All Questions
Tagged with ordinals elementary-set-theory
457
questions
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Showing that $\bigcap A$ is the least element for the set $A$ where $A$ is a set of ordinals.
The notes I am reading define a set $x$ to be an ordinal provided $x$ is transitive and every element in $x$ is transitive.
Let $A$ be a set of ordinals. I have shown that $\bigcap A$ is an ordinal. I ...
- 629
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0
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Ordinals multiplication: Does $a^2 b^2=b^2a^2$ imply $ab=ba$? [duplicate]
I found this question in one of set theory past exam: If $\alpha$ and $\beta$ are two ordinals such that $\alpha^2\beta^2=\beta^2\alpha^2$, does it necessarily imply $\alpha\beta=\beta\alpha$?
Clearly ...
- 19
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Does the equality $\omega\cdot(\omega+1)=(\omega+1)\cdot\omega$ hold?
By this answer I knkow that the equality
$$
(\omega+1)\cdot\omega=\omega^3
$$
holds whereas by the definition of ordinal multiplication I know that the equality
$$
\omega\cdot(\omega+1)=\omega^2+\...
- 4,906
1
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1
answer
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How can different representations of the same integer be equivalent?
I recently read about a way to define the set of integers as the set of all equivalence classes for some equivalence relation $\simeq$ satisfying $(a,b)\simeq(c,d)$ for $(a, b),\;(c,d)\in\mathbb{N}\...
- 508
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Hessenberg sum/natural sum of ordinals definition
I was given the following definition of Hessenberg sum:
Definition. Given $\alpha,\beta \in \text{Ord}$ their Hessenberg sum $\alpha \oplus \beta$ is defined as the least ordinal greater than all ...
- 509
0
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0
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45
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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 ...
- 11.9k
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2
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87
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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 ...
- 11.9k
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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<\...
- 191
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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) $\...
- 191
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49
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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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0
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46
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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,...
- 295
2
votes
1
answer
143
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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 ...
- 1,350
2
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1
answer
58
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Ordinal equations using Cantor's normal form
Find all possible ordinals $\alpha,\beta$ satisfying the equations:
$\alpha+\beta=\omega$
$\alpha+\beta=\omega^2+1$
For 1), discarding the trivial cases $\alpha=0$ or $\beta=0$, we have that $\...
- 357
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Reordering a Sequence of Sets Whose Union is the Whole Set
I have a set $ B $ that can be written as $ B = \cup_{\nu < \lambda} B_{\nu} $ where $ \kappa $ is the cardinality of $ B $, that is uncountable, and $ \vert B_{\nu} \vert < \kappa $ with $ \...
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$cf(\aleph_{\alpha}) = cf(\alpha)$ for all limit ordinals $\alpha$
I am trying to prove the following Lemma:
Lemma: Let $\alpha$ be a limit ordinal. Then $cf(\aleph_{\alpha}) = cf(\alpha)$.
I am using the following definitions:
Definition: Let $(\mathbb{P}, \...
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