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 ...
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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 ...
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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 ...
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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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Jech's proof of canonical well-ordering of $\alpha\times\alpha$.
I'm reading Jech's Set Theory. The canonical well-ordering of $\mathrm{Ord}\times\mathrm{Ord}$ is defined as
$$( \alpha ,\beta ) < ( \gamma ,\theta ) :\begin{cases}
\max\{\alpha ,\beta \} < \max\...
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TREE(3) and the Goodstein sequence
TREE(3) is an extremely large number that requires ordinal arithmetic to prove it is finite. For what value of n would $G(n)>TREE(3)$? The length of the
Goodstein sequence $G(n)$ is how many ...
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Cardinality of a set V vs cardinality of its ordinal Type ord(V)
I know that ordinal numbers use to position (rank) the members of a set. The order type is a relation between two sets A and B that are order isomorphic, there is a bijection between them. Sometimes ...
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would diagonalization work in this scenario?
consider a countably infinite list of infinite strings.. such that each string has an ordinal of $ \ \bf ɷ.2 \ $, and the entire list also has an ordinal of $ \ \bf ɷ.2 \ $. Can we use cantor's ...
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Trouble understanding Proof Wiki's proof that every well-ordered set is order isomorphic to an ordinal.
I'm struggling to understand a couple of steps in this proof showing that every well-ordered set is order isomorphic to an ordinal. Calling the following steps
first and second respectively, my ...
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How is the following theorem the princple of *complete* induction?
The following is from page $19$ of Holz' Introduction to Cardinal Arithmetic.
As I understand it (see here), ordinary and complete induction (on $\omega$) work as follows:
Ordinary Induction: if $P(...
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Alpha recursion - Constructible universe and Analytical hierarchy
Alpha recursion and Constructible universe are very intertwined, because the first is based on the concept of admissible ordinal $\alpha$ which is defined as an ordinal such that $L_\alpha$ - a set ...
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The existence of a function on $\kappa$
Let $\kappa>\omega$ be a regular cardinal. Let $C\subseteq\kappa$ be a club. Prove that there exists a function $f:\kappa\rightarrow\kappa$ such that $C_f=\{0<\alpha<\kappa:\forall\xi<\...
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Prove that these sets are stationary.
Define Lim($\omega_1$)={$\delta<\omega_1\ :\ \delta$ is a limit ordinal}. Assume that $\left<A_\alpha:\alpha\in\text{Lim}(\omega_1)\right>$ is a sequence satisfying the following:
(1) $\...
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can there be fractional values in the Veblen hierarchy?
I know that $\varphi_0(0)=\omega$ and $\varphi_1(0)=\epsilon_0$, but what would the value of $\varphi_{\frac{1}{2}}(0)$ be. it seems intuitive that it would be smaller than $\epsilon_0$, being $$\frac{...
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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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Limit Countable Ordinal - is it a limit of a intuitive sequence of ordinals?
I am studying set theory, ordinal part.
Set theory is new to me.
I know that commutativity of addition and multiplication
can be false in infinite ordinal world.
$ \omega $ = limit of sequence $\, 1,2,...
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Can a ZF universe ever contain "every conceivable ordinal"?
Note: For the purposes of this question, I’m considering only set universes whose ordinals are well-founded, as discussed here and here
It seems like no $\mathsf{ZF}$ universe can ever really be said ...
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Is there such an ordinal as ω-1? [closed]
Sorry for the daft question, but is there such an ordinal 'ω-1' such that ω = ('ω-1')⁺?
Still trying to learn set theory and ordinal arithmetic is absolutely boogling me 😅
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Ordinal Arithmetic: $\omega\cdot \omega^{\omega}=\omega^{\omega}$
Someone please can help me out ? Please.
I am trying to use the definition of product ordinal and exponent ordinal.
Product ordinal:
$\beta \cdot \alpha=\sup \{\beta \cdot \delta: \delta<\alpha\}$
...
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a proof idea: Every well-ordered set has an order-preserving bijection to exactly one ordinal.
I have seen a proof of the statement, and its usually by transfinite induction. And I'm trying to find out why my proof doesn't work, it seems too simple:
Let $X$ be a well-ordered set. Define $X^{<...
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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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Can $ZF$ construct this function?
Is it possible in the $ZF$ theory to construct a function with domain $\omega_1$ (the set of all countable ordinals) that maps each countable ordinal $\alpha$ to a bijection between $\alpha$ and $\...
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A couple of well ordering proofs.
I'm having trouble understanding a couple of things when studying well orderings and ordinals.
I know that given a well ordering $(A,<)$ there is no $a\in A$ s.t. $(A,<)\cong (A_a,<)$ where $...
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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 ...
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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 ...
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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_{\...
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