# Tag Info

### 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 ...

### 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 ...

### 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 ...

### 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 [...

### 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:...
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 ...