Questions tagged [ordinal-analysis]
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57 questions
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Is there a computable ordinal encoding the proof strength of ZF? Is it knowable?
In comments on Quora (see, for example, here, here, here), Ron Maimon has repeatedly expressed the strong opinion that Hilbert's program was not killed by Gödel's results in the way typically ...
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Why do stacked quantifiers in PA correspond to ordinals up to $\epsilon_0$?
I am trying to understand why induction up to exactly $\epsilon_0$ is necessary to prove the cut-elimination theorem for first-order Peano Arithmetic; or, as I understand, equivalently, why the length ...
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Why isn't this a computable description of the ordinal of ZF?
In a previous MO question, I was told by several commenters that
(a) it's known that there exists a computable ordinal $\alpha_{ZF}$ that "encodes the strength of ZF set theory" (i.e., a least ...
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Can Gentzen-style proofs give omega-consistency and beyond?
In 1936, Gentzen famously showed that Primitive Recursive Arithmetic, plus the assumption that the ordinal $\epsilon_0$ is well-founded, is able to prove Con(PA). But of course, Con(PA) doesn't yet ...
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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 ...
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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 ...
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How can any theory prove well-foundedness of ordinals above $\omega_1^{\text{CK}}$?
$\newcommand{\omegaoneck}{\omega_1^{\text{CK}}}$
Pardon the extremely basic question - this isn't quite my area - but I'm confused about the definition of proof theoretic ordinals.
The proof ...
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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 ...
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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 $\mathsf{PA}$, ...
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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 ...
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What is the proof-theoretic ordinal of bare $\mathsf{NFU}$?
On the Stanford Encyclopedia of Philosophy article on alternative axiomatic set theories, it is stated without reference that bare $\mathsf{NFU}$ (i.e., $\mathsf{NFU}$ without the axiom of infinity) ...
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Why is there a need for ordinal analysis?
Consider the Peano axioms. There exists a model for them (namely, the natural numbers with a ordering relation $<$, binary function $+$, and constant term $0$). Therefore, by the model existence ...
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What system suffices to show the strength of PRA is $\omega^\omega$?
Russell O'Connor wrote in 2009 (link):
PRA has consistency strength equivalent to the well-foundness of $\omega^\omega$, which can be stated again as the termination of some other program on all ...
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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 ...
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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 ...
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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 ...
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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, $ECP$ ...
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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 ...
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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 ...
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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 ...
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Formalizations of The Matchstick Diagram Representation of Ordinals
The matchstick diagram is a really interesting and intuitive method of representing countable ordinals. However, because of how difficult it is to graphically represent ordinals with it, I started ...
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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 ...
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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, ...
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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 ...
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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 ...
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Iterated Gentzen: or, a Sith objection to the proof of consistency of PA
$\DeclareMathOperator\PRA{PRA}\DeclareMathOperator\WF{WF}\DeclareMathOperator\Con{Con}\DeclareMathOperator\PA{PA}$Preamble: In the year … in a galaxy far far away, a nasty Sith named Darth Dubious (...
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Natural examples of recursive pseudowellorderings
Question: What are some natural examples of recursive pseudowellorderings?
By natural, I mean in the style of reasonable ordinal notation systems as opposed to dependent on a Gödel numbering or an ...
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Existence of well-ordering of epsilon_0 in weak theories
In a discussion on a youtube video on the hydra game I jokingly mentioned how everyone was assuming that $\varepsilon_0$ was well-ordered. This lead to a bit of disagreement (in a nice way!) about the ...
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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 ...
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How to build large recursive ordinals using Dillator and/or Ptykes?
I've only recently learned about Girard's theory of Dilators and Ptykes, and I find this theory very elegant, but it is not clear at all to me whether/how it can be used to produce ordinal notations ...
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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 "$\Pi^1_1$-...
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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 $\...
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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 $\omega^\omega$.
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A question about ordinal analysis
I have several questions related to ordinal analysis.
According to [1], here are the proof-theoretic ordinal of some well-known theories (using $|T|$ do denotate the proof-theoretic ordinal of $T$):
...
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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 ...
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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 \...
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What countable ordinals are called $\kappa_\alpha$?
Jervell has a notation for countable ordinals up to the small Veblen ordinal using trees:
• Herman Ruge Jervell, How to wellorder finite trees
and get good ordinal notations, Berkeley Logic ...
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How closely do ordinal collapsing functions relate to Mostowski collapse?
Ordinal collapsing functions (such as Rathjen's $\psi_\pi$-functions, not the Levy collapse function) name large countable ordinals by mapping larger ordinals below some "large" ordinal, ...
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What's the purpose of $\mathsf M\text-\mathsf P$-expressions?
In ordinal notations such as Stegert's (Ordinal Proof Theory of Kripke–Platek Set Theory Augmented by Strong Reflection Principles) and Rathjen's (An Ordinal Analysis of parameter free $\Pi_2^1$-...
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Higher order arithmetic, hierarchies and proof theoretic ordinals
I asked this question on MSE some days ago but I have not received any answer so I have decided to post it here.
I would like to consider a generalization of the notation $\Pi$ and $\Sigma$ used for ...
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$Π_2$ 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 ...
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Does Peano's axioms prove $\alpha$-induction for primitive recursive sequences for every concrete $\alpha < \varepsilon_0$?
It is well-known that Peano's axioms (PA) cannot prove $\varepsilon_0$-induction for primitive recursive sequences (PRWO($\varepsilon_0$)), because PA + PRWO($\varepsilon_0$) proves the consistency of ...
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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 ...
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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.
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Is there a notion of "predicative given the real numbers"?
A definition is called impredicative if it involves quantification over a domain that contains the thing being defined. For instance, if you define hereditary property to be a property which applies ...
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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 ...
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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 ...
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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 ...
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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) | \...
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How closely do ordinal collapsing functions relate to Skolem hulls?
Ordinal collapsing functions appear in proof theory, and they are used to name large countable ordinals by using uncountable ordinals. Previously I posted a question about why $\psi(\alpha)$ is ...
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