All Questions
288
questions
1
vote
0
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29
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A sequence of continuum hypotheses
The continuum hypothesis asserts that $\aleph_{1}=\beth_{1}$. Both it and its negation can be consistent with ZFC, if ZFC is consistent itself.
The generalised continuum hypothesis asserts that $\...
- 290
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 $\beth_{\alpha} =\begin{cases}
|\mathbb{N}| & \text{if } \alpha=0 \\
2^{\beth_{\alpha-1}} & \text{if } \alpha \text{ is a ...
- 171
1
vote
1
answer
116
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$\omega$-th or $(\omega + 1)$-th when putting odd numbers after even numbers?
The following clip is taken from Chapter 4 - Cantor: Detour through Infinity (Davis, 2018, p. 56)[2]. When putting odd numbers after even numbers, what should the index for the first odd number ($1$ ...
- 103
1
vote
0
answers
67
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How is transifnite recursion applied?
I've been struggling to understand how ordinal addition, multiplication, and exponentiation, along with the Aleph function $\aleph$, are defined using Transfinite Recursion in Jech's Set Theory or ...
- 5,166
2
votes
1
answer
90
views
What is cardinality of ordinal exponentiation?
Using von Neumann definition of ordinals, is it true that for all cardinal numbers $a$ and $b$ the following equation holds:
$$
a^b = |a^{(b)}|
$$
where on the left side is the cardinal exponentiation ...
- 59
2
votes
1
answer
166
views
What ways are there to define $\aleph$?
I've seen some posts on this website that consist in providing and comparing different proofs of a theorem (e.g. for Taylor's Theorem or trigonometric identities). Currently I'm reading Holz's ...
- 5,166
2
votes
1
answer
58
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Why does regularity of an ordinal $\gamma$ imply the existence of a sequence $(\delta_n)$ such that $\delta_n<\gamma$ for all $n$?
Source: Set Theory by Kenneth Kunen.
Lemma III.6.2: Let $\gamma$ be any limit ordinal, and assume that $\kappa:=cf(\gamma)>\omega$. Then the intersection of any family of fewer than $\kappa$ club ...
- 1,467
0
votes
1
answer
61
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Proof of the Reflection Theorem in Kunen?
I'm reading Kunen's Set Theory and the last line of the proof of the Reflection theorem (page 131) is a bit puzzling to me. To those not in possession of Kunen at the moment, the book states verbatim:
...
- 481
0
votes
0
answers
27
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Being unbounded in a limit ordinal implies order type is also a limit?
Whilst trying to follow a proof from my lecture notes, I stumbled upon the following:
$$\gamma \text{ limit, }A\subseteq \gamma, \sup A=\gamma\implies \text{type}(A)\text{ limit}$$ It sounds true, ...
- 481
1
vote
1
answer
63
views
Why do we define cardinality only for well-orderable sets?
I'm revising Kunen's Set Theory and he mentions that when we define the cardinality of a set, we should do it with well-orderable sets. Why is this the case? He points to a section which contains many ...
- 481
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
1
vote
2
answers
62
views
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 ...
- 191
8
votes
1
answer
109
views
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-...
-3
votes
1
answer
73
views
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 ...
- 983
4
votes
1
answer
183
views
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\...
- 89