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Noah Schweber
Noah Schweber
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This was originally an MSE question, but I was told to ask it on MathOverflow. Does there exist a class $C$ of $L$-structures for a finite signature $L$, which is finitely axiomatizable in first-order logic with equality, but whose equality-free theory $Th(C)$ is not finitely axiomatizable?

Non-OP edit: To clarify, the question is whether the finite $\mathsf{FOL}$-axiomatizability of $Th_\mathsf{FOL}(C)$ necessarily implies the finite $\mathsf{FOL_{w/o=}}$-axiomatizability of $Th_{\mathsf{FOL_{w/o=}}}(C)$, regardless of whether the $\mathsf{FOL}$-deductive closure of $Th_{\mathsf{FOL_{w/o=}}}(C)$ coincides with $Th_\mathsf{FOL}(C)$ or not. (That is, we don't care whether $Mod(Th_{\mathsf{FOL_{w/o=}}}(C))\not=C.$)

This was originally an MSE question, but I was told to ask it on MathOverflow. Does there exist a class $C$ of $L$-structures for a finite signature $L$, which is finitely axiomatizable in first-order logic with equality, but whose equality-free theory $Th(C)$ is not finitely axiomatizable?

This was originally an MSE question, but I was told to ask it on MathOverflow. Does there exist a class $C$ of $L$-structures for a finite signature $L$, which is finitely axiomatizable in first-order logic with equality, but whose equality-free theory $Th(C)$ is not finitely axiomatizable?

Non-OP edit: To clarify, the question is whether the finite $\mathsf{FOL}$-axiomatizability of $Th_\mathsf{FOL}(C)$ necessarily implies the finite $\mathsf{FOL_{w/o=}}$-axiomatizability of $Th_{\mathsf{FOL_{w/o=}}}(C)$, regardless of whether the $\mathsf{FOL}$-deductive closure of $Th_{\mathsf{FOL_{w/o=}}}(C)$ coincides with $Th_\mathsf{FOL}(C)$ or not. (That is, we don't care whether $Mod(Th_{\mathsf{FOL_{w/o=}}}(C))\not=C.$)

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Noah Schweber
Noah Schweber
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This was originally a MSE questionan MSE question, but I was told to ask it on MathOverflow. Does there exist a class $C$ of $L$-structures for a finite signature $L$, which is finitely axiomatizable in first-order logic with equality, but whose equality-free theory $Th(C)$ is not finitely axiomatizable?

This was originally a MSE question, but I was told to ask it on MathOverflow. Does there exist a class $C$ of $L$-structures for a finite signature $L$, which is finitely axiomatizable in first-order logic with equality, but whose equality-free theory $Th(C)$ is not finitely axiomatizable?

This was originally an MSE question, but I was told to ask it on MathOverflow. Does there exist a class $C$ of $L$-structures for a finite signature $L$, which is finitely axiomatizable in first-order logic with equality, but whose equality-free theory $Th(C)$ is not finitely axiomatizable?

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user107952
user107952
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Is there a finitely axiomatizable class of structures whose equality-free theory is not finitely axiomatizable?

This was originally a MSE question, but I was told to ask it on MathOverflow. Does there exist a class $C$ of $L$-structures for a finite signature $L$, which is finitely axiomatizable in first-order logic with equality, but whose equality-free theory $Th(C)$ is not finitely axiomatizable?