All Questions
Tagged with well-orders model-theory
10
questions
2
votes
1
answer
127
views
Is the first-order theory of the class of well-ordered sets the same as the first-order theory of the class of ordinals?
Consider the class $K$ of well-ordered sets $(W,\leq)$. Although that class is not first-order axiomatizable, it has an associated first-order theory $Th(K)$. Now consider the class of ordinals $On$, ...
- 21.5k
2
votes
1
answer
64
views
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
1
vote
1
answer
49
views
Is the class of well-ordered sets a pseudo-elementary class? [closed]
I know that the class of well-ordered sets is not an elementary class. But, is it at least a pseudo-elementary class? If not, what is the proof?
- 21.5k
1
vote
1
answer
41
views
What is the class of models of the $\forall$-theory of the class of well-ordered sets?
I know that the class $W$ of well-ordered sets is not an axiomatizable class. However, it does have an associated first-order theory $Th(W)$. Now, consider the $\forall$-part of the theory $Th(W)$, ...
- 21.5k
1
vote
3
answers
93
views
Are suborders of pseudo-well-ordered sets themselves pseudo-well-ordered?
The class $W$ of well-ordered sets is not first-order axiomatizable. However, it does have an associated first-order theory $Th(W)$. I define a pseudo-well-ordered set to be an ordered set $(S;\leq)$ ...
- 21.5k
0
votes
1
answer
170
views
How does compactness show that every set can be linearly ordered without also proving choice (via the well-ordering theorem)?
This answer to this question contains a proof that every set can be linearly ordered, but doesn't prove choice along the way as this comment points out.
I'm trying to figure out why the axiom of ...
- 11.9k
1
vote
1
answer
100
views
Why does model theory treat well-ordered but uncountable languages?
In model theory, results such as the Compactness theorem and the Löwenheim–Skolem theorem hold true, not only in countable languages, but in well-ordered languages.
I'm not questioning this idea in ...
- 1,779
3
votes
3
answers
268
views
Two questions about the first order theory of well ordered sets.
Consider the class of well-orderings $W$. Although that class is not first-order axiomatizable, it has an associated first order theory $Th(W)$. Is it finitely axiomatizable? I conjecture that, in ...
- 21.5k
3
votes
1
answer
231
views
Isomorphic subsets of countable total orders
Suppose $(\Omega,\leq)$ is a totally ordered set, with $\Omega$ infinite and countable. If $S$ is an infinite subset of $\Omega$, then $(S,\leq)$ denotes the induced totally ordered set.
Are there ...
- 799
2
votes
1
answer
77
views
What is the difference between constructing and axiomatizing a well-ordering?
In my lecture on mathematical logic, we said that the constructible universe $(\mathbb{L}, \in)$ is a model for the Axiom of Choice because all well-orderings are constructible, that is, definable by ...
