All Questions
Tagged with ordinals set-theory
1,049
questions
0
votes
1
answer
51
views
Is $\operatorname{rank}(A)\subseteq\operatorname{TC}(A)$?
This question came to me when I was thinking about rank and transitive closures. Let $A$ be a set of rank $\alpha$ and let $\operatorname{TC}(A)$ denote the transitive closure of $A$. Is it true then ...
- 448
2
votes
1
answer
85
views
Can $\sf{ST}$ construct an infinite class wellordered ordered by $\in$?
Assume the axioms of Extensionality, Empty Set, and Adjunction (meaning that $S\cup\{x\}$ forms a set for any $S,x$). Notice that we do not have Specification as an axiom, which makes this theory very ...
- 2,119
1
vote
1
answer
83
views
Can cardinality $\kappa$ exist where $\forall n\in\mathbb{N} \beth_n<\kappa$,$\kappa<|\bigcup_{n\in\mathbb{N}}\mathbb{S}_n|$,$|\mathbb{S}_n|=\beth_n$
The Wikipedia article on Beth numbers defines $\beth_\alpha$ such that $\beth_{\alpha} =\begin{cases}
|\mathbb{N}| & \text{if } \alpha=0 \\
2^{\beth_{\alpha-1}} & \text{if } \alpha \text{ is a ...
- 171
0
votes
0
answers
65
views
Can the class of ordinals be extended even further? [duplicate]
Is it possible for anything to come after all ordinals? I don't see why not. For example, one can take a non-ordinal set $S$, and then add in all the ordered pairs $(\alpha, S)$ to $ON$, where $\alpha$...
- 21.5k
2
votes
0
answers
73
views
Set of ordinals isomorphic to subsets of total orders
Background. Given a poset $(S,<)$ we'll indicate with $\tau(S,<)$ the set of all the ordinals which are isomorphic to a well ordered subset of $(S,<)$. We're in $\mathsf{ZFC}$.
Questions.
...
- 509
3
votes
1
answer
73
views
Constructible subsets of an ordinal - an alternate definition?
Work over ZFC. Recall the standard definition of $\mathrm{Def}(X)$, the set of constructible subsets of the set $X$. This can be considered as taking all those subsets of $X$ that satisfy some first-...
- 2,396
1
vote
0
answers
46
views
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
1
vote
1
answer
93
views
Intuition of Ordinal Number
I am learning the concept of ordinal numbers.
In the book of Set Theory and Metric Spaces by I. Kaplansky (Sec. 3.2, pg. 55), the author states
We attach to every well-ordered set an ordinal number; ...
- 513
2
votes
0
answers
28
views
References that give the cofinality of ordinal addition, multiplication and exponentiation
$\newcommand{\cf}{\operatorname{cf}}$
Let $\alpha$, $\beta$ be ordinals. I believe that we have
\begin{align*}
\cf(\alpha+\beta)=\cf(\beta),\quad\beta\neq 0;\quad\cf(\alpha\cdot\beta)=\begin{cases}\cf(...
- 1,923
2
votes
1
answer
104
views
Define the sum of transfinite ordinal sequences according to finite ordinal sum.
I know it is possibile to define finite sum of ordinals: if $(\alpha_i)_{i\in n}$ is a sequence of lenght $n$, with $n$ in $\omega$, then the symbolism $\sum_{i\in n}\alpha_i$ has a (formal) meaning; ...
- 4,906
-2
votes
1
answer
57
views
Prove that the order type of $\alpha\cdot\beta$ is the antilexicographic order in $\alpha\times\beta$. [closed]
This question is related to this one, but not a duplicate, since I am struggling with injectivity and monotonicity, rather than proving that $\{\alpha\cdot\eta + \xi:\eta<\beta\textrm{ and }\xi<\...
- 4,906
4
votes
1
answer
182
views
Colourful class function
Background. We're in $\mathsf{ZFC}$, and I can use the principle of $\epsilon$-induction, but not (directly) the $\epsilon$-recursion.
Problem. Let $F : V \to V$, where $V$ is the class of all the ...
- 509
0
votes
0
answers
75
views
How Should I Do the Inductive Case of this Proof on Ordinals?
Question
Prove that for all ordinals $\alpha$, $V_\alpha=\{x:\rho (x)<\alpha\}$.
Note:
The function $\rho$ is a rank function.
Attempt
I did the proof by induction on ordinals. I started with the ...
- 451
2
votes
0
answers
137
views
Strictly decreasing function from an ordinal to $\mathbb{R}$
Context: We are working in $\mathsf{ZFC}$.
Problem: Given a poset $(P,<)$, let $\text{Dec}(P)$ be the set of all strictly decreasing function $f : \alpha \to P$, where $\alpha$ is an ordinal number....
- 509
4
votes
1
answer
126
views
Is there a smarter way to calculate $F(\omega^{15} + 7)$? (Ordinal arithmetic)
Given the function $F : \text{Ord} \to \text{Ord}$ definite by transfinite recursion as it follows:
$$F(0) = 0 \qquad F(\alpha + 1) = F(\alpha) + \alpha \cdot 2 + 1 \qquad F(\lambda) = \sup_{\gamma &...
- 509