All Questions
Tagged with well-orders proof-explanation
25
questions
1
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137
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Question about a proof that any well-ordered set is isomorphic to a unique ordinal
I am studying a proof that every well-ordered set is isomorphic to a unique ordinal. However, I don't understand why $A = pred(\omega)$ (see yellow).
One direction is clear:
Let $x \in pred(\omega)$, ...
- 1,089
2
votes
1
answer
184
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CH, countable additivity, and Ulam's Theorem
I am seriously struggling in properly getting various steps in the proof of Theorem 1.12.40 (Ulam's Theorem) in Bogachev's first volume on measure theory which concerns the classical statement that ...
- 4,113
4
votes
1
answer
295
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Analyzing the proof of Kirszbraun's theorem
Kirszbraun's theorem 1934
Let $A \subset \mathbb{R}^n$. If $f \colon A \rightarrow \mathbb{R}^m$ is a $L$-Lipschitz function, then there exists an extension $F \colon \mathbb{R}^n \rightarrow \mathbb{...
- 541
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1
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Prove without the well ordering principle that no m exists such that $n < m < n + 1$ for positive integers $m$ and $n$. [duplicate]
I'm trying to prove without the well ordering principle that no integer $m$ exists such that $n < m < n + 1$ for positive integers $m$ and $n$.
I know there's a proof here that uses the well ...
- 383
3
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1
answer
2k
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Every metric space is paracompact (an elegant proof)
I'm studying a different proof to show that each metric space is paracompact in the book: Singh, Tej Bahadur-Introduction to Topology. It is a very elegant construction unlike the inductive method ...
- 89
1
vote
2
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145
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Well Ordering implies Induction Proof doubt
I’m trying to understand the proof for the fact that that the Principle of Well-Ordering implies the Principle of Mathematical Induction; that is, if S ⊂ N such that 1 ∈ S and n + 1 ∈ S whenever n ∈ S,...
- 143
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0
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286
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Well Ordering Principle proof for amount of postage
I have encountered this task for WOP practicing and struggle with forming set of counterexamples and moving on from there.
(a) Prove using the Well Ordering Principle that, using $6¢$, $14¢$, and
$21¢...
2
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0
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Proof of "For each integer m, the set S = {i ∈ Z : i ≥ m} is well-ordered." [duplicate]
This is question 6.17 from Mathematical Proofs by Chartrand/Polimeni/Zhang.
Prove: For each integer $m$, the set $S = \{i ∈ Z : i ≥ m\}$ is well-ordered.
The given proof in the back of the book ...
- 85
2
votes
2
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150
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Well-Ordering Proof in Cunningham
In Cunningham's Proof [1] of the Well-Ordering Principle (assuming $\textsf{AC}$), he begins by considering the set $\mathcal{W} = \{ \preceq | \preceq \text{a well-ordering on a subset of } A \}$ ...
- 439
1
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1
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73
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Sets well ordered under different operations Proving
$(a)\quad R^+ \cup \{0\}, <$
$(b)\quad [0,1], >$
$(c)\quad \text{The set of integers divisible by 5}, <$
$(d)\quad \{\{0,1,...,n\}|n ∈ N\},⊆$
I believe that:
(a) Is not well-ordered ...
- 233
1
vote
0
answers
83
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How to approach solving this induction problem?
I've been practicing induction and I came across this problem:
Consider the following series, 1, 2, 3, 4, 5, 10,20, 40, ..., which
starts as an arithmetic series, but after the first 5 terms ...
- 303
2
votes
0
answers
1k
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Zorn's lemma, every set can be well-ordered
My task is to prove the well-ordering theorem:
Every non empty set $X$ can be well-ordered
I want to do so in the following steps, which involve Zorn's lemma:
Step 1):
Let $X$ be non empty and $\...
- 11.2k
1
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0
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503
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Well Ordering Principle sum of natural Numbers help
I was reading over the following solution for the sum of all natural numbers using the well-ordering principle.
Let Fact 1 be:
$$\forall n:\ (1+2+3+...n)=\frac{n(n+1)}{2}$$
By contradiction, ...
- 117
1
vote
1
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I don't understand how this set can be contained in $\Bbb N$
In my lecture notes there is a proof for the division algorithm which sets $S=\{a-xb|x\in \Bbb Z, a-xb \geq 0 \}$ then says $S\subset\Bbb N$ so we can use the well ordering principle.
There's a ...
- 2,815
0
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3
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Prove that for every natural number $a$ there are integers $t \geq 0$ and $r$ such that $a = 3^t r$ and 3 does not divide r.
Prove that for every natural number $a$ there are integers $t \geq 0$ and $r$ such that $a = 3^tr$ and $3 \not | r$.
Would I use the well ordering principle for this? There’s so many variables so I’m ...
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