# 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.

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### An infinite linear system of equations with an uncountable number $A$ of equations

I will start with an example to make things clear and avoid confusion : Take all $x>0$ and $$\exp(x) = \sum_{-1<n} a_n x^n$$ Now finding $a_n$ can be described as an infinite linear system of ...
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### Is there an unbounded countable set of ordinals?

Let $\text{Ord}=\{0,1,\ldots,\omega,\ldots\}$ be the set of ordinals. Does $\text{Ord}$ have an unbounded countable subset? In particular, is $$\{\omega, \omega^\omega, \omega^{\omega^\omega},...\}$$ ...
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### Inequality involving ordinals

Recently, I came across a difficult question in my set theory module: Let $\alpha$ be an infinite ordinal. Suppose A ∪ B = $\alpha$, A ∩ B = ∅, and otp(A) = otp(B) = $\beta$. Give an example to show ...
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### What is a sequence of ordinals?

What is a sequence of ordinals? The concept of a sequence of ordinals shows up here and here, and in the definition of cofinality in Jech, third edition, page 31. $\text{cf}(\alpha) =$ the least ...
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### Solution verification: proof that the supremum of a set of cardinals is a cardinal

I'm trying to show that if $X$ is a set of cardinals, then $\sup X$ is a cardinal. This is part (ii) of Lemma 3.4 in the third edition of Jech on page 29. The proof in Jech is pretty terse and I don't ...
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### Why does proof of Zorn's lemma need to use the fact about ordinals being too large to be a set?

I'm not understanding why its necessary to invoke the knowledge about ordinals in order to prove Zorn's lemma. The Hypothesis in Zorn's lemma is Every chain in the set Z has an upper bound in Z Then ...
1 vote
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### Defining fundamental sequence in advance

My question is about large countable ordinal numbers and their fundamental sequence. It looks as our knowledge of large countable ordinal number theory grows, brand-new fundamental sequence for brand-...
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### Express small, large veblen ordinal using ...... (repeat symbol) form

Sometimes it's better to see once than to say a thousand words. Following link shows what ordinals really looks like : Ordinal tetration: The issue of ${}^{\epsilon_0}\omega$ And following is a ...
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### Ordinal arithmetic proof: $a,b <\omega^2$ $\implies$ $a+b<\omega^2$ [closed]

i need to prove the following If $a$, $b$ are ordinals such that $a,b <\omega^2$ $\implies$ $a+b<\omega^2$. I think it could be done using injective functions but have no idea on how to do it.
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### Diamond principle for functions $\omega_1\to\omega_1$

EDIT: I now see that the question was asked and answered more directly, but it's not clear to me that that answer makes sense and it will take me a while to parse. In Kunen's Set Theory, chapter II ...
1 vote
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### Show that $\{\alpha<\omega_1 : L_\alpha \prec L_{\omega_1}\}$ is closed unbounded in $\omega_1$.

I was doing this exercise and there is a hint to consider the Skolem functions for $L_{\omega_1}$. However, I did not find any general definition of what a Skolem function may be in Kunen (1980), and ...
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### What does the cardinality alone of a totally ordered set say about the ordinals that can be mapped strictly monotonically to it?

For any cardinal $\kappa$ and any totally ordered set $(S,\le)$ such that $|S| > 2^\kappa$, does $S$ necessarily have at least one subset $T$ such that either $\le$ or its opposite order $\ge$ well-...
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### Ordinal Arithmetic: Cantor Normal Form Algorithm

Is there a known algorithm/procedure to take an arbitrary expression of an ordinal $\alpha$ and convert it to Cantor normal form?
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### In a fast-growing hierarchy, what conditions on $(\alpha,\beta,n)$ imply $f_\alpha(n)<f_\beta(n)$?

Consider any specific fast-growing hierarchy $(f_\alpha)$ defined with one of the commonly used systems of fundamental sequences (e.g., any of those mentioned at the given link, using either Cantor ...
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### Ordinal vs. Cardinal $0$ [closed]

From my files ... $1, 2, 3, ...$ are cardinals, they count as in $9$ trucks, $12$ voles, etc. First ($1^{st}$), Second ($2^{nd}$), etc. are ordinals, they're used to order stuff. Now, I've heard of ...
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### Using a form of the Axiom of Choice to prove the Well Ordering Theorem

I have been studying ordinals on my own for a few weeks now, and I have just gone through the proof for Hartogs' lemma, that given $S$ a set, $\exists \alpha$ an ordinal s.t. there is no injective ...
1 vote
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### Writing infinitely long expressions in set theory

Using just the three symbols {, }, and , we can write any hereditarily finite set as a finite sequence of those three symbols. However, I have thought of something recently. What if we allow ...
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### Morley Rank and Cardinality paradox?

