All Questions
Tagged with well-orders real-analysis
21
questions
0
votes
2
answers
38
views
Confusion about the validity of the proof of Trichotomy of order for natural numbers in Tao's Analysis
It's well-known that in Tao's Analysis I P28, he provides a provement of Trichotomy of order for natural numbers as follows.
Denote the number of correct propositions among the three (i.e. $a<b,\ ...
1
vote
0
answers
113
views
Does any non-maximal element of a well-ordered set have a unique successor?
$\newcommand{\setcomplement}[2]{#1 \setminus #2}$
$\newcommand{\singleton}[1]{\left\{#1\right\}}$
$\newcommand{\segment}[2]{\operatorname{Seg}_{#1}\left(#2\right)}$
I wanted to find the smallest ...
- 1,840
7
votes
1
answer
184
views
Any Well-Ordering of $\mathbb{R}$ has no Corresponding Metric
Background
I have been doing some reading over winter break so far and found the idea that $\mathbb{R}$ has a well-ordering strange. So I have been thinking about it a little and was wondering if my ...
- 552
0
votes
3
answers
113
views
Well ordering principle in Folland chapter 0
In Folland chapter 0.2 Orderings, the well-ordering principle is stated as:
Every nonempty set $X$ can be well ordered.
I read over the proof, which seems fine to me. However, I also understand that,...
- 573
2
votes
2
answers
114
views
Show that $\cup_{x\in\Omega}\mathcal{F}_x$ is a $\sigma$-algebra
Let $X$ be an arbitrary set and let $\Omega$ denote the minimal uncountable well-ordered set. Given $\mathcal{E}\subset \mathcal{P}(X) $ with $\emptyset\in \mathcal{E}$ we define a collection of ...
- 4,827
2
votes
0
answers
51
views
Show that $f$ is an order isomorphism from $\mathbb{N}$ to $I_{\omega}$
Am reading Folland's Real Analysis book, and on page 10 he introduces the set $\Omega$ of countable ordinals. This set is the unique (up to order isomorphism) uncountable well ordered set such that ...
- 4,827
1
vote
0
answers
34
views
Filling the details in Proposition 0.17 in Folland
On page 9 of Folland's Real Analysis book we can read:
0.17 Proposition. If $X$ and $Y$ are well-ordered, then either $X$ is order isomorphic to $Y$, or $X$ is order isomorphic to an initial segment ...
- 4,827
3
votes
2
answers
361
views
Infinite subset of natural numbers
I want to show that if the set $A$ is an infinite subset of natural numbers $N$, then $\text{card}(A)=\text{card}(N)$. To do so, it suffices to find an injective function $f:N\rightarrow A$.
One ...
- 991
5
votes
2
answers
554
views
Any set can be well-ordered; doesn't that imply $(0,1)$ has a smallest real number?
I came across the above theorem ( also known as Zermelo's theorem) in real analysis. Since any set can be well-ordered so can the set (0,1) which means there must exist a least element of the set.
...
- 85
4
votes
2
answers
176
views
Is my proof that the Sharkovsky Ordering is a total ordering, correct?
The Sharkovsky ordering is an ordering of the natural numbers $\mathbb{N}$, where
$3$ $\prec$ $5 $ $\prec$ $7 $ $\prec$ $9$ $\prec$ ...
$2*3$ $\prec$ $2*5$ $\prec$ $...
- 85
0
votes
2
answers
386
views
Prove the Well-Ordering Principle by Mathematical Induction.
So I want to prove that every non-empty subset of the natural numbers has a least element. I used induction but I'm not sure if doing that proves the statement for infinite subsets of $\mathbb{N}$. ...
- 337
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
0
votes
0
answers
99
views
Well order on $ℚ$ doubt [duplicate]
I know the set of rational numbers can be well ordered since there exists a bijection from the set of natural numbers and then set of rational numbers by ordering $ℚ$ in this way
my question is, in ...
user782485
0
votes
2
answers
592
views
Proof of the well-ordering principle
I tried to prove Well-Ordering Principle by myself, and I finally did it. However, I'm not sure if this proof is correct. Can anyone evaluate my proof?
Proof:
Since the set of natural numbers, $\...
- 123
4
votes
1
answer
161
views
Well ordered subsets of $\mathbb{R}$
At the entrance exam of a french school, the following problem was given :
Characterize well-ordered subsets of $\mathbb{R}$
The only property I found was that such a subset must be at most ...
- 933