# Tagged Questions

An ordinal is the order type of a well-ordered set. The first few ordinals are $0, 1, 2, \dots, \omega, \omega+1, \dots$ where $\omega$ is the order type of $\mathbb{N}$, and $\omega+1$ is the order type of $\mathbb{N}$ together with a maximum element.

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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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### Order type of the smallest set containing the identity function and closed under exponentiation

Let $E$ be the smallest set of functions $\mathbb{N}^+\to\mathbb{N}^+$ containing the identity function $n \mapsto n$ and closed under exponentiation $(f,g) \mapsto \left(n \mapsto f(n)^{g(n)}\right)$...
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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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### Are Conway's omnific integers the Grothendieck group of the ordinals under commutative addition?

This is a question in two parts. Say that $\mathbf{On}$ is the proper class of all ordinal numbers in ZFC. We can define a binary operator over $\mathbf{On}$ which corresponds to the commutative ...
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### Looking for a copy of Leo Harrington's unpublished notes on the first nonprojectible ordinal

Sometime around 1975, Leo Harrington wrote a set of notes, apparently 13 pages long, entitled Kolmogorov's $R$-operator and the first nonprojectible ordinal. I do not know how widely they were ...
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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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### Transfinitely extending $\sf PA$ — can we get stronger than $\sf ZFC$?

Let $\sf PA$ denote the theory of natural numbers with constants $(0, 1)$ and binary operators $(+,\times)$ based on the first-order predicate calculus with equality, having the following axioms, ...
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### Is there a modern account of Veblen functions of *several* variables?

Veblen $\phi$ functions extend the $\xi \mapsto \phi(\xi) := \omega^\xi$ and the $\xi \mapsto \phi(1,\xi) := \varepsilon_\xi$ functions on the ordinals by repeatedly taking fixed points (I won't ...
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### What is the idea behind stationary sets?

Let $\kappa$ be a cardinal (of uncountable cofinality). A subset $S \subseteq \kappa$ is called stationary if it intersects every club, i.e. closed unbounded subset of $\kappa$. Now my question is ...
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### Ordinals and complexity classes

What is the least recursive ordinal $\alpha$ such that there is no algorithm in complexity class $\mathsf{P}$ which implements a well-ordering of $\mathbb{N}$ with order type $\alpha$? (where the size ...
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### The Theory of Transfinite Diophantine Equations [closed]

The theory of Diophantine equations is one of the main stream research areas in number theory. There are many known results and unknown conjectures about the existence of non-trivial solutions for ...
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### How this set of functions is ordered?

Notation: $k, m, n$ are non-negative integers $f, g, h$ are functions $\mathbb{N} \to \mathbb{N}$ $f^k$ is $k$-th iterate of the function $f$: $f^0(n)=n, f^{k+1}(n)=f^k(f(n))$ $f \prec g$ means ...
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Philip Ehrlich's paper “The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small”, The Bulletin of Symbolic Logic 18 (1) 2012, pp. 1-45. claims as a theorem that, in NBG, $\... 1answer 778 views ### Ordinal category theory? Just out of curiosity: Is there a notion of$\alpha$-category for an ordinal number$\alpha$, extending the given notions for$\alpha \leq \omega$? If there is none, which one would you propose? Feel ... 1answer 320 views ### Ordinals in constructive mathematics ? (references) I'm looking for references presenting a constructive treatment of the theory of ordinals. By constructive I mean valid in the internal logic of a topos (so no axiom of choice and no law of excluded ... 4answers 534 views ### Direct axiomatization of ordinal and cardinal numbers Again, this question is related (**) to a previous one: in standard books on basic set theory, after stating the axioms of ZFC, ordinal numbers are introduced early on. Afterwards cardinals appear: ... 1answer 330 views ### Which ordinals are proof-theoretic ordinals? Few months ago I have posted this question on MO, but I must admit that at the time, admittedly, I had no idea on how technical proof-theoretic considerations can be. I have decided to revise this ... 1answer 288 views ### Arithmetic strength of Peano + the Howard ordinal Consider the theory$\mathrm{PA}+\mathrm{BHO}$consisting of first-order Peano arithmetic ($\mathrm{PA}$) enriched by an axiom scheme which allows well-founded induction up to any ordinal less than [a ... 0answers 160 views ### How “small” can an ordinal be made by forcing? I know that forcing essentially does not change the ordinals, but by small I mean in comparison with other ordinals whose definition might not be stable under forcing, like the smallest uncountable ... 3answers 542 views ### Is it consistent with ZFC that for all ordinals$\alpha, \beta < \omega$it holds that$2^{\aleph_\alpha} = 2^{\aleph_\beta}$? Let$\gamma=\omega$(the first transfinite ordinal). Is it consistent with ZFC that for all ordinals$\alpha, \beta < \gamma$it holds that$2^{\aleph_\alpha} = 2^{\aleph_\beta}$? If yes, can the ... 1answer 356 views ### value of Theta in ZF+AD Since I found out about it, I've always been interested in the Axiom of Determinacy rather than the Axiom of Choice. Along these lines, I've kept flipping back to http://en.wikipedia.org/wiki/%CE%98_%... 1answer 123 views ### Anything known about the Grundy Ordinal of Sylver's Coinage Sylver's coinage is an example of an unbounded finite (if slightly modified) combinatorial impartial game. Quoth wikipedia: The two players take turns naming positive integers that are not the ... 0answers 103 views ### Mapping graphs to ordinals Robertson-Seymour theorem implies that graph minor relation is a well-quasi-ordering, which means (among other things) that this relation can be extended to a well-order, and other result says that ... 4answers 702 views ### Prime numbers and limit ordinals As a set, i.e. as a von Neumann ordinal, the$\omega$-th limit ordinal$\omega^2$is fairly complex and not so easy to visualize (for the novice). But as an explicit well-ordering of$\mathbb{N}$, ... 2answers 2k views ### Foundations: Existence of uncountable ordinals. This isn't really a research question, but at least it's research-level mathematics. I'm talking with some other people about the first uncountable ordinal, and I want some facts to inform this ... 1answer 218 views ### Reference request, proof about well-ordering of ordinals I am looking for a second order proof that ordinals are well-ordered up to$\epsilon_0$. So, starting with encoding those ordinals in numbers, defining the < relation on those encoded ordinals and ... 1answer 416 views ### Cardinality of cofinal set of normal functions$f \colon \omega_1 \to \omega_1$What is the cardinality of the set$F$of all normal functions$f \colon \omega_1 \to \omega_1$, where$\omega_1$is the first uncountable ordinal? What is the least cardinality of a subset of$F$... 2answers 317 views ### Heights of several interesting posets Let the height of a poset$P$be the supremum of ordinals that are order types of all well-ordered subsets of$P$(with order inherited from$P$). Define several sets of total functions, in each ... 1answer 798 views ### Cantor's Normal Form and Aleph_1 The Cantor Normal Form Theorem states that every ordinal$\alpha > 0$can be uniquely expressed in the form $$\omega^{\beta_1}k_1 + \omega^{\beta_2}k_2 + \dots + \omega^{\beta_n}k_n$$ for some$n \...
Consider the Grothendieck ring $K[\Omega]$ of the semiring $\Omega$ of all ordinals under the operations of natural sum and product. Its quotient field $K(\Omega)$ is naturally a subfield of the ...
If $k$ is an ordered field, the least ordinal $s(k)$ which doesn't embed in $(k,<)$ is regular. This is because every interval of an ordered field embeds in every infinite interval so given a ...