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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Are all sets partially ordered?
If all sets can be well-ordered, does this also mean that all sets can be partially ordered?
Can someone give me an example of a set that is not partially ordered?
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How to define a well-order on $\mathbb R$?
I would like to define a well-order on $\mathbb R$. My first thought was, of course, to use $\leq$. Unfortunately, the result isn't well-founded, since $(-\infty,0)$ is an example of a subset that ...
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A way to well-order real line
How is well-ordering in real line possible?
I know that the axiom of choice provides possible well-ordering, but intuitively, this does not seem to make sense.
How can you compare 1.111111.... and 1....
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Isomorphism between well ordered sets
Let A be well ordered by an order relation G.
Let A be well ordered by an order relation G'.
G ≠ G'.
Then does there exist an isomorphism f between A ordered by G and A ordered by G'?
Plus, is there ...
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Existence of a particular well-ordering of [0,1]
How do you show, assuming the Axiom of Choice and the Continuum Hypothesis, that there exists a well-ordering on $[0,1]$ such that for all $x$, there are only countably many $y$ such that $y \leq x$?
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Totally ordering the power set of a well ordered set.
Let's say I take a set $S$, where $S$ can be well ordered. From what I understand, one can use that well ordering to totally order $\mathscr{P}(S)$.
How does a body actually use the well ordering of $...
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The well ordering principle
Here is the statement of The Well Ordering Principle: If $A$ is a nonempty set, then there exists a linear ordering of A such that the set is well ordered.
In the book, it says that the chief ...
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Explicit well-ordering of $\mathbb{N}^{\mathbb{N}}$
Is there an explicit well-ordering of $\mathbb{N}^{\mathbb{N}}:=\{g:\mathbb{N}\rightarrow \mathbb{N}\}$?
I've been thinking about that for awhile but nothing is coming to my mind. My best idea is ...
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questions about well-order
In Wikipedia, well-order is
defined as a strict total order on a set $S$
with the property that every
non-empty subset of $S$ has a least
element in this ordering.
But then later, well-order is
...
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Showing that well-ordered subsets of $P(\omega)$ are countable
I have the following problem:
Show that no uncountable subset of $P(\omega)$ is well-ordered by the inclusion relation.
I think they want me to do by embedding it in a separable complete dense ...
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Is there a known well ordering of the reals?
So, from what I understand, the axiom of choice is equivalent to the claim that every set can be well ordered. A set is well ordered by a relation, $R$ , if every subset has a least element. My ...
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