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It's a very nice question! The answer is negative. For a counterexample, consider: Let $T$ be the theory of a partial order $\leq$, that is, a reflexive, transitive, antisymmetric relation. Let $H$ b …

It is a remarkable fact, due to Hajnal and Kertész and explained very well in this MathOverflow answer by user Ashutosh, that the axiom of choice is equivalent to the assertion that every nonempty set …

There is no such model. Suppose that $M$ is a countable transitive model like that. Since $M$ is countable, it has only countably many subsets of $\omega$. Let $f:\omega\to\omega$ be a bijective funct …

I heard from a reputable mathematician that ZFC proves that there is a minimal countably saturated real-closed field. I have several questions about this. Is there a soft model-theoretic construction …

One can easily make a model of ZF-Reg with numerous Quine atoms. Simply begin with a model of ZFCU, with numerous urelements, and then turn the urlements into Quine atoms, which are singleton sets $a= …

Here is a positive way to interpret your question, and I think this is probably what you had intended to ask about. Theorem. There is no computable model $M$ on domain $\mathbb{N}$ in a language $L$ i …

The minimal model, when it exists, also known as the Shepherdson-Cohen model, is the smallest transitive model of ZFC. This model will have the form $L_\alpha$ for some countable ordinal $\alpha$, and …

I understand your theory to be the set of all sentences in the first-order language of set theory that are true in every model of $\newcommand{\ZFC}{\text{ZFC}}\ZFC_2$. This is a perfectly sensible no …

Let me observe that every total function $f$ that is representable in PA has your property with respect to every formula $\alpha$ expressible in the language of arithmetic. This includes every primiti …

The answer is no, because such ultrapowers are always $\aleph_1$-saturated, but $\mathbb{R}$ is not. More concretely, the ultraproduct will be an ordered field with uncountable cofinality — every cou …

Yes, for the reasons you mention, it is important to define your Gödel coding in such a way that the syntactic operations you want to undertake with assertions in the language are indeed expressible i …

Yes, there are a variety of mathematical systems that are able to serve as a foundation of mathematics, whether one uses ZFC set theory, ZFC plus large cardinals, or ZF, or PA, or category theory, typ …

The answer is yes. Let $T$ be the theory of the model $\langle\mathbb{Q},<,c_i\rangle$, where we choose constants $c_i$ as follows: every natural number $n$ is the value of a constant, and for each $n …

The confusion seems to arise from your statement "the mantle of $V$ which is the smallest ground for $V$." But this not quite right. In a bottomless model of ZFC, the mantle is not a ground. It is the …

No, because the latter theories has parametrically definable automorphisms that swap two equivalent classes, but the former theory is definably rigid (no need for class choice). If the theories were b …