For question in Proof Theory, where "proofs" themselves are the object of mathematical investigation. It is not to be used to request a proof of some result.
-4
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0answers
55 views
Theory of Numbers. PROOF using division algorithm and well ordering theorem [closed]
I really help with the following two questions.
1) Assume that b>0. Show that there exists k an element of Z, s.t a+kb>0. (Use the division algorithm)
2)Use Theorem 3.8 (well ordering theorem) to ...
8
votes
0answers
304 views
What is known about the reverse mathematics of algebraic number fields?
I know work on the reverse mathematics of countable algebraic field extensions including Galois theory, notably including Dorais, Hirst, and Shafer http://arxiv.org/pdf/1209.4944v2.pdf. But algebraic ...
13
votes
1answer
420 views
Can ZFC prove it cannot derive an inconsistency in $n$ steps?
Let $Con(\mathtt{ZFC}, n)$ denote the statement "$\mathtt{ZFC}$ cannot prove the contradiction within $n$ steps (or better within $n$ symbols) within a given proof system (say a natural deduction to ...
2
votes
1answer
66 views
Notion of strongness in cut rule [closed]
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 ...
1
vote
2answers
251 views
What is the consistency strength of a standard model of ZF versus a transitive model?
A standard model of ZF need not be transitive, of course, and Joel David Hamkins' answer to Large cardinal axioms and Grothendieck universes gives Tarski sets as an interesting example.
I should ...
5
votes
1answer
173 views
Higher order arithmetic and fragments of ZFC
Zbierski "Models for Higher Order Arithmetics" (BULL. DE L'ACAD. POLONAISE DES SCIENCES Serie des sciences math., astr. et phys. - Vol. XIX, No. 7, 1971) defines ZF$_n$ as ZFC with the power set axiom ...
12
votes
1answer
745 views
Von Neumann's consistency proof
In the paper Zur Hilbertschen Beweistheorie, John Von Neumann has proposed a consistency proof for
a fragment of first-order arithmetic (the fragment without induction and with
the successor axioms ...
9
votes
2answers
262 views
When was Bounded Zermelo set theory first formulated?
Bounded Zermelo set theory, and many variants named for MacLane in some way, are used in equiconsistency proofs for Simple Theory of Types plus infinity, and for the Elementary Theory of the Category ...
7
votes
2answers
276 views
Peano arithmetic vs. fast-growing hierarchy with pathological fundamental sequences
Fundamental sequence for a countable limit ordinal $\alpha$ is an increasing sequence $\{\alpha[i]\}$ of ordinals of length $\omega$ such that $\lim_{i\rightarrow\omega}\alpha[i]=\alpha$. There are ...
7
votes
1answer
194 views
Which ordinals can be proof-theoretic ordinals of a reasonable theory?
When talking to a friend recently he asked a question - are there any reasonable first-order theories which have proof theoretic ordinal equal to small or large Veblen ordinal? I have then extended ...
1
vote
2answers
261 views
An interpretation of not-Con(PA)
Edit After Andreas Blass answer below and comments below the original post I have changed it to accommodate posters' remarks. I hope it is clear and makes more sense now.
Let $\mathrm{PA}$ be the ...
12
votes
1answer
584 views
Time in Girard's Geometry of Interaction
Jean-Yves Girard writes at the end of his paper
"Towards a Geometry of Interaction", page 105, that we have three intuitions about the nature of time:
time is logic modulo the order of rules,
time ...
6
votes
2answers
177 views
Models of PRA/EFA with induction on $X$ but not $\omega^X$
As I currently understand it, induction on formulas containing $N+1$ first-order quantifiers is required to prove the well-ordering of the ordinal $(\omega \uparrow\uparrow N) < \epsilon_0$, that ...
6
votes
1answer
306 views
Essential incompleteness via diophantine formulas?
Work in the first order language of number theory, consisting of the symbols $\mathbf{0}$, $\mathbf{S}$, $\boldsymbol{+}$, and $\boldsymbol{\cdot}$, and let $Q$ denote Robinson's arithmetic.
By a ...
3
votes
1answer
104 views
Results where complexity bounds implies finite number of test cases
We have all been there, when a formula works for the first 30 parameters,
but it is not sufficient for a proof. My question is where one can actually just check a finite number of cases, to conclude ...
7
votes
2answers
314 views
Proof complexity of two directions of equivalency?
This question is not precise, but I believe has a precise formulation.
Consider a mathematical theorem which gives an equivalency between two conditions. As an extreme example:
\begin{theorem}
A ...
11
votes
4answers
1k views
Does formalizing math require search and creativity, or is it near-mechanical?
I remember reading somewhere that it takes about a week to convert a page of math into something a proof-assistant like Isabelle or HOL Light would accept.
Is this type of conversion something that ...
11
votes
1answer
362 views
Is it possible for a theorem to be constructive only in a non-constructive metatheory?
There are several theorems in category-theoretic logic which say something like, "any proposition in X logic that is provable in topos logic assuming (the law of excluded middle and) the axiom of ...
4
votes
1answer
158 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 ...
0
votes
1answer
150 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 ...
4
votes
0answers
78 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 ...
6
votes
1answer
285 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 ...
23
votes
2answers
487 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
1answer
192 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
1answer
206 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 ...
8
votes
3answers
672 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
4answers
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
1answer
277 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 ...
7
votes
0answers
181 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
1answer
122 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
0answers
89 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 ...
2
votes
3answers
347 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 ...
5
votes
1answer
296 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?") ...
9
votes
1answer
301 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
1answer
239 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
0answers
326 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
0answers
126 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 ...
1
vote
0answers
419 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
2answers
262 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 ...
1
vote
0answers
165 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 ...
4
votes
2answers
385 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
1answer
265 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
1answer
670 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 ...
1
vote
1answer
209 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 ...
17
votes
2answers
841 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
1answer
230 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
1answer
479 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
2answers
247 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
0answers
220 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
1answer
489 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 ...
