All Questions
Tagged with cardinals order-theory
60
questions
4
votes
1
answer
82
views
Induction does not preserve ordering between cardinality of sets?
Consider building a binary tree and consider it as a collection of points and edges. Here is one with five levels, numbered level $1$ at the top with $1$ node to level $5$ at the bottom with $16$ ...
- 6,360
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-...
-1
votes
2
answers
128
views
The rationals and their initial segments, edited version [closed]
This post was inspired by an Alon Amit post on Quora. The Quora
problem posed to AA was something like, only slightly more confused
than, this: How can the set of initial segments of the rational ...
- 23
0
votes
1
answer
78
views
Special Aronszajn tree actually has continuum cardinality?
Wikipedia gives the following construction of a special Aronszajn tree. Supposedly, this tree has $\aleph_1$ nodes, as each level and each branch is countable. However, it seems to me that this ...
- 5,830
1
vote
1
answer
90
views
Cardinality of subsets of $\mathbb{Q}$ (or $\mathbb{R}$) which are order isomorphic to $\mathbb{Q}$ (respectively $\mathbb{R}$)
I am supposed to solve these two exercises for a set theory course. I should have done the first one, but I do not have good ideas for the second one.
Find the cardinality of the subsets of $\mathbb{...
- 1,828
2
votes
0
answers
58
views
What's wrong with this proof that $|\omega_1|\ge2^{\aleph_0}$?
According to this question, the size of $S_{\mathbb N}$, the set of all bijections on $\mathbb N$, is at least $2^{\aleph_0}$. So $|S_{\mathbb N}| \ge 2^{\aleph_0}$
For each bijection $f:\mathbb N\...
- 22.9k
1
vote
1
answer
201
views
Cardinality of the set of strict total orders on $\mathbb{R}$
A strict total order on $\mathbb{R}$ is a relation $R \subseteq \mathbb{R} \times \mathbb{R}$ such that:
$\forall x \in \mathbb{R}, (x,x) \not \in R$
$\forall x,y,z \in \mathbb{R}, (x,y) \in R \text{ ...
- 469
1
vote
0
answers
56
views
How to linearly order the set of all subsets of real numbers?
I wondered if there are linearly ordered sets of any cardinality. As I understand it, there are. But I want to see at least one concrete example of a linearly ordered set which cardinality is greater ...
- 21
2
votes
1
answer
250
views
Is almost every definable number uncomputable?
We know that almost all real numbers are undefinable.
We also know that almost all real numbers are uncomputable.
We also know that there are numbers that can be defined but not computed.
However, ...
0
votes
1
answer
76
views
How to show that $\omega+\omega_1=\omega_1$? [duplicate]
I'm just learning ordinal and cardinal arithmetic for the first time, and after learning that $n+\omega=\omega$ for finite $n$, I started to wonder whether $\omega+\omega_1=\omega_1$. This seems ...
- 6,672
1
vote
1
answer
119
views
Why does the back and forth method fail to prove that, for each cardinality, any dense linear order without endpoints is unique up to isomorphism?
First of all I must say I'm not not very knowledgeable about set theory beyond the very basics, so please bear with me if I've made some obvious mistakes in my reasoninig.
I've looked at and ...
- 13
0
votes
1
answer
42
views
Is the proper segment of a TOSET is initial segment? [closed]
I can't fimd its answer.
I have learnt that for a well ordered set proper segment is the initial segment. But i am unable to find segment of a toset which is proper and not initial.
0
votes
1
answer
76
views
Left Compatible with Ordinal Addition (Multiplication) for alephs
I have a question about strictly increasing properties of addition and multiplication on Ord. Do they work for alephs?
My attempt:
Subset is Left Compatible with Ordinal Addition:
$\forall \alpha, \...
- 85
1
vote
1
answer
71
views
Cardinality of order preserving functions from totally ordered set with dense subset
I know that in the category of continuous functions, if $X$ and $Y$ are Hausdorff topological spaces and $D\subseteq X$ is a dense subset of $X$, then the cardinality of the continuous functions from $...
- 7,805
0
votes
2
answers
52
views
Suppose |$A$| = |$B$| and let $f: A \to B$ be a surjective function. Can we/ should we define the ratio of size of set $A$ : set $B$ wrt function $f$?
I've not seen this idea anywhere before, but I'm not a set theory buff either.
Let $A$ and $B$ be sets with the same cardinality and let $f: A \to B$ be a surjective function. Suppose, for every $y \...
- 20.7k