If the Morley rank $RM(\phi)$ of a definable set $X$=$\phi(\mathcal{M})$ is $\alpha$, based on the inductive definition of RM (I am using David Marker's book Model Theory: An Introduction, definition ...
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### Predicate for identifying von Neumann naturals

I am trying to write a formula in the language of ZF that checks for von Neumann naturals. This is how far I got: The predicate "Ind(a)" is true, if and only if $a$ is inductive. Ind(a) \...
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### Why do we define the proof-theoretic ordinal of a theory the way we do when there are unnatural well-orderings out there?

The proof-theoretic ordinal of first-order arithmetic ($\mathsf{PA}$) is $\varepsilon_{0}$. However, in pages 3 and 4 of Andreas Weiermann's Analytic combinatorics, proof-theoretic ordinals, and phase ...
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### Is there a model of KP with no $\emptyset'$ ordinal notation for the Bachmann-Howard ordinal?

The Bachmann-Howard ordinal (BHO) is a large recursive ordinal defined using ordinal collapsing functions. Kripke-Platek set theory (KP) is a fragment of ZF obtained by removing powerset, swapping ...
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### Well ordered set similar to an ordinal is an ordinal?

I'm wondering if a well-ordered set $B$ which is similar to an ordinal $A$ is necessarily an ordinal ? I think it may be not as the $\in$ relation in the ordinal number does not necessarily be the ...
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### The set of predecessors of a node in a tree is finite?

I came across the following definitions of tree and transitive tree, respectively, in the book Modal Logic by P. Blackburn: Definition 1. A tree is a relational structure $(T, R)$, where: $T$, the ...
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### Help with basic ordinal arithmetic: What is the supremum of the sequence $\omega^{2} + \omega$, $\omega^{2}*2 + \omega$, $\omega^{2}*3 + \omega$, ...?

I need some help with some basic ordinal arithmetic. I am trying to determine the supremum of the sequence $\omega^{2}*1 + \omega$, $\omega^{2}*2 + \omega$, $\omega^{2}*3 + \omega$, ... where the ...
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### The range of any continuous function from $[0, \omega_1] \times [0, \omega_1] \rightarrow \mathbb{R}$ is countable

I came across this question: “Prove that the range of any continuous function from $[0, \omega_1] \times [0, \omega_1] \rightarrow \mathbb{R}$ is countable” I’m quite new to ordinals and I’m having a ...
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### Dubious proof in Halmos's book: Two similar ordinal numbers are always equal

I'm studying ordinal numbers using Naive set theory of Halmos. I think there was a small mistake in his proof of the statement "if two ordinal numbers are similar, then they are equal" The ...
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### How axiom of substitution implies unique function for ordinals in Halmos book?

I'm studying set theory using Halmos's book. I'm stopping at the chapter 19 about ordinal numbers. I have 2 questions please: Is the axiom of substitution necessary in the definition of ordinal ...
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### When we speak of ordinal, do we always refer to natural numbers and theirs successors?

I'm studying set theory using Halmos'book. In the book, the definition of an ordinal is : A set $S$ is an ordinal if and only if $S$ is strictly well-ordered with respect to set membership and every ...
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### On the proof that the successor of an ordinal is also an ordinal

Definition. A set $\alpha$ is an ordinal if it is transitive and strictly well-ordered by $\in$. The proofs I have seen (e.g. Lemma 2.5 in Jech and Hrbacek or p.32 in these notes) go like this: First ...
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### Does the number $\frac{1}{\omega}$ exist?

Edit : Ok, "is not a real number" is a very different statement from "does not exist.". My initial thought is "does not exist.". Sorry, I never know about something like ...
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### How can initial segment of the set of all natural numbers is equal to itself

I'm studying ordinal numbers in Halmos's book (page 75). After the definition of ordinal number, he proves that $w^+$ is an ordinal number where $w$ is the set of all natural numbers. In the last part ...
### Which countable ordinals are common to every $\omega$-model of ZF?
Consider the progression of larger and larger countable ordinals ($\omega$, $\omega^{\omega}$, $\epsilon_0$, the Veblen hierarchy, etc., described nicely here). On the one, hand, it seems clear that ...
### Proving directly that $\aleph_{\alpha+1} \cdot \aleph_{\alpha+1} = \aleph_{\alpha+1}$ by recursion
I have defined the alephs by induction : $\aleph_{\alpha+1}$ is by definition the smallest cardinal superior to $\aleph_{\alpha}.$ Is there a quick way to show that \$\aleph_{\alpha+1} \cdot \aleph_{\...