We show how to extend any commutative ring (or semiring) so that division by any element, including 0, is, in a sense, possible. The resulting structure is called a wheel. Wheels are similar to rings, but 0x = 0 does not hold in general; the subset {x | 0x = 0} of any wheel is a commutative ring (or semiring), and any commutative ring (or semiring) with identity can be described as such a subset of a wheel. The main goal of this paper is to show that the given axioms for wheels are natural, and to clarify how valid identities for wheels relate to valid identities for commutative rings and semirings.

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It is well known that the extensional axiom of choice (ACext) implies the law of excluded middle (EM). We here prove that the converse holds as well if we have the intensional ('type-theoretical') axiom of choice ACint, which is provable in Martin-Löf's type theory, and a weak extensionality principle (Ext−), which is provable in Martin-Löf's extensional type theory. In particular, EM is equivalent to ACext in extensional type theory.The following is the principle AC int of intensional choice: if A, B are sets and R a relation such thatIt follows from AC ext that surjective functions have right inverses: If = B is an equivalence relation on B andsurjectivity is an instance of the premise needed to apply intensional choice (switch A and B). It says that there is a function g :This, however, does not mean that g is extensional, that is, that it preserves equivalence relations. If = A is an equivalence relation on A and = B is an equivalence relation on B, it might very well happen that f preserves them but g does not. If both A and B are the set of Cauchy sequences of rational numbers, for instance, and = A is pointwise equality, while = B is real number equality, then the identity function is clearly extensional from A/= A to B/= B , but not in the other direction. Thus it has a right inverse (itself) which is not extensional. In fact, we have no reason to expect that we should be able to construct an extensional right inverse in this case.The principle AC ext of extensional choice states that there is an extensional function with the property required. To be precise, it states that if R is an extensional relation (that is, it respects the equivalence relations) and (∀x : A)(∃y : B) R(x, y) is true, then there is an extensional function f : A −→ B such that (∀x : A) R(x, f (x)) is true. As the before-mentioned example with real numbers indicates, one cannot justify AC ext constructively. In fact, it implies the principle of excluded middle. 1) Thus there is, from a constructive point of view, a big difference in status between the intensional and extensional axioms of choice. However, they are equivalent in ZF and other theories with sufficiently strong axioms for quotient sets, which explains why we are used to hear the name 'axiom of choice', with no mention of the extensionality.At the beginning we remark the following well-known proposition: * e-mail: jesper@math.su.se, http://www.math.su.se/∼jesper 1) This was left as an exercise by Bishop [2, p. 58]. It was proved for toposes by Diaconescu [3], for constructive set theory by Goodman and Myhill [4], and for some intensional type theories e. g. by Lacas and Werner [5] and Maietti [6,7]. We give a somewhat different proof in this paper.

Natural deduction systems with indefinite and definite descriptions (ε-terms and ι-terms) are presented, and interpreted in Martin-LÖf's intensional type theory. The interpretations are formalizations of ideas which are implicit in the literature of constructive mathematics: if we have proved that an element with a certain property exists, we speak of ‘the element such that the property holds’ and refer by that phrase to the element constructed in the existence proof. In particular, we deviate from the practice of interpreting descriptions by contextual definitions.

A version of Birkhoff's theorem is proved by constructive, predicative, methods. The version we prove has two conditions more than the classical one. First, the class considered is assumed to contain a generic family, which is defined to be a set‐indexed family of algebras such that if an identity is valid in every algebra of this family, it is valid in every algebra of the class. Secondly, the class is assumed to be closed under inductive limits. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)