# Tag Info

Accepted

### Why is there a need for ordinal analysis?

The axioms of first-order arithmetic include the induction schema, which says that, for every formula $A(x)$ with free variable $x$, the conjunction of $A(0)$ and $\forall x\,(A(x)\rightarrow A(x+1))$ ...

### Why is there a need for ordinal analysis?

In addition to the reasons Andreas gives, Gentzen's theorem gives additional information that's interesting even if you don't have any qualms about consistency. In particular, ordinal analysis gives ...
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### Regarding Gentzen's note regarding 'Godel-points' (i.e., "Where is the Godel-point hiding?")

Your question seems to boil down to (after fixing an error) the following: Any model $\mathfrak{M}$ of ACA$_0$ has a first-order part $Num(\mathfrak{M})$, which satisfies PA; why doesn't this mean ...
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### How can any theory prove well-foundedness of ordinals above $\omega_1^{\text{CK}}$?

Just slightly expanding the comment of Emil Jeřábek: on one hand, in ZFC we can define some objects we call ordinals and prove transfinite induction of each of them. This is not what we ...
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### Correspondence between proof-theoretic ordinals and fast growing functions?

Yes: for many theories, $\alpha$ is the proof-theoretic ordinal of T exactly when T proves that $f_\beta$ is total for all $\beta<\alpha$, but does not prove $f_\alpha$ is total. (Where $f_\alpha$ ...
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### What system suffices to show the strength of PRA is $\omega^\omega$?

First, this is not quite the right question. The implication as such is going to be provable in a trivial base theory; the most important question is what “well-foundedness of $\omega^\omega$” means ...
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### Existence of well-ordering of epsilon_0 in weak theories

The existence of $\epsilon_0$ and its order is not a problem, its well-foundedness is. Cantor normal forms (recursively expanded) of ordinals below $\epsilon_0$ can be written as strings over a ...

### Formalizations of The Matchstick Diagram Representation of Ordinals

I prefer to think of the matchstick representations a little differently than you present them. Namely, your function $f$ provides the amount to jump up by at each step, the difference between each ...

### Formalizations of The Matchstick Diagram Representation of Ordinals

Yes, $\omega_1^{md}$ is just $\omega_1$. Fix some countable limit ordinal $\alpha\ge\omega^2$ (so that there are infinitely many limit ordinals $<\alpha$ - ordinals $<\omega^2$ can be handled ...
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### Gentzen's result on PA

Yes, Gentzen found a single "induction instance" which is PA-unprovable: more-or-less $\varphi(\alpha)$ = "Every sentence with a proof of cut-rank $\alpha$ has a cut-free proof." Now, this $\varphi$ ...
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### What does "can almost be proven in PA" mean regarding Theorem 2 of Timothy Chow's expository article, "The Consistency of Arithmetic"?

I believe you're overthinking what Chow means by "almost provable." We have an arithmetic statement of the form $$(*)\quad\forall x\varphi(x),$$ which while not provable in PA has the property that ...
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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?

Let me show that for extensions $T\supseteq\mathsf{ACA}_0$ the usual proof-theoretic ordinal $|T|_{WO}$ coincide with $|T|_{WPO}$ that is the suprema of $\mathsf{o}(X)$, for recursive wpo $X$, for ...
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### What is the proof-theoretic ordinal of KPh?

Let $\mathsf{KPh}$ denote the theory $\mathsf{KP}+\textrm{The recursively inaccessibles are unbounded}\! "$. I haven't found an explicit analysis of $\mathsf{KPh}$ in Rathjen's preprints, but there ...
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### Does this restriction on powersets in ZF have a proof theoretic ordinal?

Your theory interprets $\mathsf{ZF}^-$. In fact, $\mathsf{ZF}-$ (namely, $\mathsf{ZF}$ without Powerset) interprets $\mathsf{ZF}^-+(V=L)$. It answers your questions negatively since the proof-...