New answers tagged model-theory
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Type realized by $n$-tuple (Chang & Keisler)
This is essentially because formulas can be negated. You can take that as a hint and try to write the proof yourself, or continue reading.
Let $\gamma(x_1, \ldots, x_n)$ be any formula that is not in $...
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Countable elementary substructure of $H_\kappa$
But the witness $A$ in $X$ must be $M$, so $M\in X$, and $X\models\varphi(M)$.
No, being that $H_{\omega_1}$ is a definable element of $H_\kappa,$ $H_{\omega_1}$ is an element of $X,$ and serves as ...
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Is reflexivity modally definable?
Reflexivity is modally definable (as expressed by its characteristic formula called axiom T: $\Box\phi\rightarrow\phi$); irreflexivity is not.
Recall that $M_{1} =\langle W_{1}, R_{1}, V_{1}\rangle$ ...
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Why is this proof of the Tarski-Vaught criterion incorrect?
No issues with the proof. The problem is you are proving the wrong thing: the statement you have written is not the Tarski-Vaught criterion.
The Tarski-Vaught criterion is the more useful statement ...
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Determining whether a certain class of algebras is a quasi-variety
I'll answer the simplified version of your question I proposed in the comments:
Is there a variety of algebras $V$ whose signature contains unary functions $f$ and $g$ and a constant $*$ such that (a)...
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Analog of Birkhoff's HSP theorem regarding ultraproducts and elementary sublattices
Note that the notion of "axiomatic class" here is different than the one in Birkhoff's HSP theorem. The HSP theorem is about classes of algebras axiomatized by equations (i.e., sentences of ...
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Analog of Birkhoff's HSP theorem regarding ultraproducts and elementary sublattices
Keisler's survey on ultraproducts is a good source, incidentally.
This is Theorem 2.13 (and the subsequent comment) of Frayne/Morel/Scott, Reduced direct products. I found out about this result from ...
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Why does any theory expanding $T^*$ have the witness property?
The solution suggested in spaceisdarkgreen's answer works, but there is a solution which, in my opinion, is much simpler and more elegant.
You can do exactly the same steps as you describe in your ...
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Why does any theory expanding $T^*$ have the witness property?
As mentioned by Chris Eagle in the comments, this doesn't work: for instance, if you start the construction on the empty theory, you just get the empty theory back, and while this does technically ...
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Non-standard model of arithmetic
The official name for this is the reduct. You take the structure $\mathfrak A'$ and make a structure $\mathfrak A$ in signature $\mathcal L_{NT}$ by using the same underlying set as $\mathfrak A'$ and ...
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Can we always prove whether a statement is undecidable by adding consistency strength?
Your question is a bit unclear to me since you are not putting any restrictions on what you mean by "consistency strength". For example let's assume you are asking the following:
Question: ...
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A countable inductive first-order theory which has non existentially-closed (e.c) model has a non e.c. model of any infinite size
I'd like to answer my question. The answer is an adaption of the suggestion made by David Gao is the comment above, and the details can be found fully in Hodges' book in the section on relativisation. ...
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5
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Definability of the class of Hausdorff topological spaces using the closure operator in a first-order way.
You want a formulation in terms of the partial order of all subsets together with the closure operator.
We get the elements of the space as the atoms of the partial order. Containment of elements ...
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what's means V is model of zfc?
There are a couple of subtleties here. (For clarity I'm going to distinguish the actual elementhood relation $\in$ from a canonical symbol representing it $\varepsilon$)
First of all, and I think this ...
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3
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Closure operators and uniqueness of the size of basis
This isn't true in general.
I'll give a simple version of a counterexample due to Vlastimil Dlab. A more complex version of this construction, which produces a closure operator with bases of sizes ...
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Model unspecific Witness of an Existence Proposition
Let's say we want to reason logically about fraggles. You and I may have different philosophical convictions about the nature and reality of fraggles, but we have a shared interest in proving things ...
