All Questions
102
questions
2
votes
1
answer
89
views
Does a cardinal with uncountable cofinality imply that the cardinal is regular?
In our book we use for our classes, we often require cardinals to be uncountable regular cardinals (when proving stuff with cofinality/stationarity...). We often use that by creating some sort of ...
- 1,917
2
votes
0
answers
142
views
Does worldly cardinal exist if $\mathsf{ZFC}$ is consistent?
A worldly cardinal is a cardinal $\kappa$ such that $V_\kappa$ is a model of $\mathsf{ZFC}$. Please forgive me if this is very silly, but if $\mathsf{ZFC}$ is consistent (so there exists a model of $\...
- 1,923
-1
votes
1
answer
94
views
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 ...
user1234970
0
votes
1
answer
148
views
Cardinality of the set of measure zero sets
I have been thinking about this question for a while now and found nothing on the matter so far.
Assuming the continuum hypothesis (or maybe also for the case that we assume that it is false), what is ...
-1
votes
2
answers
46
views
Cardinal number of the iterated set $A^{*}$, where $A=\{ a,b,c\}$. Why can't I use Cantor's diagonal argument? [duplicate]
I have a question which may sound silly to you, but I'm confident that I don't understand Cantor's diagonal argument very well to use it. Any provided insight would be appreciated.
I was tasked with ...
- 187
1
vote
0
answers
68
views
Let $T=\{c_m\neq c_n: m,n\in \mathbb{N}, m\neq n \}$ in the language $L=\{c_n : n\in \mathbb{N}\}$. Prove that T is not $\aleph_0$-categorical.
as the title says I am asked to prove the following:
Let $T=\{c_m\neq c_n: m,n\in \mathbb{N}, m\neq n \}$ in the language $L=\{c_n : n\in \mathbb{N}\}$, where each $c_n$ is a constant symbol. Prove ...
- 391
0
votes
1
answer
286
views
What does it mean "cardinality of a model" and "cardinality of a language" ? Are they the same thing? (Model Theory)
I'm studying the lowenheim- skolem theorem but i am a bit confused this when it comes to cardinality, in some definitions they use the cardanality of the language, in others they use the cardinality ...
- 1
1
vote
1
answer
45
views
Cardinality of well-formed formulae on a multi-sorted signature
I had some questions regarding the cardinality of the sets of well-formed formulae and terms on a multi-sorted signature. I will give first the usual definitions.
A signature Σ is a triple〈S;F;R〉 ...
- 75
0
votes
1
answer
121
views
Is it meaningful to invert large cardinals?
As cardinal numbers involve the notion of ever increasing multitudes of things,
is there a mathematically useful concept of ever decreasing multitudes?
We already have rationals tending to zero, and ...
- 289
0
votes
1
answer
120
views
Cofinality of functions $\mathbb N\to\mathbb N$ under eventual domination
As the question suggests, I'm interested in the cofinality of functions $\mathbb N\to\mathbb N$ under eventual domination, i.e.
$$f\prec g \quad\Longleftrightarrow\quad\exists N,\ \forall n>N,\ f(n)...
- 4,764
2
votes
1
answer
141
views
About the definition of hereditary cardinality
I've seen these two definitions of sets that are hereditarily of cardinality $< \kappa$.
$x$ is hereditarily of cardinality $< \kappa$ iff $|trcl(x)| < \kappa$
$x$ is hereditarily of ...
- 12k
3
votes
1
answer
92
views
$\operatorname{cf}(\sup(B \cap \omega_m))$ is the natural numbers
Let $\mathcal{L}=\{\in, \preceq\}$. Let $\mathcal{A}=(V_\theta, \in)$ (where $\theta > \omega_\omega$) be an $\mathcal{L}$-structure which interprets $\preceq^\mathcal{A}$ by some fixed well-...
- 1,847
1
vote
1
answer
51
views
Cardinality of tail transformation
I am reading this article, "A non-measurable tail set" by Blackwell and Diaconis.
In this article we don't have axiom of choice, we just suppose that there exists a free ultrafilter on $\...
- 1,595
3
votes
3
answers
90
views
Compact cardinal cannot be successor?
This is a follow-up question to $\kappa$ is compact $\implies$ $\kappa$ is regular. The definition I'm using for "compact" is the same as there.
I am trying to show if $\kappa$ is compact, ...
- 1,847
1
vote
1
answer
119
views
cofinality of $\prod_{n \in \omega} \aleph_n$ under everywhere dominance
In Don Monk's notes on set theory (http://euclid.colorado.edu/~monkd/full.pdf), it is written on page 784 (the second item after Proposition 33.6) that the cofinality of $\prod_{n \in \omega} \aleph_n$...
- 337