# Questions tagged [well-orders]

For questions about well-orderings and well-ordered sets. Depending on the question, consider adding also some of the tags (elementary-set-theory), (set-theory), (order-theory), (ordinals).

458 questions
Filter by
Sorted by
Tagged with
58 views

### Continuation of well ordered sets

We shall say that a well ordered set $A$ is a continuation of a well ordered set $B$, if, in the first place, $B$ is a subset of $A$, if, in fact, $B$ is an initial segment of $A$, and if, finally, ...
91 views

### 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 ...
51 views

### Is there an order-preserving injection from $\alpha\cdot\alpha+\omega$ to $\mathcal{P}(|\alpha|)$?

Let $\alpha$ be an infinite ordinal. Is there an order-preserving injection from $\langle\alpha\cdot\alpha+\omega,\in\rangle$ to $\langle\mathcal{P}(|\alpha|),\subsetneq\rangle$? I tried to construct ...
79 views

### Are $\Bbb{R}\times\Bbb{Q}$ and $\Bbb{Q}\times\Bbb{R}$ order-isomorphic?

Consider the two following linear orders, $X=\langle \Bbb{R}\times\Bbb{Q}, <_\text{lex} \rangle$ and $Y=\langle \Bbb{Q}\times\Bbb{R}, <_\text{lex} \rangle$, where $<_\text{lex}$ is the ...
1 vote
52 views

### Are suborders of pseudo-well-ordered sets themselves pseudo-well-ordered?

The class $W$ of well-ordered sets is not first-order axiomatizable. However, it does have an associated first-order theory $Th(W)$. I define a pseudo-well-ordered set to be an ordered set $(S;\leq)$ ...
61 views

### Order type of a set that is ordered but not well-ordered

I looking at ordered sets in the Kolmogorov (Introductory Read Analysis) book. Section 3.5 Definition 2 says the order type of a well ordered set is called an ordinal number. If the set is infinite, ...
30 views

90 views

### How exactly does Halmos define an ordered pair?

Halmos writes in "Naive Set Theory": Given the set ${X}$, consider the collection $W$ of all well-ordered subsets of $X$. Explicitly: an elememt of $W$ is a subset $A$ of $X$ together with ...
17 views

### What does the second property in Halmos definition of continuation mean?

Halmos defines in his book "Naive Set Theory" that a well-ordered set A is a continuation of a well-ordered set B if B is a subset of A B is an initial segmemt of A the ordering of the ...
55 views

1 vote
48 views

50 views

### Relation between a fixed point and being a well-order

I've been trying to prove the following, but with no particular success: Given a linear order $\leq$ on $A$, define $\pi:2^A\to 2^A$ by $X\mapsto\{y\in A: (\forall x < y)(x\in X) \}$. Let $A_0$ be ...
70 views

### Supremum in well orderings

Problem *x7.27 from Moschovakis' textbook is asking to define a definite operation $\sup \mathscr E$, such that for every family $\mathscr E$ of well-ordered sets, $\sup \mathscr E$ has the following ...
212 views

### Maximal element of infinite well ordered set

Can an infinite well ordered set $A \subseteq \mathbb{R}$ have a greatest element? I've read it a couple of times that the answer is yes, but I just couldn't agree with that. If set $A$ is well ...
In my notes the construction of the least ordinal follows from taking the supremum of the image of the set $$R=\{ A\in \mathcal{P}(\mathbb{N}\times\mathbb{N}) : A\text{ is a well-ordering of a subset ... 1 vote 1 answer 43 views ### Comparability theorem for well ordered sets using transfinite recursion Similar questions have already been asked here and here. But I am asking for verification of my proof. Halmos leaves out an "easy" transfinite induction argument, which I have struggled to ... 1 vote 1 answer 53 views ### Partial ordered can be extended to total orders I am studying Halmos' Naïve Set Theory, and in Section 17, Well Ordering, he mentions the following exercise: Prove that if R is a partial order in a set X, then there exists a total order S in ... 3 votes 1 answer 69 views ### A chain of well ordered sets defines a "unique" well order on the union of chains? Halmos in his Naïve Set Theory, states the following result in Section 17, Well Ordering: If a collection \mathscr C of well ordered sets is a chain with respect to continuation, and if U is the ... 5 votes 1 answer 159 views ### Union of powers of a well-ordered set is well-ordered. While studing a certain type of rings I was trying to solve the following exercise (which is what I need to prove that this ring is well defined): Let S be a well-ordered subset of an ordered group ... 0 votes 1 answer 39 views ### Why to include set inclusion in the definition of continuation of well ordered sets? In his Naïve Set Theory, Halmos defines, in Section 17, Well Ordering, the following: We shall say that a well ordered set A is a continuation of a well ordered set B, if, in the first place, B ... 1 vote 1 answer 119 views ### Zorn's lemma and maximal elements This continues a discussion begun at checking Zorn's lemma on an example where an example was offered to help understand maximal elements and Zorn's lemma. That example used the set {1,...,100} ... 6 votes 2 answers 228 views ### Strange property of Well-Ordering on an Uncountable Set Background I was messing around with well-orderings of uncountable sets, and proved the following theorem. It seems a little strange to me, and I am curious if anyone here has encountered something ... 1 vote 2 answers 49 views ### 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,... 7 votes 1 answer 132 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 ... 0 votes 0 answers 48 views ### Please Check My Proof of Well-Ordering Principle using Induction Well-Ordering Principle: \exists m \in A[\forall n \in A (m \in n \vee m = n) ] for all A \subseteq w where w is the set of natural numbers and A \neq \phi. Base Case: A_{1} = \{ e_{1} \} is ... 5 votes 1 answer 125 views ### Proof of Comparability Theorem for well-ordered sets using transfinite recursion I am reading through Halmos's "Naive Set Theory". In Section 18, Halmos uses transfite recursion to prove the Comparability Thereom for well-ordered sets: The assertion is that if \langle ... 0 votes 0 answers 61 views ### Prove n^2 < 2^{n+1} by well-ordering Please help me prove n^2 ≤ 2^{n+1} by the well-ordering (not induction) n\in\Bbb N \setminus \{{0\}} I've assumed that n=1 is not counter example so m > 1 and now try with n = m-1.$$(m-1)...
I'm having trouble with this problem: $P(k)$ is a propositional function in the natural numbers $\mathbb{N}$. $P(1)$ is true. Further, \$\forall j,k \in\mathbb{Z}(\, [P(j) \land P(k)] \implies P(j+...