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Kleene's $O$ is a way to use natural numbers as notations for recursive ordinals. I’m wondering what happens if you modify the definition of Kleene’s $O$ to allow for arithmetical truth as an oracle. Let $T$ be the set of Godel numbers of true statements in the language of first-order arithmetic. Let $0$ be a notation for $0$, and if $i$ is a notation for $\alpha$, then $2^i$ is a notation for $\alpha+1$. If $\phi_e^T$ (the $e^{th}$ partial recursive function with access to $T$ as an oracle) is a total $T$-recursive function enumerating ordinal notations in strictly increasing order (as ordinals), then let $3\cdot 5^e$ be a notation for the least upper bound of the ordinals denoted by the range of $\phi_e$. Let $O_T$ be the set of all ordinal notations obtained in this way.

My question is, what is the smallest ordinal that does not have a notation in $O_T$? I realize that ordinal might be difficult to describe exactly, but can we at least put some upper and lower bounds on how big it is?

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  • $\begingroup$ I really recommend reading Sacks' book - many of your questions are directly answered in it, and it's freely and legally available online. $\endgroup$ – Noah Schweber Nov 8 '19 at 17:20
  • $\begingroup$ @NoahSchweber Thanks, I'll check it out. $\endgroup$ – Keshav Srinivasan Nov 8 '19 at 17:21
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It's a theorem of Spector that every hyperarithmetic ordinal is computable (see Sacks' book). True (first-order) arithmetic is tiny in the hyperarithmetic hierarchy, so it doesn't get us any further than $\omega_1^{CK}$.

  • In fact, a stronger result is true: $L_{\omega_1^{CK}}\cap\mathcal{P}(\omega)=HYP$.

(Incidentally, it's also true that every computable ordinal is polynomial-time computable, so $\omega_1^{CK}$ is the first non-polytime-computable ordinal and the first non-hyperarithmetic ordinal. It's a really robust notion!)


Note that as usual, it's far simpler to talk about computable ordinals than notations. There are situations where notations are useful, but in my experience more as a tool than the initial definition.

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  • $\begingroup$ What do you mean by talking about computable ordinals rather than notations? How do I phrase my question without talking about notations? $\endgroup$ – Keshav Srinivasan Nov 8 '19 at 17:21
  • $\begingroup$ @KeshavSrinivasan A computable ordinal is an ordinal which has a computable copy: that is, an ordinal $\alpha$ is computable iff ($\alpha$ is finite or) there is some computable binary relation $R$ on $\omega$ which well-orders $\omega$ with ordertype $\alpha$. Similarly, a hyperarithmetic ordinal is one with a hyperarithmetic copy, etc. The classic text for this topic is Ash/Knight Computable structures and the hyperarithmetic hierarchy - unfortunately that one's harder to find. $\endgroup$ – Noah Schweber Nov 8 '19 at 17:24
  • $\begingroup$ The fact mentioned in my answer to your other recent question - that "constructive" ordinals (= ordinals with notations) and computable ordinals coincide, and that this relativizes - is the key to switching from notations to general copies. In particular, your question can be rephrased as: "What is the least ordinal with no copy computable from true first-order arithmetic?" $\endgroup$ – Noah Schweber Nov 8 '19 at 17:25
  • $\begingroup$ Cool, Ash/Knight is the textbook of the logic course I'm taking next semester. $\endgroup$ – Keshav Srinivasan Nov 8 '19 at 17:27
  • $\begingroup$ Does truth in the language of first-order set theory in the standard model of $ZFC$ get you anywhere higher than the Church-Kleene ordinal? Or is "the standard model of $ZFC$" too vague to answer that? $\endgroup$ – Keshav Srinivasan Nov 8 '19 at 17:31

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