# Questions tagged [ordinal-numbers]

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.

144 questions
Filter by
Sorted by
Tagged with
271 views
+50

### Veblen function with uncountable ordinals & beyond

Disclaimer: I am not a professional mathematician. Background: I have been researching large countable ordinals for awhile & I think the Veblen function is particularly eloquent. My understanding ...
346 views

### Does every cofinal branch through Kleene's O compute true arithmetic?

My question concerns cofinal branches through Kleene's $O$, which is a set of natural numbers and a computably enumerable relation $<_O$ on this set that provides ordinal denotations for any ...
282 views

### From a constructive perspective, what are the ordinal numbers?

From a constructive and computational perspective, what are the ordinal numbers? On the one hand, it seems you can represent ordinal numbers symbolically using something like Cantor Normal Form ...
616 views

### Has the exponentiation of ordinals a nice geometric model?

It is well known that the sum $\alpha+\beta$ of two ordinals $\alpha,\beta$ can be defined "geometrically" as the order type of the sum $(\{0\}\times \alpha)\cup(\{1\}\times\beta)$ endowed with the ...
196 views

120 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 ...
122 views

### What is the bijection from $\omega_1^{CK}$ to the natural numbers?

A while back, I learned about the Church-Kleene ordinal and I wondered, since we know that it is countable, what is the bijection from $\omega_1^{CK}$ to $\mathbb{N}$?
81 views

### Can I find the definition of Jager's ordinal collapsing functions?

http://cantorsattic.info/index.php?title=J%C3%A4ger%27s_collapsing_functions_and_%CF%81-inaccessible_ordinals&action=edit Sadly, Cantor's attic is making an error. This is all I know about Jager'...
369 views

### Examples of proofs using induction or recursion on a big recursive ordinal

There are many proofs use induction or recursion on $\omega$, or on an arbitary (may be uncountable) ordinal. Are there some good examples of proofs which use a big but computable ordinal? The ...
187 views

### Smallest ordinal modelling $\aleph_1$?

Let $X_1$ be the class of all ordinals $\alpha$ such that there exists a transitive model $M$ of ZF(C) such that $M$ thinks that $\alpha$ is $\aleph_1$. Every class of ordinals has a minimum element (...
209 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 $\omega^\omega$. ...
166 views

### Club filter basis in $\omega_1$

My question is about existing of basis of club filter club($\omega_1$) with cardinality $c$. Does it exist?
368 views

### Which ordinal is larger, the supremum of ordinals writable by iterated Infinite Time Turing Machines or the smallest $\Sigma_2^1$-reﬂecting ordinal?

The ordinal $\tau_1$ corresponds to $\lambda^{\textit{it}}$ (the supremum of all ordinals writable by iterated ITTMs) — see Deﬁnition 3.1 in the paper “ITTMs with Feedback” [Robert S. Lubarsky]. ...
167 views

### Is it an open problem whether fast-growing hierarchies can be defined without fundamental sequences?

Googleology Wiki says this, concerning the relation between fast-growing hierarchies defined for all countable ordinals, and the existence of a system of assigning a canonical fundamental sequence to ...
406 views

### Relation between $\eta$ and $\omega^L_1$

I posted this question on MSE (link: Eventual Writability (general)) about 10 days ago. The current version of this question is a highly abridged version of the one posted there. Let's write "...
294 views

### Why do ordinal collapsing functions use regular cardinals?

Inaccessible cardinals are defined as regular strong limit cardinal, and weakly inaccessible cardinals as regular weak limit cardinal. These cardinals are used by some ordinal collapsing functions. My ...
151 views

### Weaker versions of Gandy ordinals

Gostanian's paper "The next admissible ordinal" (see https://www.sciencedirect.com/science/article/pii/0003484379900251 ), is concerned with the supremum of the $\alpha$-recursive ordinals for various ...
532 views

### Compact definition of ordinals

This question is about von Neumann's informal definition of ordinals as "sets of all smaller ordinals" and was discussed in this math.stackexchange question. When trying to formalize this definition, ...
570 views

### Computable models of the ordinal numbers

It's known, for example in the answer to this question: Is there a computable model of ZFC? that ZFC has no computable model. My questions is: is there a model of ZFC for which the order relation on ...
216 views

### Generalizing König's Lemma

In some recent work, I need a strengthening of König's Lemma to "trees" of arbitrary ordinal heights. Trees, in this context, are really just well-founded partially ordered sets. See, for instance, ...
211 views

### Capturing the $\omega_1^{\mathrm{CK}}$-th stage of Gödel's constructible hierarchy

For an ordinal $\alpha$, let $L_\alpha$ be the $\alpha^{th}$ set of Gödel's constructible hierarchy and let $\omega_1^{\mathrm{CK}}$ be the first non-recursive ordinal or the first admissible ordinal ...
194 views

### Is the least ordinal containing all countable ordinals defined by a formula an element of itself? (Out of date, see below)

[UPDATE] The question has been updated and is mostly unrelated to the question above the line. The updated question is below the line. The following argument supports a "yes" answer; is it convincing?...
94 views

### Coloring triples in trees

Definitions Let us say a tree is a partially ordered set $(P, \leqslant )$ such that for any $t\in P$, the ancestor set $\{s\in P: s\leqslant t\}$ is finite and linearly ordered. We let $MAX(P)$ ...
330 views

451 views

### Which finite sets could be packed into a square?

This question is inspired by an interesting visualization of the finite levels of von Neumann's hierarchy on Adam P. Goucher's blog, Complex Projective 4-Space. The problem starts with a two-...
216 views

### Is the measurable space $(\omega_1,\mathcal{P}(\omega_1))$ separable?

Here $\omega_1$ is the first uncountable ordinal, and $\mathcal{P}(\omega_1)$ denotes the power set of $\omega_1$. Separable means countably generated as a $\sigma$-algebra.
226 views

### Reference Request: Existence of Ordinal Rank Theory?

Notations: Recall that $\omega_1$ is the first uncountable ordinal. Let $X$ be a Polish space (completely metrizable and separable) and $F(X)$ be the collection of all real-valued functions on $X.$ ...
447 views

### Ordinal vs. cardinal dimension

$\newcommand{\Ord}{\operatorname{Ord}}$When does it make sense to define the dimension of a space to be an infinite ordinal, instead of restricting to infinite cardinals? We would essentially have to ...
122 views

233 views

### Large cardinals disproving J=L

Given $J_\alpha$ denotes the $1+\alpha$th term in the Jensen Hierarchy and $J=\bigcup_{\alpha\in On}J_\alpha$, are there any known large cardinals $\mathfrak{K}$ such that $\text{ZFC}+\mathfrak{K}$ is ...
104 views

### Ordinal corresponding to well-quasi-order on graphs

Let $K$ be an infinite cardinal. Then, by the Robertson–Seymour theorem, the set of graphs with fewer than $K$ vertices and edges form a well-quasi-order. In terms of $K$, what is the maximal order ...
274 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) | \...
603 views

### Ramsey's theorem for the first uncountable ordinal

Sierpiński proved that a version of Ramsey's theorem for colourings of pairs of countable ordinals fails miserably by comparing the ordering of $\omega_1$ with the linear ordering of (a subset of) the ...
I have several questions related to ordinal analysis. According to , here are the proof-theoretic ordinal of some well-known theories (using $|T|$ do denotate the proof-theoretic ordinal of $T$): ...