polynomial got very nice answers on Gelfond's theorem. But for work on proof theory of arithmetic I want a proof in arithmetic. The ...

**9**

votes

**0**answers

394 views

### “Hard” separation results in reverse mathematics (or similar)

This is a fairly broad question. In particular, I specify 5 questions (Q1, Q2.1, Q2.2, Q3, Q4) which for me all fall under one umbrella. Since this is unreasonably broad, I'm really interested in an ...

**6**

votes

**0**answers

136 views

### cut-elimination for infinitary logic

Takeuti (1987, 223) deduces a cut-elimination theorem for infinitary logic from the corresponding soundness-and-completeness theorems. However, is there a way to adapt the basic Gentzen-style ...

**2**

votes

**0**answers

544 views

### What is the role of the (formalized) omega rule in Ramified Analysis?

In the 1960's, Feferman and Schutte did groundbreaking proof-theoretic work to find out the strength of predicative systems of second-order arithmetic. They used the ramified theory of types, a ...

**3**

votes

**2**answers

379 views

### What is the proof-theoretic ordinal of Hyperarithmetical Comprehension?

As I discuss in my answer here, it seems to me that Solomon Feferman shows, on pages 10-11 of his seminal 1964 paper "Systems of Predicative Analysis", that if you consider predicative second-order ...

**2**

votes

**0**answers

186 views

### Has the Ramified Theory of Types been applied to Predicative Set Theories?

Questions of predicativity are well-studied in the context of arithmetic. We have a base theory, first-order Peano arithmetic. Some people, like Edward Nelson (in chapter 1 of his book) and Charles ...

**5**

votes

**2**answers

443 views

### Did Gödel prove that the Ramified Theory of Types collapses at $\omega_1$?

Second-Order Arithmetic is considered impredicative, because the comprehension scheme allows formulas with bound second-order variables that range over all sets of natural numbers, including the set ...

**4**

votes

**1**answer

374 views

### Can the Burgess-Hazen analysis of Predicative Arithmetic be extended to Transfinite Types?

Around page 300 of his book "Mathematical Thought and its Objects", Charles Parsons discusses the work of Edward Nelson, who believes that mathematical induction is impredicative, because it can be ...

**0**

votes

**1**answer

700 views

### Is there any danger far from home? (Edited & Revised Version) [closed]

The notion of formal proof is defined by finite sequences ($<\omega$ - sequences) of sentences. In some sense if a sentence $\sigma$ is (finitely) provable from the theory $T$ it is very "near" to ...

**0**

votes

**1**answer

220 views

### Does this algorithm terminate in all scenarios?

Let $x \in \mathbb{R}^p$ denote a $p$-dimensional data point (a vector). I have two sets $A = \{x_1, \dots, x_n\}$ and $B = \{x_{n+1}, \dots, x_{n+m}\}$, so $|A| = n$, and $|B| = m$. Given $k \in \...

**20**

votes

**2**answers

966 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

298 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 ...

**13**

votes

**1**answer

838 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

267 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

301 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

711 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

210 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

290 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

683 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 x\phi_{1}\to\phi_{1}...

**1**

vote

**2**answers

297 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

766 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?

**7**

votes

**7**answers

388 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

710 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

278 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

**6**

votes

**1**answer

419 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

404 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

1k 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

340 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

306 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

469 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 ...

**28**

votes

**0**answers

968 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

340 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

231 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 $\beth_0\dots\...