All Questions
Tagged with order-theory real-numbers
63
questions
2
votes
1
answer
89
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Abelian subgroups of order automorphism group $({\rm Aut}(\mathbb R,\le), \circ )$
I am searching for any results regarding Abelian subgroups of $({\rm Aut}(\mathbb R,\le), \circ )$, the order automorphism group of $\mathbb R$ (order automorphisms of $\mathbb R$ with the composition ...
- 169
0
votes
1
answer
68
views
How to Define Order Relations in Irrational Numbers Compared to $\Bbb N,~ \Bbb Z$, and $\Bbb Q$
I'm curious about how order relations are defined among irrational numbers. While the ordering of integers and rational numbers is straightforward, in the sense:
After constructing natural numbers $\{...
- 775
0
votes
1
answer
53
views
Characterizing the subsets of $\mathbb{R}$ with LUB property
Say a subset $S \subset \mathbb{R}$ has $\textbf{least-upper-bound property (LUBP)}$ if every nonempty bounded above subset $T \subset S$ has supremum that exists in $S$.
I want to characterize the ...
- 547
-1
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2
answers
128
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The rationals and their initial segments, edited version [closed]
This post was inspired by an Alon Amit post on Quora. The Quora
problem posed to AA was something like, only slightly more confused
than, this: How can the set of initial segments of the rational ...
- 23
3
votes
1
answer
112
views
Is every dense complete endless linearly ordered subset of real order-isomorphic to real?
Let $(M,\leq)$ be a non-empty
dense ($\forall a<b\in M,\exists c\in M,a<c<b$),
complete (every non-empty subset that is bounded above has a supreme)
endless (there is no minimal or maximal ...
- 1,785
1
vote
0
answers
78
views
If $X$ is totally ordered, has the least upper bound property, and has a countable dense subset, then is $X$ isomorphic to an interval of $\mathbb R$?
Let $X$ be a totally ordered set which has the least upper bound property and has an at most countable dense subset. Assume that $X$ has more than one element. Is $X$ isomorphic to some interval of $\...
- 1,499
1
vote
2
answers
138
views
Example of a complete unbounded dense linearly ordered set that isn't isomorphic to $\mathbb{R}$
I know as a fact that $\mathbb{R}$ is the unique (upto isomorphism) complete linearly ordered field. But if we remove the "field" condition and replace it with "dense unbounded set"...
- 782
0
votes
1
answer
81
views
Characterizations of the reals
I know that one characterization of the reals is that it is the only Dedekind-complete ordered field. Are there any other characterizations of the reals as a field?
- 4,057
1
vote
0
answers
67
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What's the proof that the only Dedekind-complete field is the reals? [duplicate]
I know that the field of the rational numbers is ordered but not Dedekind-complete. What's the proof that the only Dedekind-complete field is the reals?
- 4,057
1
vote
1
answer
339
views
What is an example of a nonempty subset of $\mathbb{R}$ that is bounded above that does not contain its least upper bound?
What is an example of a nonempty subset of $\mathbb{R}$ that is bounded
above that does not contain its least upper bound? This is an on-a-review sheet for my final. I thought the completeness axiom ...
- 11
1
vote
0
answers
56
views
How to linearly order the set of all subsets of real numbers?
I wondered if there are linearly ordered sets of any cardinality. As I understand it, there are. But I want to see at least one concrete example of a linearly ordered set which cardinality is greater ...
- 21
1
vote
1
answer
70
views
Can we uniquely define for arbitrary, real-valued, finite sequence $X$, infinitely many pairs (real-valued $f(X)$, rank order of elements of $f(X)$)?
For an arbitrary sequence $X$ of $n$ distinct real numbers, can we uniquely and exhaustively define a set of infinitely many pairs of the form: $[f_{j},$ order$(f_{j}(x))]$, where $f_{j}$ is a real-...
- 165
0
votes
0
answers
24
views
How to construct an increasing $\aleph_1$ sequence of real numbers. [duplicate]
We have $\aleph_1\leq |\mathbb{R}|$. Do we know if there exists an increasing $\aleph_1$ sequence of real numbers? (That is, a set $\{a_\theta\in\mathbb{R}:\theta<\omega_1\}$ such that $a_{\theta_1}...
- 782
3
votes
1
answer
132
views
A property of partitions of the real numbers
Let a strict linear order $C = (V, <)$, be an irreflexive and transitive relation < defined on $V$, and call a section of $C$ a partition of $V$ into two sets $A, B$, such that $x < y$, ...
- 1,413
1
vote
0
answers
53
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Order relation on the geometric line as defined in Kock's synthetic differential geometry
I'm trying to figure out to what an order relation $<$ would look like on the geometric line $R$ as defined in Kock's synthetic differential geometry.
If I understand correctly (in constructive ...
- 306