All Questions
Tagged with ordinal-analysis proof-theory
32 questions
16
votes
1
answer
2k
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 ...
- 161
15
votes
1
answer
605
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 ...
- 21.1k
14
votes
1
answer
1k
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 ...
- 28.2k
13
votes
1
answer
982
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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 $\mathsf{PA}$, ...
- 151
13
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3
answers
1k
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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 ...
- 28.2k
10
votes
1
answer
735
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 ...
- 32.5k
10
votes
1
answer
258
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Is there a relation between type (maximum linearization) of a computable WQO and the ordinal strength of a theory needed to prove it?
Background:
Given a well partial order $X$ (more commonly studied with antisymmetry dropped as well-quasi-orders, but I'm going to say well partial order to make this definition simpler, obviously ...
- 2,585
10
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1
answer
358
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$Π^0_1$ Proof Ordinals
Natural theories extending EFA (exponential/elementary function arithmetic) are well-ordered by $Π^0_1$ provability, and we would like a formal definition of the well-ordering that is robust yet as ...
- 7,545
10
votes
1
answer
373
views
Complexity of induction formulas in proof theoretic ordinals
According to The Art of Ordinal Analysis, the proof theoretic ordinal of a theory $T$ is the least ordinal $\alpha$ such that:
$${\bf ERA}+TI(\alpha,ECP)\vdash Con(T)$$
In above definition, $ECP$ ...
- 1,689
9
votes
1
answer
643
views
Gentzen's result on PA
The Wikipedia states that Gentzen proved that "in modern terms, the proof-theoretic ordinal of PA is $\varepsilon_0$." Further down, the article defines what the "proof theoretic ordinal" of a theory ...
- 1,174
9
votes
1
answer
1k
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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 ...
9
votes
0
answers
264
views
Consequences of recent claims of Ordinal Analysis of $Z_2$
Recently Toshiyasu Arai submitted "An ordinal analysis of $\Pi_{N}$-Collection" and Henry Towsner submitted "Proofs that Modify Proofs", both of which claim ordinal analysis of ...
- 161
8
votes
2
answers
560
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 ...
8
votes
2
answers
824
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, ...
- 1,439
8
votes
1
answer
295
views
Correspondence between proof-theoretic ordinals and fast growing functions?
For theories with well known proof-theoretic-ordinals, (what) is there a correspondence between their proof-theoretic-ordinal and (ordinal indexed families of?) fast growing functions provable total ...
- 225
8
votes
1
answer
706
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Which ordinals are proof-theoretic ordinals?
Few months ago I have posted this question on MO, but I must admit that at the time, admittedly, I had no idea on how technical proof-theoretic considerations can be. I have decided to revise this ...
- 28.2k
6
votes
1
answer
292
views
Does $\text{ACA}_0$ + True Arithmetic prove the well-foundedness of every recursive ordinal?
As discussed in Noah Schweber's answer to What is the proof-theoretic ordinal of true arithmetic?, it is somewhat ambiguous what “the proof-theoretic ordinal of True Arithmetic” might mean. In one ...
- 4,587
6
votes
0
answers
181
views
Iterated $\Pi^1_1$-reflection and non-Gandiness underrepresented in ordinal analyses?
This is a copy of Math StackExchange question #4395977, which I felt was more appropriate for MathOverflow.
Note on terminology: "admissible", "$(^+)$-stable", and "$\Pi^1_1$-...
- 2,031
6
votes
0
answers
183
views
Ordinal strength of iterated truth theories
Consider the theory ${\rm PA}^{\mathbb{T}}$ obtained by adding a truth predicate to Peano arithmetic, applicable to sentences of the unaugmented language and satisfying the compositionality axioms $\...
- 42.8k
6
votes
0
answers
421
views
What is proof-theoretic ordinal of weak first-order arithmetic?
According to Wikipedia(https://en.wikipedia.org/wiki/Ordinal_analysis) and nlab(https://ncatlab.org/nlab/show/ordinal+analysis), a proof-theoretic ordinal of $\mathsf{PRA}$ is $\omega^\omega$.
...
- 178
5
votes
1
answer
720
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 ...
- 4,587
5
votes
1
answer
360
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A possible flaw in Theorem 14.17 in Kurt Schütte's -Proof Theory-
Reading Chapter V, pages (73-97) in Proof Theory (Springer, 1977), by Kurt Schütte, I have encountered a peculiar problem which puzzles me.
On page 96, a map $\rm{Nr}:\overline{\rm{OT}}\rightarrow \...
- 2,136
5
votes
0
answers
265
views
$Π_2$ strength of KP
I am looking for a characterization of the $Π_2$ statements provable in KP.
Here, KP (often denoted KPω) is the Kripke-Platek set theory, including infinity and full induction on ordinals. Here is ...
- 7,545
4
votes
1
answer
887
views
Going beyond the strength of Peano arithmetic "without sets"
First-order arithmetic is fairly weak, as measured for example by its consistency strength. When a stronger theory is desired, it is common to work with (fragments of) second-order arithmetic or set ...
- 3,612
4
votes
0
answers
367
views
Ordinal analysis and nonrecursive ordinals
Ordinal analysis is typically described as characterizing recursive ordinals in a theory $T$, but there is a sense in which it can characterize all $T$-ordinals, even those that are nonrecursive.
...
- 7,545
3
votes
2
answers
814
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 ...
- 4,587
3
votes
0
answers
144
views
Partial well-ordering of formulas
Given a theory $T$, for arbitrary formulas $φ$ and $ψ$ that provably in $T$ denote an ordinal, set $[φ]_T < [ψ]_T$ iff provably in $T$, the ordinal denoted by $φ$ is less than the ordinal denoted ...
- 7,545
3
votes
0
answers
853
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 ...
- 4,587
2
votes
1
answer
542
views
What is the proof-theoretic ordinal of KPh?
If we work in this notation:
$$C_0 (\alpha, \beta) = \beta \cup \lbrace 0 \rbrace$$
$$C_{n+1} (\alpha, \beta) = \lbrace \gamma + \delta, \omega^\gamma, \Omega_{\gamma}, I_{\gamma}, \psi_\pi(\eta) | \...
- 524
1
vote
1
answer
466
views
What does "can almost be proven in PA" mean regarding Theorem 2 of Timothy Chow's expository article, "The Consistency of Arithmetic"?
In his expository article, "The Consistency of Arithmetic" (MSN), Prof. Chow has the following theorems:
Theorem 1. If $a_1, a_2, a_3,\dotsc$ is a sequence of ordinals and $a_i \ge a_j$ whenever $...
- 6,132
1
vote
0
answers
122
views
Is there an error in W. Buchholz's paper "A simplified version of local predicativity"?
I want to self-learn proof theory. It seems that the operator controlled derivation method is important in this field, and the paper in the title is the first paper that uses this method.
So I'm ...
- 1,490
0
votes
0
answers
157
views
How to define BHO alternatives below admissible ordinals?
Bachmann-Howard ordinal is a recursive ordinal. It's not that large compared to those proof-theoretic ordinals of stronger theories, but the definition of BHO is sufficient to illustrate how ...
- 1,490