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 ⇔ ACext holds in extensional type theory. The following is the principle ACint of intensional choice: if A, B are sets and R a relation such that (∀x: A)(∃y: B)R(x, y) is true, then there is a function f: A → B such that (∀x: A)R(x, f(x)) is true. It is provable in Martin-Löf’s type theory [8, p. 50]. It follows from ACint that surjective functions have right inverses: If =B is an equivalence relation on B and f: A → B, we say that f is surjective if (∀y: B)(∃x: A)(y =B f(x)) is true. With R(y, x) def = (y =B f(x)), surjectivity

Abstract The universal Turing machine is an anticipatory theory of computability by any digital or quantum machine. However the Church-Turing hypothesis only gives weak anticipation. The construction of the quantum computer (unlike classical computing) requires theory with strong anticipation. Category theory provides the necessary coordinate-free mathematical language which is both constructive and non-local to subsume the various interpretations of quantum theory in one pullback/pushout Dolittle diagram. This diagram can be used to test and classify physical devices and proposed algorithms for weak or strong anticipation. Quantum Information Science is more than a merger of Church-Turing and quantum theories. It has constructively to bridge the non-local chasm between the weak anticipation of mathematics and the strong anticipation of physics.

The axiomofchoice and the law of excluded middle in weak set theories

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The first author gratefully acknowledges the support he has received from the NSERC of Canada for the last thirty seven years. Received by the editors 2005-03-01. Transmitted by F. W. Lawvere, W. Tholen and R.J. Wood. Reprint published on 2005-06-15.

This article is a commentary on remarks made in a recent book [12] that perpetuate several myths about Brouwer and intuitionism. The footnote on page 279 of [12] is an unfortunate, historically and factually inaccurate, blemish on an otherwise remarkable book. In that footnote, in which Ok discusses Brouwer (who, incidentally, was normally known not as “Jan ” but as “Bertus”, a shortening of his second name, Egbertus), 1 he says:...later in his career, he [Brouwer] became the most forceful proponent of the so-called intuitionist philosophy of mathematics, which not only forbids the use of the Axiom of Choice but also rejects the axiom that a proposition is either true or false (thereby disallowing the method of proof by contradiction). The consequences of taking this position are dire. For instance, an intuitionist would not accept the existence of an irrational number! In fact, in his later years, Brouwer did not view the Brouwer Fixed Point Theorem as a theorem. These sentences contain a number of outdated but still common misconceptions

Abstract. Taking a `set ' to be a type together with an equivalence relation and adding an extensional choice axiom to the logical framework (a restricted version of constructive type theory) it is shown that any `set' can be well-ordered. Zermelo's rst proof from 1904 is followed, with a simpli cation to avoid using comparability of well-orderings. The proof has been formalised in the system AgdaLight. 1