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.
1,981
questions
3
votes
1
answer
62
views
How to well order the set of all functions with domain an ordinal and codomain a set?
Source: Set Theory by Kenneth Kunen.
Theorem I.12.14 ($ZFC^-$) Let $\kappa$ be an infinite ordinal. If $\mathcal{F}$ is a family of sets with $|\mathcal{F}|\leq\kappa$ and $|X|\leq\kappa$ for all $X\...
- 1,467
3
votes
1
answer
204
views
Ascertaining whether "absolute" standardness of $\omega$ is actually possible in ZFC
Books on set theory seem to at least strongly imply that there can exist some $\mathsf{ZFC}$ universe whose $\omega$ is "standard" in an "absolute" sense (order-isomorphic to the ...
- 1,024
0
votes
1
answer
51
views
$\gamma\mapsto \sup_{\xi \in \gamma}f(\xi)$ is not strictly bounded
This is motivated by Exercise 15(c) in Cori, Lascar, Pelletier.
Given a map $f:\beta\to \alpha$ whose image is not strictly bounded (i.e. the upper bounded of the image of $f$ is the ordinal $\alpha$)....
- 4,223
1
vote
1
answer
194
views
How to define ordinal addition
From Jech's Set Theory:
We shall now define addition, multiplication and exponentiation of ordinal numbers, using Transfinite Recursion.
Definition 2.18 (Addition). For all ordinal numbers $\alpha$
$...
- 5,166
2
votes
0
answers
67
views
Step in Jech's proof that any well-ordered set is isomorphic to an ordinal
From Jech's Set Theory:
Proof. The uniqueness follows from Lemma 2.7. Given a well-ordered set $W$,
we find an isomorphic ordinal as follows: Define $F(x) = \alpha$ if $\alpha$ is isomorphic
to the ...
- 5,166
1
vote
1
answer
67
views
What exactly is the $\in$-Induction Principle?
Throughout the post $\mathcal{L}_\text{ZF}$ is the language of $\text{ZF}$, while $\text{fld}(R)$ is the field of the relation $R$, and $\text{wf}(R)$ means the relation $R$ is well-founded i.e.
$\...
- 5,166
3
votes
2
answers
111
views
Showing there is an inductive set that contains no limit ordinals
If we define a natural number $n$ as
$$n\in\bigcap\{x:x\text{ is an inductive set}\}$$
then showing any natural number is either zero or a successor ordinal is equivalent to showing there is an ...
- 5,166
1
vote
1
answer
163
views
Are the reals a "subset" of the class of ordinals
I am not sure if it's even correct to use subset in this context but I'm sure it gets the point across. I just want to know if the class of ordinals includes non-integer elements like $4.5$, $\pi$, $e$...
- 35
5
votes
1
answer
157
views
Is the Stone Cech compactification $\beta\omega$ of $\omega$ radial?
$\beta\omega$ is sequentially discrete, that is, every convergent sequence is eventually constant. Thus, since the space is not discrete, the space is not sequential, that is, it's untrue that every ...
- 6,384
0
votes
0
answers
125
views
Is omega minus one an ordinal?
Is there an 'signed' ordinal system 'stronger' than the 'original' ordinals, but 'weaker' than the surreals, with subtraction and division? Like one with negative integers and $\omega -1$ but not $0.5$...
- 239
3
votes
1
answer
60
views
Cofinalities of ordinals $\omega_1 \omega, \omega_2 \omega_1, \omega_2 \omega_1 \omega$?
I am trying to understand cofinalities so I would really appreciate it if you could confirm whether the following line of reasoning is correct. (All multiplication is ordinal multiplication)
$$
cf(\...
- 481
4
votes
1
answer
352
views
Constructing rationals from countable ordinals
Take the set of all countable ordinals $\omega_1$. Define $+$ and $\cdot$ by the Hessenberg sum and product. In a similar fashion to the naturals, construct the integers and the rationals from these ...
0
votes
0
answers
71
views
An infinite linear system of equations with an uncountable number $A$ of equations
I will start with an example to make things clear and avoid confusion :
Take all $x>0$ and
$$\exp(x) = \sum_{-1<n} a_n x^n$$
Now finding $a_n$ can be described as an infinite linear system of ...
- 16.4k
2
votes
1
answer
143
views
Is there an unbounded countable set of ordinals?
Let $\text{Ord}=\{0,1,\ldots,\omega,\ldots\}$ be the set of ordinals. Does $\text{Ord}$ have an unbounded countable subset? In particular, is
$$\{\omega, \omega^\omega, \omega^{\omega^\omega},...\}$$
...
- 984
7
votes
1
answer
160
views
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 ...
- 339