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Register a new userAffine logic for constructive mathematics
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Michael Shulman
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We show that numerous distinctive concepts of constructive mathematics arise automatically from an antithesis translation of affine logic into intuitionistic logic via a Chu/Dialectica construction. This includes apartness relations, complemented subsets, anti-subgroups and anti-ideals, strict and non-strict order pairs, cut-valued metrics, and apartness spaces. We also explain the constructive bifurcation of some classical concepts using the choice between multiplicative and additive affine connectives. Affine logic and the antithesis construction thus systematically constructivize classical definitions, handling the resulting bookkeeping automatically.
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Interactive theorem provers based on dependent type theory have the flexibility to support both constructive and classical reasoning. Constructive reasoning is supported natively by dependent type theory and classical reasoning is typically supported by adding additional non-constructive axioms. However, there is another perspective that views constructive logic as an extension of classical logic. This paper will illustrate how classical reasoning can be supported in a practical manner inside dependent type theory without additional axioms. We will see several examples of how classical results can be applied to constructive mathematics. Finally, we will see how to extend this perspective from logic to mathematics by representing classical function spaces using a weak value monad.
Using the tools of reverse mathematics in second-order arithmetic, as developed by Friedman, Simpson, and others, we determine the axioms necessary to develop various topics in commutative ring theory. Our main contributions to the field are as follows. We look at fundamental results concerning primary ideals and the radical of an ideal, concepts previously unstudied in reverse mathematics. Then we turn to a fine-grained analysis of four different definitions of Noetherian in the weak base system $mathsf{RCA}_0 + mathsf{I}Sigma_2$. Finally, we begin a systematic study of various types of integral domains: PIDs, UFDs and Bezout and GCD domains.
We introduce Z-stability, a notion capturing the intuition that if a function f maps a metric space into a normed space and if the norm of f(x) is small, then x is close to a zero of f. Working in Bishops constructive setting, we first study pointwi
In this paper we study the reverse mathematics of two theorems by Bonnet about partial orders. These results concern the structure and cardinality of the collection of the initial intervals. The first theorem states that a partial order has no infinite antichains if and only if its initial intervals are finite unions of ideals. The second one asserts that a countable partial order is scattered and does not contain infinite antichains if and only if it has countably many initial intervals. We show that the left to right directions of these theorems are equivalent to ACA_0 and ATR_0, respectively. On the other hand, the opposite directions are both provable in WKL_0, but not in RCA_0. We also prove the equivalence with ACA_0 of the following result of Erdos and Tarski: a partial order with no infinite strong antichains has no arbitrarily large finite strong antichains.
We investigate the strength of a randomness notion $mathcal R$ as a set-existence principle in second-order arithmetic: for each $Z$ there is an $X$ that is $mathcal R$-random relative to $Z$. We show that the equivalence between $2$-randomness and being infinitely often $C$-incompressible is provable in $mathsf{RCA}_0$. We verify that $mathsf{RCA}_0$ proves the basic implications among randomness notions: $2$-random $Rightarrow$ weakly $2$-random $Rightarrow$ Martin-L{o}f random $Rightarrow$ computably random $Rightarrow$ Schnorr random. Also, over $mathsf{RCA}_0$ the existence of computable randoms is equivalent to the existence of Schnorr randoms. We show that the existence of balanced randoms is equivalent to the existence of Martin-L{o}f randoms, and we describe a sense in which this result is nearly optimal.
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