Lindenbaum algebras of intuitionistic theories and free categories

Abstract We consider formal theories synonymous with various free categories (logoi and topoi). Their Lindenbaum algebras may be described as the lattices of subobjects of a terminator. These theories have intuitionistic logic. We show that the Lindenbaum algebras of second order and higher order arithmetic (topoi), and set theory are not isomorphic to the Lindenbaum algebras of first order theories such as arithmetic (logoi). We also show that there are only five kernels of representations of the free Heyting algebra on one generator in these Lindenbaum algebras.

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