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Results tagged with lo.logic
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user 83901
first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
2
votes
Accepted
Can we have a theory $T$ that is complete for simple sentences in the language of $T$ that a...
Unless I'm missing something, every sentence is equivalent to a simple sentence. To see this, take a sentence $\varphi$ and produce an equivalent sentence $\psi$ by the following procedure: Let $x$ be …
- 13.2k
3
votes
Is there a notion of Skolemization for continuous logic?
Many of the things you might want to do with Skolemization (such as producing Ehrenfeucht-Mostowski models) can be done by passing to a discretization of your theory and then Skolemizing there. Withou …
- 13.2k
7
votes
Accepted
Can we re-write every effective first order theory using finitely many primitives?
The theory of the (full) random hypergraph is a counterexample. (Full meaning we are allowing any arity.)
The language consists of a relation symbol $E_n$ for each $n \geq 1$ (sometimes people start …
- 13.2k
1
vote
Does every uncountably categorical theory have a $\varnothing$-definable strongly minimal im...
I honestly thought about a lot of different possible counterexamples before realizing that a small modification of the structure I already mentioned gives a counterexample. Specifically if we consider …
- 13.2k
3
votes
A generalization of Vaught's two cardinal theorem
In case anyone else is wondering about this very specific question, I found a relevant paper but also a counterexample (which I believe actually invalidates one of the results in that paper).
The cou …
- 13.2k
5
votes
ω-categorical, ω-stable structure with trivial geometry not definable in the pure set
This is a follow-up to Szymon Toruńczyk's answer in which I will prove the claims in it.
Proposition.
$\mathrm{Th}(M)$ is totally categorical with trivial geometry.
Proof. First, note that $M = \mathr …
- 13.2k
8
votes
Accepted
Is the set of ordinals in Double Extension Set Theory really a set?
Perhaps I'm misunderstanding something, but it should be the case that $\beta \in_2 \mathbf{S}_1$ is equivalent to $S_1(\beta)$. Therefore the defining formula of $\mathbf{ORD}$ can be written as
$$\m …
- 13.2k
7
votes
Measure of the numbers with length of $n$ for a nonstandard number $n$
As requested this is a slightly more detailed answer to your first question. I really don't know how to approach your second question.
By a mild variation Vaught's two cardinal to get a model $N$ of …
- 13.2k
1
vote
Accepted
Complexity of deciding if an incomplete first-order theory has a stable completion
EDIT: Thanks to tomasz's comment I realized I was making this more complicated than it needed to be. Here is a simpler construction:
Let $\mathcal{L}=\{\leq_i\}_{i<\omega}$ be a countable sequence of …
- 13.2k
3
votes
Construction of a model of $ZFC+\neg Con(ZFC)$
As has been discussed in the comments, there is no inner model or forcing-like construction that will give you a model, since the collection of natural numbers needs to grow.
You'll have to tell me if …
- 13.2k
6
votes
Accepted
Sizes of "nearly amorphous" models
Amorphicity implies strong minimality and $\omega$-categoricity, which together imply $\kappa$-amorphicity for any $\kappa$.
Assume that $T$ is amorphic. To see that $T$ is $\omega$-categorical, we pr …
- 13.2k
5
votes
Hyperimaginaries and continuous logic
In general a $T^{eq}$ construction in continuous logic, as it's typically defined, can only possibly add hyperimaginaries corresponding to equivalence relations defined by countable partial types. If …
- 13.2k
14
votes
Can ultraproducts avoid all "factor structures"?
I realized there's an easier example that doesn't need a measurable cardinal.
Consider the language $\def\Lc{\mathcal{L}}\Lc$ which consists of unary predicates $U_n$ for each $n<\omega$. Consider the …
- 13.2k
6
votes
Accepted
Variable elimination for propositional formulas in Heyting algebras
The answer for $\bigwedge_t$ is no. Perhaps the idea here can be adapted to $\bigvee_t$.
Consider the propositional formula $t \vee (t \to x)$ in the complete Heyting algebra of open subsets of $\math …
- 13.2k
8
votes
Quantifier complexity of definition of compactness
Often the way you prove that something isn't formalizable in first-order logic is (ironically enough) with a compactness proof. This is how you show, for instance, that there isn't a first-order theor …
- 13.2k