All Questions
Tagged with ordinal-analysis ordinal-numbers
28 questions
4
votes
1
answer
269
views
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 ...
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
2
votes
0
answers
194
views
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 ...
- 2,031
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
6
votes
1
answer
365
views
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 ...
- 42.4k
5
votes
1
answer
310
views
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, ...
- 2,031
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
5
votes
0
answers
262
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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 ...
- 161
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
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
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
8
votes
0
answers
287
views
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 ...
- 7,545
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
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
5
votes
1
answer
411
views
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 $T$):
...
- 615
8
votes
4
answers
1k
views
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 ...
- 1,252
10
votes
1
answer
358
views
$Π^0_1$ 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
6
votes
1
answer
474
views
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 ...
- 35.4k
5
votes
1
answer
409
views
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 ...
- 22.3k
8
votes
1
answer
706
views
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
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
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
27
votes
1
answer
2k
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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 ...
- 9,823
19
votes
0
answers
590
views
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 ...
- 9,823
40
votes
2
answers
4k
views
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 ...
- 9,823
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
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
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