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I am wondering if there are some first order theories of algebraic structures or structures of interest to model theorists of large cardinal consistency strength or at least unexpectedly high consistency strength if the large cardinal heirarchy is not reached. I am very much interested in theories of groups, modules, lattices, etc with possibly extra structure. I am not as much interested in theories such as Peano arithmetic or some other variant of arithmetic. Also, I prefer finitely axiomatized theories.

Example

In Jech and Dougherty's paper Finite left-distributive algebras and embedding algebras, the authors develop some basic facts about two kinds of structures which they call two-sorted embedding algebras and extended two-sorted embedding algebras. The two-sorted embedding algebras and extended two-sorted embedding algebras are both axiomatized by finitely many first order axioms and such axiomatizations seem very natural to me. Every extended two-sorted embedding algebra is a two-sorted embedding algebra and every two sorted embedding algebra in which every ordinal is a critical point can be extended to an extended two-sorted embedding algebra. Let $A_{n}$ denote the $n$-th classical Laver table. If $(1)_{n\in\omega}\in\varprojlim_{n\in\omega}A_{n}$ freely generates a left-distributive algebra, then there exists a two-sorted embedding algebra. On the other hand, it is unknown (or at least was unknown) whether the existence of a two-sorted embedding algebra implies that $(1)_{n\in\omega}\in\varprojlim_{n\in\omega}A_{n}$ freely generates a left-distributive algebra. Therefore, the consistency strength of the theory of two-sorted embedding algebras is completely unknown and falls somewhere below the statement (for all $n\in\omega$ there is an $n$-huge cardinal).

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    $\begingroup$ Maybe some works of Harvey Friedman be related. In particular see his papers:(1) Extremely large cardinals in the rationals, (2) Countable model theory and large cardinals, (3) Applications of Large Cardinals to Graph Theory. They can be found in Harvey's Foundational Adventures. $\endgroup$ – Mohammad Golshani Jan 11 '16 at 4:18
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You might want to take a look at pp.1-2 (and the top quarter of pg. 3) of Harvey Friedman's paper "Restrictions and Extensions" (the rest of the paper (the paper is all of six pages) deals with the systems Weak Second-order Logic, $L_{\omega_1},_{\omega}$, and Second-order Logic). This paper deals specifically with generalized algebras ("By an algebra we mean a nonempty set dom(M) together with a finite list of functions on dom(M) of various finite arities $\ge$0. The signature of M is the list of arities the functions of M"). It can also be found in "Harvey's Foundational Adventures".

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