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Axiom of choice and excluded middle in categorical logic
 Abstract: Bull. Symb. Logic
, 1995
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Elementary constructive operational set theory. To appear in: Festschrift for Wolfram Pohlers, Ontos Verlag
"... Abstract. We introduce an operational set theory in the style of [5] and [17]. The theory we develop here is a theory of constructive sets and operations. One motivation behind constructive operational set theory is to merge a constructive notion of set ([1], [2]) with some aspects which are typical ..."
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Abstract. We introduce an operational set theory in the style of [5] and [17]. The theory we develop here is a theory of constructive sets and operations. One motivation behind constructive operational set theory is to merge a constructive notion of set ([1], [2]) with some aspects which are typical of explicit mathematics [14]. In particular, one has nonextensional operations (or rules) alongside extensional constructive sets. Operations are in general partial and a limited form of self–application is permitted. The system we introduce here is a fully explicit, finitely axiomatised system of constructive sets and operations, which is shown to be as strong as HA. 1.
EM + Ext − + ACint is equivalent to ACext
, 2004
"... 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 (‘typetheoretical’) axiom of choice ACint, which is provable in MartinLöf’s type theory, and a weak extensionality princip ..."
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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 (‘typetheoretical’) axiom of choice ACint, which is provable in MartinLöf’s type theory, and a weak extensionality principle (Ext−), which is provable in MartinLöf’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 MartinLöf’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
Zermelo's WellOrdering Theorem in Type Theory
"... 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 wellordered. Zermelo's rst proof from 1904 is followed, wi ..."
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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 wellordered. Zermelo's rst proof from 1904 is followed, with a simpli cation to avoid using comparability of wellorderings. The proof has been formalised in the system AgdaLight. 1
WORLD THEORY
"... Abstract. In this paper a general mathematical model of the World will be constructed. I will show that a number of important theories in Physics are particularizations of the World Theory presented here. In particular, the worlds described by the Classical Mechanics, the Theory of Relativity and th ..."
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Abstract. In this paper a general mathematical model of the World will be constructed. I will show that a number of important theories in Physics are particularizations of the World Theory presented here. In particular, the worlds described by the Classical Mechanics, the Theory of Relativity and the Quantum Mechanics are examples of worlds according to this definition, but also some theories attempting to unify gravity and QM, like String Theory. This mathematical model is not a Unified Theory of Physics, it will not try to be a union of all the results. By contrary, it tries to keep only what is common and general to most of these theories. Special attention will be payed to the space, time, matter, and the physical laws. What do we know about the laws governing the Universe? What are the most general assumptions one can make about the Physical World? Each theory in Physics and each philosophical system came with its own vision trying to describe or explain the World, at least partially. In the following, I will try to keep the essential, and to establish a mathematical context, for all these visions. The purpose of this distillation is to provide a mathematical common background to both physical and metaphysical
Did Brouwer Really Believe That?
, 2007
"... 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 ..."
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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 socalled 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
Independence, Randomness and the Axiom of Choice
, 1992
"... We investigate various ways of introducing axioms for randomness in set theory. The results show that these axioms, when added to ZF, imply the failure of AC. But the axiom of extensionality plays an essential role in the derivation, and a deeper analysis may ultimately show that randomness is incom ..."
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We investigate various ways of introducing axioms for randomness in set theory. The results show that these axioms, when added to ZF, imply the failure of AC. But the axiom of extensionality plays an essential role in the derivation, and a deeper analysis may ultimately show that randomness is incompatible with extensionality.