# Questions tagged [ordinals]

In the ZF set theory ordinals are transitive sets which are well-ordered by $\in$. They are canonical representatives for well-orderings under order-isomorphism. In addition to the intriguing ordinal arithmetics, ordinals give a sturdy backbone to models of ZF and operate as a direct extension of the positive integers for *transfinite* inductions.

1,753 questions
Filter by
Sorted by
Tagged with
43 views

45 views

1 vote
28 views

### closed set and the supremum in order topology

Suppose $\alpha$ is an ordinal endowed with the order topology, i.e. its basic open sets are generalized open intervals. Given $C \subseteq \alpha$, I want to show (1) implies (2) : (1) For all ...
1 vote
27 views

1 vote
94 views

1 vote
64 views

### Explain why transfinite induction does not assume that a property must be true for zero.

THIS QUESTION IS NOT A DUPLICATE OF THIS ONE!!! So I would like to discuss the following proof of transfinite induction which is taken by the text Introduction to Set Theory wroten by Karell Hrbacek ...
33 views

### Find the smallest cardinal $\kappa$ such that $\omega + \omega$ is the supremum of $\kappa$ smaller ordinals

I'm asked to find the smallest cardinal $\kappa$ such that $\omega + \omega$ is the supremum of $\kappa$ smaller ordinals. By definition, $\omega + \omega = \sup \{\omega + n : n < \omega\}$. I ...
79 views

### About the definition of hereditary cardinality

I've seen these two definitions of sets that are hereditarily of cardinality $< \kappa$. $x$ is hereditarily of cardinality $< \kappa$ iff $|trcl(x)| < \kappa$ $x$ is hereditarily of ...
48 views

### The axiom of regularity in a fact about $V_\omega$

I'm working out the details in this proof (from here), and I have some questions about the second part ("For the other direction, ..."). First, why is it possible to assume WLOG that $A$ is ...
1 vote
75 views

1 vote
35 views

### Link between a theory’s proof-theoretic ordinal and the fastest-growing function it can prove total

When I’ve tried to read up on proof-theory I’ve come across this point multiple times - that given a well-founded fast-growing hierarchy, the index of the fastest-growing function f that T can prove ...
57 views

41 views

215 views

### Is there a measurable function from $[0,1]$ to $ω_1$?

Does there exist a measurable function from $[0,1]$ (with the Lebesgue measure) to $ω_1$ that induces the Dieudonné measure? Definitions: $ω_1$ is the set of all countable ordinals, equipped with its ...
69 views

### Iterated $\Pi^1_1$-reflection and non-Gandiness underrepresented in ordinal analyses?

The admissible ordinals $\alpha$ s.t. the supremum of $\alpha$-recursive well-orderings is less than the next admissible $>\alpha$ are called non-Gandy, and according to Madore's "A Zoo of ...
26 views

### Definitions of ordinal addition and multiplication

For any two sets $\alpha$ and $\beta$, let $\alpha+\beta = \{\alpha', \alpha+\beta'\}$ and $\alpha\cdot\beta = {\alpha\cdot\beta'+\alpha'}$, where $\alpha'$ and $\beta'$ are variables taking as values ...
72 views

### Questions about $\omega_1$ as a space (The Set of All Countable Ordinals)

The Set of All Countable Ordinals, $\omega_1$ I'm trying to understand several things regarding $\omega_1$ and trying to get a better intuition. I have four questions regarding this space (in bold), ...
1 vote
44 views

### The cardinality of the preimage of an ordinal

Define a cardinal as an ordinal $\kappa$ such that for all ordinals $\alpha < \kappa$, $\alpha$ injects into $\kappa$ but is not bijective to $\kappa$. Let $\kappa$ be an infinite cardinal. I'm ...
28 views

### How do you proove that TREE(3) is finite in layman terms?

Apologies in advance. I am just a layman who knows about ordinals and know about TREE(n) but I don't know how to prove it is finite. Is there a simple transfinite ordinal proof? I don't know anything ...
23 views

46 views

### Isomorphism between the first the uncountable ordinal $\omega_{1}$ and real numbers

Recently, i have studied the proof of why the set of all countable ordinals $\Gamma$ is an ordinal and why it is an uncountable. (Basically, since it contains all countable ordinals as predecessors, ...
68 views

### Supremum in well orderings

Problem *x7.27 from Moschovakis' textbook is asking to define a definite operation $\sup \mathscr E$, such that for every family $\mathscr E$ of well-ordered sets, $\sup \mathscr E$ has the following ...
Working in $\sf ZF-Reg.$ can we define the unary predicate "is an ordinal", denoted by "$\operatorname {od}$", meaning is a von Neumann ordinal, in a recursive manner? The usual ...
### Are the ordinals $\alpha$ so that the stable ordinals $< \alpha$ are unbounded in $\alpha$ equivalent to the nonprojectible ordinals $\alpha$
An ordinal $\alpha$ is called nonprojectible if and only if $\forall \beta (\beta \in X \cap \alpha \implies \min\{\gamma \in X: \beta \in \gamma\} \in \alpha)$, where \$\delta \in X \iff L_\delta \...