All Questions
Tagged with ordinals model-theory
36
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
156
views
Is this a valid basis for a transfinite number system?
I've been curious about transfinite number systems including infinite ordinals, hyperreals, and surreal numbers. The hyperreals in particular seem particularly appealing for introducing a hierarchy of ...
1
vote
2
answers
62
views
Show that $\{\alpha<\omega_1 : L_\alpha \prec L_{\omega_1}\}$ is closed unbounded in $\omega_1$.
I was doing this exercise and there is a hint to consider the Skolem functions for $L_{\omega_1}$. However, I did not find any general definition of what a Skolem function may be in Kunen (1980), and ...
- 191
3
votes
1
answer
130
views
Morley Rank and Cardinality paradox?
If the Morley rank $RM(\phi)$ of a definable set $X$=$\phi(\mathcal{M})$ is $\alpha$, based on the inductive definition of RM (I am using David Marker's book Model Theory: An Introduction, definition ...
- 61
3
votes
1
answer
87
views
Definition of the ordinal rank of sentences of the natural number structure.
Let our structure be $(\mathbb{N};+,\cdot,0,1)$. I want to define the ordinal rank of a sentence regarding this structure. Informally, the ordinal rank is the number of steps you need to either verify ...
- 21.5k
0
votes
1
answer
47
views
equivalency of distinct ordinals
I've come across this statement:
it is readily checked that distinct ordinals $\alpha\neq\beta$ are not $L_{∞,ω}$-equivalent
Why ?
Does it even hold that as I'd guess they are not $L_{\kappa,\omega}$...
- 3,978
0
votes
1
answer
71
views
Proof in a model via supposition externally made about model
I'm working with only models satisfying a particular schema, using a result that from this schema we can deduce a theorem, and attempting to show this theorem holds in all such models. Can I do this ...
- 1,311
4
votes
1
answer
155
views
Minimal models in strong set theories
Given some theory $T$, let $M(T)$ denote the height of the minimal model of $T$, i.e. $\min\{\eta: L_\eta \vDash T\}$. Obviously there are some famous examples, e.g. $M(\textsf{KP}) = \omega$, and $M(\...
- 279
0
votes
1
answer
26
views
Can we conclude $rng(f) \subseteq \beta$ for some $\beta \in B \cap \omega_1$?
Consider the language $\mathcal{L}=\{ \in\}$. Let $\mathcal{A}$ be the $\mathcal{L}$-structure $(V_\theta, \in)$ for some $\theta> \omega$. Let $\mathcal{B}$ be an elementary substructure of $\...
- 1,847
0
votes
1
answer
109
views
Is this set of ordinals an ordinal?
Consider the language $\mathcal{L}=\{ \in\}$. Let $\mathcal{A}$ be an $\mathcal{L}$-structure whose domain is some "sufficiently large" Von-Neumann universe and which interprets $\in$ in the ...
- 1,847
13
votes
1
answer
501
views
Which ordinals can be "mistaken for" $\aleph_1$?
I've just finished working my way through Weaver's proof of the consistency of the negation of the Continuum Hypothesis in his book Forcing for Mathematicians. One of the key points in this proof is ...
1
vote
1
answer
214
views
Ordinals in $L$, the constructible universe
I am trying to understand the constructible universe $L$. Based on the way it is constructed, it is clear that every ordinal is included in $L$, i.e., $\alpha \subset L$ for any ordinal $\alpha$. ...
user856387
2
votes
1
answer
122
views
Does $\mathbb{N}\cup\{\omega\}$ satisfy true arithmetic?
Does $\mathbb{N}\cup\{\omega\}$ satisfy true arithmetic?(whereas $\omega$ is greater than any natural number) Even if not the standard model of natural number?
I believe it does, but I am not quite ...
user846115
1
vote
1
answer
119
views
Why does the back and forth method fail to prove that, for each cardinality, any dense linear order without endpoints is unique up to isomorphism?
First of all I must say I'm not not very knowledgeable about set theory beyond the very basics, so please bear with me if I've made some obvious mistakes in my reasoninig.
I've looked at and ...
- 13
4
votes
0
answers
156
views
Troubles with Kanamori's section on $0^{\sharp}$
I am having trouble understanding the following lemma in Kanamori's book. Some notations: $\mathcal{M}(T,\alpha)$ denotes the (unique up to isomorphism) model of $T$ generated by a set of ...
- 480