# 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).

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### 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, ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 $(S;\leq)$ ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ... • 2,085 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 ... • 2,085 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 ... • 2,085 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 ... • 1,462 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 ... • 2,085 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} ... • 33 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,... • 133 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 ... • 15 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 ... • 562 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+...