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Persistent finite axiomatizability, relational edition

Let me collect some partial results and some speculation that may be useful here. The set $T_{w/o=}$ is axiomatized by $A_I$ in some cases. For example, if $A$ is preserved under quotient by the ...
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6 votes

How hard must "no high-degree irreducibles" proofs be?

I don’t know how to make use of the full power of the intermediate value theorem, hence I will work instead with a weaker axiomatization: let $\def\rcf{\mathrm{RCF}}\rcf'_k$ denote the first two ...
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6 votes

Free algebras from model theory perspective

Here are some papers. (1) Baldwin, J. T.; Shelah, S. The structure of saturated free algebras. Algebra Universalis 17 (1983), no. 2, 191-199. From the Math Review (written by Steve Comer): The authors ...
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An exercise in fuzzy logics built from a t-norm

My teacher has provided a solution: Take a $[0, 1]_*$-interpretation with $I(\phi \rightarrow \phi * \phi ) = 1$, and say $a:=I(\phi)$. Define the function \begin{align*} h: [0, 1] & \rightarrow [...
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Is there a finitely axiomatizable class of structures whose equality-free theory is not finitely axiomatizable?

EDIT: As pointed out by Emil Jerabek in the comments, the argument for relational languages fails in the last step. However, the question is already answered by the example with function symbols. EDIT:...
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5 votes
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Are equinumerous size preserving models of a theory isomorphic?

There is no version of this question I can think of which has an affirmative answer. Let $\alpha,\beta$ be distinct countable ordinals such that $L_\alpha\equiv L_\beta\equiv L_{\omega_1^L}$ (which ...
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8 votes
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How often are forcing extensions of countable computably saturated models of $\mathsf{ZFC}$ computably saturated?

The answer to Question 1 is positive (thus the answer to Question 2 is also positive). More explicitly, the positive answer to Question 1 follows from the following well-known facts: Lemma 1. $(M,\...
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3 votes

Tarski's original proof of quantifier elimination in algebraically closed fields

Passmore's PhD thesis has an introductory section developing an elementary quantifier elimination algorithm for algebraically closed fields of characteristic zero "from scratch" with much ...
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5 votes
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Tarski's original proof of quantifier elimination in algebraically closed fields

I doubt you’ll find a shorter proof than Swan’s which is equally elementary. In particular: For algebraically closed fields, you can stop in the middle of page 10 of the document, which should make ...
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0 votes

Smallest ordinal modelling $\aleph_1$?

About well-definedness of $X_1$, "being a model of ZFC" is definable since ZFC is a recursive theory, so we could construct some $\Sigma_1^0$ predicate $\textrm{isZFCAxiom}(e)$ for $e$ a ...
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