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Register a new userThe Strength of the Gratzer-Schmidt Theorem
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The Gratzer-Schmidt theorem of lattice theory states that each algebraic lattice is isomorphic to the congruence lattice of an algebra. We study the reverse mathematics of this theorem. We also show that the set of indices of computable lattices that are complete is $Pi^1_1$-complete; the set of indices of computable lattices that are algebraic is $Pi^1_1$-complete; the set of compact elements of a computable lattice is $Pi^{1}_{1}$ and can be $Pi^1_1$-complete; and the set of compact elements of a distributive computable lattice is $Pi^{0}_{3}$, and there is an algebraic distributive computable lattice such that the set of its compact elements is $Pi^0_3$-complete.
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We analyze the strength of Hellys selection theorem HST, which is the most important compactness theorem on the space of functions of bounded variation. For this we utilize a new representation of this space intermediate between $L_1$ and the Sobolev space W1,1, compatible with the, so called, weak* topology. We obtain that HST is instance-wise equivalent to the Bolzano-Weierstrass principle over RCA0. With this HST is equivalent to ACA0 over RCA0. A similar classification is obtained in the Weihrauch lattice.
In this paper, we show that $mathrm{RT}^{2}+mathsf{WKL}_0$ is a $Pi^{1}_{1}$-conservative extension of $mathrm{B}Sigma^0_3$.
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