All Questions
Tagged with ordinal-analysis set-theory
17 questions
6
votes
0
answers
181
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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 "$\Pi^1_1$-...
- 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
1
vote
0
answers
123
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Is $\sf \Gamma_0$ the proof theoretic ordinal of this kind of predicative class theory?
Adopting the approach of Mono-sorted $\sf NBG$, define sets as elements of classes, then axiomatize:
Extensionality, Predicative Class comprehension, emptyset, in the usual manner along mono-sorted $\...
- 11.3k
1
vote
0
answers
127
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What is the proof theoretic ordinal of this kind of predicative type-set theory?
The following is a kind of Predicative Type Set Theory.
The question is about what is exactly the proof theoretic ordinal of this theory? Is it lower than the one expected for predicative theories, i....
- 11.3k
5
votes
1
answer
310
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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, ...
- 2,031
5
votes
0
answers
115
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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 $\Pi_2^1$-...
- 704
1
vote
1
answer
251
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Does this restriction on powersets in ZF have a proof theoretic ordinal?
If we add to the language of set theory a total one place function symbol $\mathcal P$ standing for powerset operator, and then add to ZF-Power the following axioms:
Power: if $\phi$ is a formula in ...
- 11.3k
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
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
4
votes
0
answers
367
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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.
...
- 7,545
2
votes
1
answer
542
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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) | \...
- 524
5
votes
1
answer
411
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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 $T$):
...
- 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
14
votes
2
answers
1k
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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 ...
- 907
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
40
votes
2
answers
4k
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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 ...
- 9,823