All Questions
Tagged with cardinals model-theory
51
questions
1
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2
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62
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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
-1
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2
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107
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When we say "cardinality of first order language L" and "cardinality of a structure or model" what we are meaning? [closed]
I ask about for what set we are referring for these cardinalities?
- 11
0
votes
1
answer
57
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Bound for the cardinality of the model of a set of formulas
I don't know if the proof of the theorem below is correct. The goal is to do the proof from the beginning, without using compactness or Löwenheim-Skolem. Only the Tarski-Vaught test (maybe).
We are ...
- 414
0
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0
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41
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Are cardinals always partial-well-ordered without AC? [duplicate]
With AC, we have that cardinals are well-ordered. Without AC, we can have two sets which are incomparable, meaning that, at best, cardinals are only partially ordered.
But do we still have, without AC,...
- 8,000
0
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0
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73
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Help with an exercise on the cardinality of the set of formulae of a First-Order Language.
Let $\mathcal{L}$ be a first order language consisting of:
$P_i$ predicate symbols, $f_j$ function symbols & $c_k$ constant symbols indexed respectively by th sets $I,J$ & $K$. Furthermore ...
- 623
0
votes
1
answer
95
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Are there any proposed operations to actually construct an inaccessible set?
When we postulate the smallest infinite set, we define it using an interative process involving iterations on the empty set.
When we postulate the existence of a set of continuum cardinality, we again ...
- 1,437
1
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0
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68
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Let $T=\{c_m\neq c_n: m,n\in \mathbb{N}, m\neq n \}$ in the language $L=\{c_n : n\in \mathbb{N}\}$. Prove that T is not $\aleph_0$-categorical.
as the title says I am asked to prove the following:
Let $T=\{c_m\neq c_n: m,n\in \mathbb{N}, m\neq n \}$ in the language $L=\{c_n : n\in \mathbb{N}\}$, where each $c_n$ is a constant symbol. Prove ...
- 391
1
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1
answer
60
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elementary embeddings $j$ in set theory with $V$ and $M$
I'm confused by a variety of elementary non-trivial elementary embedings $j$ we might have.
There are 9 "syntactical" possiblities;Here $M$ is a transitive model. I'll name them with a wish ...
- 3,978
0
votes
1
answer
286
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What does it mean "cardinality of a model" and "cardinality of a language" ? Are they the same thing? (Model Theory)
I'm studying the lowenheim- skolem theorem but i am a bit confused this when it comes to cardinality, in some definitions they use the cardanality of the language, in others they use the cardinality ...
- 1
1
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1
answer
95
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Reducing to regular cardinals in c.c.c. implies same cardinals and cofinalities
In Chapter 14 of Jech's Set Theory, he asserts the following:
Definition 14.33. A forcing notion $P$ satisfies the countable chain condition (c.c.c.) if every antichain in $P$ is at most countable.
...
- 1,453
3
votes
1
answer
92
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$\operatorname{cf}(\sup(B \cap \omega_m))$ is the natural numbers
Let $\mathcal{L}=\{\in, \preceq\}$. Let $\mathcal{A}=(V_\theta, \in)$ (where $\theta > \omega_\omega$) be an $\mathcal{L}$-structure which interprets $\preceq^\mathcal{A}$ by some fixed well-...
- 1,847
13
votes
1
answer
501
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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 ...
0
votes
1
answer
115
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Cardinality of a language $L_\Sigma$ over a decidable signature $\Sigma$
In the middle of a proof of a theorem I was studying, in order to prove a cardinality argument, there was the following statement:
Note that $|L_\Sigma|=|\Sigma|+ \aleph_0$
Where $L_\Sigma$ is a ...
- 373
5
votes
1
answer
131
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Uncountable Boolean algebra in ZFC with a countable ultrahomogeneity property
In ZFC, is there an uncountable atomless Boolean algebra $B$ such that for all countable subalgebras $A_1,A_2\subset B,$ every isomorphism $f:A_1\to A_2$ extends to an automorphism of $B$? $A_1$ and $...
- 1,500
5
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0
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87
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Is there a structure on $\mathbb{R}$ with a given cardinality of isomorphisms
We have, for example
$|\mathrm{Aut} (\mathbb{R})|= |\mathbb{R}^\mathbb{R}|$
$|\mathrm{Aut} (\mathbb{R,+,0})|= |\mathbb{R}^\mathbb{R}|$
$|\mathrm{Aut} (\mathbb{R,<})|= |\mathbb{R}|$
$|\mathrm{Aut} (\...
- 682