All Questions
Tagged with well-orders sequences-and-series
7
questions
3
votes
2
answers
373
views
Construction of two uncountable sequences which are "interleaved"
I believe the answer to my following question is no, but some things about uncountable sets/sequences can be really counterintuitive so I wanted to double check:
Does there exist a pair of uncountable ...
- 107
6
votes
1
answer
127
views
Sequence of quadratic surds over nonnegative integers without having to delete or sort?
I am trying find an strictly increasing iterative sequence that gives this set sorted: $$[a+\sqrt{b}: a,b \in \mathbb{N_0}].$$ These are a subset of constructable numbers. When I look at it, there are ...
- 61
0
votes
1
answer
100
views
Description of largest possible countable set / number
I am looking for an elegant / standard (if any) description of the largest countable set.
A first naive approach would be to construct this set, X, by taking the integers (0 to, but not including, ω_0,...
3
votes
2
answers
125
views
Meaning of 'set of well-ordered sequences'
I'm trying to make sense of a construction of a module given in the following research paper: A New Construction of the Injective Hull, Fleischer, 1968.
On the second page, a module $F$ is constructed,...
- 3,440
1
vote
1
answer
126
views
Cardinality of set of well-ordered sequences
We think of $A=\mathbb{R}^\mathbb{N}$ as the set of all functions $f:\mathbb{N}\to\mathbb{R}$. Consider the following subset of $A$:
$$
B=\{f\in A\mid f(\mathbb{N}) \text{ is a well-ordered subset of $...
- 29
1
vote
0
answers
79
views
Is there a well-ordered uncountable set of real numbers?
The Problem and its Solution
My approach (in a different way): We write $\mathbb{R}=\displaystyle\bigcup_{n \in \mathbb{Z}} [n-1,n]$ and let us call $[n-1,n]=I_n$.
Suppose $S \subset \mathbb{R}$ is ...
- 2,620
-1
votes
1
answer
91
views
Using circles to map $\mathbb{N}\to \mathbb{N^2}$
I am using $\mathbb{N}[i]$ for the Guassian integers that have non-negative real and imaginary components. We can create an ordering on them in the following way : First we will look to magnitude, ...
- 3,813