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Model unspecific Witness of an Existence Proposition
The way it shakes out in a formalist perspective is that any time we have proven a statement of the form $\exists !x\psi(x)$, that entitles us to definitionally extend our background theory with a new ...
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Definable types
If this is true, then (the $\varphi$-part of) $p$ is realised.
Indeed, if $p\vdash \forall y(\varphi(x,y)\leftrightarrow \psi(y))$, then since $p$ is a type, $\mathcal U\models \exists x\forall y(\...
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How to reduce to reduce to the case where the class $C$ is a set in this model theory exercise
Use the cumulative hierarchy! There must be some ordinal $\alpha$ such that for every $\mathcal{M}\in C$, there is an $\mathcal{M}'\in C\cap V_\alpha$ with $\mathcal{M}\equiv\mathcal{M}'$; this is ...
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Can you give me an example of an implicit use of Godel's Completeness Theorem, say for example in group theory?
It's worth noting that we rarely care about the existence of a first-order proof from a given set of axioms. A first-order proof is a very messy object, and if we're not already interested in them for ...
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Definable closure has the exchange property in an o-minimal structure
If $b\in{\rm dcl}(c)$ then there is a formula $\psi(x,y)\in L$ such that $\psi(c,b)\wedge\exists^{=1} y\ \psi(c,y)$. Then the required function $f(x)=y$ is defined by (after the correction of Alex) $\...
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Can you give me an example of an implicit use of Godel's Completeness Theorem, say for example in group theory?
A simple example might be something like the following:
Proposition: An element $g \in G$ of a group and any of its conjugates $hgh^{-1} \in G$ have the same order.
This is a collection of first-...
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Need help with a question about cardinality and perhaps model theory
Here's an affirmative (but nonconstructive) answer. It is quite simple and straightforward, although it may look complicated and obscure after I've transcribed it (I hope correctly but who knows) into ...
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Can you define a topology so that all the parametrically definable subsets are open?
Not an answer but too long for a comment:
As Alex Kruckman observed, once every parameterically-definable set is open we have the discrete topology. That said, there are a couple possible ways around ...
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Is there a theorem that says we don't limit our notion of conservative extension by only focussing on set models?
Short version
No, there isn't! "$T$ is model-theoretically-conservative over $S$" is a $\Pi_2$ property, and in general (and in this specific case) analysis of such properties and their ...
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An Example of a Proof-theoretic Conservative Extension that's not a Model-theoretic One
There are many examples! A convenient tool is that if $T\subseteq S$ are consistent theories and $T$ is complete, then $S$ is automatically a proof-theoretic extension of $T$. This lets us generate ...
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Understanding the witness property in the Henkin construction
You're correct that the two versions are equivalent when $T$ is complete (maximal), and your proposed argument (alternating between completing the theory and adding witnesses) works just fine.
In fact,...
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If the Collatz conjecture is undecidable, then it is true
If a theorem T is undecidable in some model M, then you can create a model M' by adding the axiom "T is true" and the model M'' by adding the axiom "T is false", without creating a ...
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If the Collatz conjecture is undecidable, then it is true
Here's an alternative approach. Rephrase the Collatz conjecture: we'll conjecture that the graph of the Collatz function must be semiconnected.
Consider some standard folklore of non-standard models ...
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If the Collatz conjecture is undecidable, then it is true
The problem, as far as I can tell, is that interpreting the Collatz conjecture in a nonstandard model $M$ involves iterating the Collatz map a nonstandard number of times. That is, if the Collatz ...
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Large invariant types realized in $\cal U$.
This happens if and only if it happens for a definable reason.
Precisely: Let $\mathcal{U}\models T$ be a monster model, $\varphi(x;y)$ a formula, and $a\in \mathcal{U}^y$. Then $p(x) = \{\varphi(x;f(...
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A first order theory of $\mathbb{R}$
The first-order theory of $\mathbb{R}$ is well-understood. It is one of the oldest and most important examples in the field of model theory.
An ordered field $R$ is real closed if every positive ...
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