-1
votes
0answers
37 views

Is an o-minimal structure equivalent to a totally ordered set?

Is the notion of o-minimality synonymous to a totally ordered set? Both notions seem to emanate from Tarski although he may not have discussed o-minimality explicitly...
3
votes
1answer
68 views

isomorphism between divisible, totally ordered, abelian groups

Let $G$, $H$ be divisible, abelian, linearly ordered groups, whose cardinalities are equal and satisfy $\mu := |G|=|H|>\aleph_{0}$. These are supposed to be (order!) isomorphic. And just about ...
4
votes
1answer
81 views

Infinite linear order with endpoints which is non-dense

In the process of answering questions about normal models, I had to prove the following: Any normal model of $\chi$ is a non-dense linear order with a least and greatest element. The next question ...
3
votes
1answer
76 views

Does model theory extend to partial functions?

I have been reading a bit about effect algebras and d-posets recently, sets $M$ on which you have a single partial binary operation (partial here meaning partially defined, i.e. the domain of this ...
8
votes
1answer
204 views

Is this a characterization of well-orders?

While grading some papers and thinking about a question related to well-orders (in particular, pointing a mistake in a solution), I came to think of a reasonable characterization for well-orders. I ...
4
votes
4answers
123 views

Show that Total Orders does not have the finite model property

I am not sure whether my answer to this problem is correct. I would be grateful if anyone could correct my mistakes or help me to find the correct solutions. The problem: Show that Total Orders ...
5
votes
1answer
73 views

$T\vDash\psi$ equivalences

$T\vDash\psi$ means $T$ satifies $\psi$ from Tarski's definition of truth, it simply means that the sentence $\psi$ is valid in $M$. I call a sentence $\psi $ universally if it is valid in every ...
5
votes
1answer
124 views

Semi-formal language - Universe has at least three elements

First of all I would like to construct a semi formal sentence, such that the universum has at least three elements. My attempt: $$\exists x\exists y\exists z (x\not=y\wedge y\not=z\wedge x\not=z)$$ ...
4
votes
1answer
215 views

Exclude operation symbols in signature

We probably know what a signature is, it contains a set $\sigma_{op}$ (the operation symbols), $\sigma_{rel}$ (the relation symbols) and a function $ar:\sigma_{op}\cup\sigma_{rel}\rightarrow\mathbb ...
4
votes
2answers
66 views

Semi-formal language

I want use semi-formal language to describe the following four points. (1) The group axioms with signature $\{*\}$ (2) The property "linear order" with signature $\{<\}$ The following properties ...
3
votes
1answer
141 views

Dense linear orderings isomorphism [duplicate]

First of all the definition: A dense linear order is a $\{<\}$-structure in which the following formulas are valid: Question: How can I prove that any two countable DLO's (dense linear order) ...
3
votes
2answers
77 views

Class of structures isomorphic to $(\mathbb{Z},<)$ in infinitary logic $L_{\omega_1 \omega}$

We want to define the class of all structures isomorphic to $(\mathbb{Z},<)$ in the infinitary logic $L_{\omega_1 \omega}$. Therefor we define strict order as usual: $\forall x,y,z ~~ (x<y ...
5
votes
2answers
248 views

Why is a commutative ring with an infinite number of idempotent elements unstable?

In a book of model theory I found the following statement: A commutative ring with an infinite number of idempotent elements unstable. I haven't manage to prove it yet. As stability in the model ...
4
votes
1answer
143 views

number of linear orders

It is well known that for every infinite cardinal $\kappa$ the number of non-isomorphic total orders of cardinality $\kappa$ is $2^\kappa$. Who first proved this, and in what context? Was it proved ...
9
votes
2answers
1k views

Showing any countable, dense, linear ordering is isomorphic to a subset of $\mathbb{Q}$

I'm trying to knock out a few of the later exercises from Enderton's Elements of Set Theory. This problem is #17, found on page 227. A partial ordering $R$ is said to be dense iff whenever $xRz$, ...