All Questions
Tagged with model-theory order-theory
73
questions
1
vote
1
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54
views
In the transitive closure of $\in$, is the empty set the least element?
Consider a model $(M,\in)$ of ZFC set theory. Consider the transitive closure of $\in$, which I will denote by $R$. Now, it can be proven that $R$ is a strict partial order. In that strict partial ...
- 21.4k
4
votes
2
answers
214
views
Regarding the "smallest" nontrivial, dense order type.
Is $\eta$, the order type of the rationals, necessarily the smallest nontrivial, dense order type?
By "smallest" I mean that there is no dense order which embeds into $\eta$, into which $\...
- 5,040
-3
votes
1
answer
159
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order property vs. antisymmetric property
The definition of order property is well known:for a first-order theory $T$ the order property means that for some first-order formula $\phi(\bar{x},\bar{y})$ linearly orders in $M$ some infinite $\...
- 3,978
0
votes
1
answer
51
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Can strict partial orders be axiomatized using just one elementary sentence?
This is a follow-up to my previous question, here: Can equivalence relations be axiomatized using just one elementary sentence?. Referring back to that question, I define an elementary sentence to be ...
- 21.4k
7
votes
1
answer
405
views
Continuous chain of $\kappa^+$ isomorphic linear orders on $\kappa\ge\aleph_1$
For $\kappa\ge\aleph_0$ an infinite cardinal, and for $\kappa<\alpha\le\kappa^+$ an ordinal, we ask the existence of a chain $\left\langle(I_i,{<}):i<\alpha\right\rangle$ of linear orders ...
- 142
5
votes
1
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112
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Can "degrees of preservability" over a theory have greatest lower bounds (nontrivially)?
For a (first-order) theory $T$, let the preservability degrees over $T$ be the following preorder $\mathsf{Pr}(T)$:
Elements of $\mathsf{Pr}(T)$ are finite sets of formulas in the language of $T$.
...
- 249k
7
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1
answer
145
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Cardinality of Dedekind Completion of Hyperreals
Let $^*\mathbb{R}$ denote the hyperreal field constructed as an ultra power of $\mathbb{R}$. At the expense of losing the field properties, we may take the Dedekind completion of $^*\mathbb{R}$ to get ...
- 2,980
0
votes
1
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77
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The $\forall$-theory of $(\mathbb{R}; \leq)$
Consider the structure $(\mathbb{R};\leq)$. What is an axiomatization of its $\forall$-theory? I conjecture that the axioms for linear orders suffice. That is, all you need are reflexivity, anti-...
- 21.4k
0
votes
1
answer
96
views
Completeness of a theory that is not $\omega$-categorical
Let $T$ be the theory of linear orders with no endpoints and let $\mathcal{L}=\{<,c_0,c_1,\dots\}$ be the language that consists of a binary relation symbol and a countable amount of constant ...
- 519
2
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0
answers
51
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Question about $\omega$-categoricity
Let $T$ be the theory of linear dense orders with no endpoints and let $\mathcal{L}=\{<,c_0,c_1,\dots\}$ be the language that consists of a binary relation symbol and a countable amount of constant ...
- 519
4
votes
1
answer
405
views
Is the theory of linear dense orders with distinct endpoints complete?
I'm trying to solve some problems about elementarily equivalent structures. For example, I know that the structures $\langle\mathbb{Q},<\rangle$ and $\langle\mathbb{R},<\rangle$ are elementarily ...
- 519
0
votes
1
answer
43
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Is the equality-free theory of linear orders the same as the equality free-theory of linear preorders?
This is a natural follow-up to my question, here:In first-order logic without equality, is the theory of partial orders the same as preorders?. My current question is, consider first-order logic ...
- 21.4k
0
votes
0
answers
49
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Finding restriction of the ultraproduct that behaves like $\mathbb{Z}$
Let $\mathcal{A}= \prod_{n \in \mathbb{N}} \mathcal{A}_n /\mathcal{U}$, where $\mathcal{A}_n=(\{0, 1, \dots, n\},<)$ and $\mathcal{U}$ is a non-principal ultrafilter of $\mathbb{N}$.
Can we find ...
- 1,847
3
votes
1
answer
113
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ultrafilters as linear orders
In Henson's Model Theory lecture notes I found an exercise quite early on (1.30, p. 12) that prove too difficult for me. It goes like this:
Let $L$ be the first order language whose only nonlogical ...
- 109
2
votes
1
answer
212
views
For every poset $(X,\le)$ there is a linear order $\preceq$ on $X$ which extends $\le$
Use the Compactness Theorem in order to show that for every partially ordered set $(X,\le)$ there is a linear order $\preceq$ on $X$ which extends $\le$, that is: for all $x,y\in X$ we have $x\le y \...
- 2,438