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).
551
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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
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Why do we need "canonical" well-orders?
(I asked this question on MO, https://mathoverflow.net/questions/443117/why-do-we-need-canonical-well-orders)
Von-Neumann ordinals can be thought of "canonical" well-orders, Indeed every ...
- 5,704
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0
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Every Non empty subset has a least element implies linear order
Suppose $(A,R)$ be structure where R is a binary relation on $A$.
Suppose $A$ has the property that every Non empty subset of $A$ has a least element w.r.t. the relation $R$. Then $R$ is a linear ...
user1072974
2
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2
answers
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What is the meaning of "induction up to a given ordinal"?
Given an ordinal $\alpha$, what does it mean: "induction up to $\alpha$"? When $\alpha=\omega$, is this is ordinary mathematical induction? Also, Goodstein's Theorem is equivalent to "...
- 4,442
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1
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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
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1
answer
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non-wellordered linear order that doesn't contain a copy of $\omega^*$ in ZF?
Obviously a wellorder cannot contain a copy of $\omega^*$ (the dual order of $\omega$), and this can be proven in ZF. In ZFC, it is easy to prove any linear order which is not a wellorder does contain ...
- 5,671
2
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1
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Intersection of a chain of closed sets
I was recently attempting proving a conjecture of mine about the existence of certain minimal nonempty closed sets in a topological space. I opted to use Zorn's Lemma, and the proof would go through ...
- 1,249
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2
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Product of Countable Well-Ordered Set with $[0,1)$ is Homeomorphic to $[0,1)$
As part of a proof that the long line is locally Euclidean, I'd like to prove the following:
Proposition. If $A$ is a countable well-ordered set, then $A \times [0,1)$ with the dictionary order is ...
- 2,221
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0
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Order type of this order over naturals
For no specific reason other than curiosity, I find myself wondering of a way to formalize a specific order of the natural numbers based on a sort of "recursive prime number factorization", ...
- 82
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1
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Initial segment of well ordered set
An initial segment (B) of an well-ordered set $(A,<)$ is a subset $X⊊A$ such that, for all $x \in X$ and for all $y∈A$ such that $y<x, y∈X$.
I want to prove that this set is equal to $A_z =$ { $...
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How do you prove that there exists a highest element of any finite, nonempty subset of Natural Numbers? Is the following algorithmic proof valid?
Since the given set, $C \subset \mathbb{N}$ is non empty, hence by well ordering principle there exists $\alpha \in C$ which is the lowest element in C. Also, since the set $C$ is finite, $\quad \...
- 308
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Partial order, well order and initial segment
I saw the following definition for initial segment in https://digitalcommons.kennesaw.edu/cgi/viewcontent.cgi?article=2161&context=facpubs
If $\leq$ is a partial order in a set $X$, then a chain ...
- 522
6
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1
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Riddle: finite set that contains one of the three numbers
A colleague told me this two-part riddle:
PART $\mathbb{N}$:
Three players sit in a circle, each having a positive integer number
$x_i$ written on their hat. Every player can see only the numbers of
...
- 3,439
1
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1
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"Name" for this order relation of binary sequence
I am trying to define a Matrix $M_n$ of dimension $2^n \times n$ where each row corresponds to a element of the set $\{0,1\}^n$. To avoid ambiguity, I using an order relation over the set $\{0,1\}^n$ ...
- 537
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1
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How many automorphisms are there for $ \langle \omega , < \rangle $?
How many automorphisms are there for $ \langle \omega , < \rangle $?
I'm not sure how to start this, although I expect there to be an upper bound of $2^{\aleph_0}$. ($\aleph_0^{\aleph_0} = 2^{\...
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