A. S. Troelstra. Principles of Intuitionism. Number 95 in Lecture Notes in Mathematics. Springer-Verlag, 1969.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....One interpretation of CT is that every object must have a concrete description or itself be concrete. In terms of classical logic, rules as universal instantiation must be accompanied by the substitution of a concrete object. The reliance on (CT) places great emphasis on decidability. Following [22], we call a notion j A a decidable property of predicate A. Notions play an important role in constructive logics. The goal is to constructively attach programs to mathematical objects as outlined in BishopBridges [5] We do not elaborate here as the details are found in [21] 3 Geometric ....

....Preliminaries Proof Rules Since every constructive intuitionistic proof is also a classical proof, we can use classical theory is guide to constructive theory. Our purpose in this paper is to illustrate how the constructive theory is formulated. There are a number of works available in this area[10, 11, 12, 14, 2, 5, 7, 17, 22, 24] . Recall, though, that many classical equivalences fail to hold in intuitionistic logic. Although we must develop axioms, we must also say what the rules of proof are. An initial insight: Greek geometers seem to have some constructive proclivities. In the commentaries, there are ample indications ....

A. S. Troelstra. Principles of Intuitionism. Number 95 in Lecture Notes in Mathematics. Springer-Verlag, 1969.

....One interpretation of CT is that every object must have a concrete description or itself be concrete. In terms of classical logic, rules as universal instantiation must be accompanied by the substitution of a concrete object. The reliance on (CT) places great emphasis on decidability. Following [43], we call a notion jA a decidable property of predicate A. Notions play an important role in constructive logics. 2.2 Our Program The goal is to constructively attach programs to constructive real number as outlined in Bishop Bridges [11] The organization of our effort, shown in Table 1, is ....

A. S. Troelstra. Principles of Intuitionism. Number 95 in Lecture Notes in Mathematics. SpringerVerlag, 1969.

....It is not clear what the status of and 0 are: they have a foot in each world. Taking a cue from analysis[21, p. 42] I call the set a a cover. 8.2. Sets. Since A is constructive, there is a function that passes on membership in A. To reduce notation, assume that A is that function. In [25], the convention is that Aa is true when a is an object in A as in (ii) above. Actually, this convention seems to be handy about one half the time; the other half A a = a would be better. Staying with the first convention, I take A A if for all a : A A a = true. In more familiar ....

A. S. Troelstra. Principles of Intuitionism. Number 95 in Lecture Notes in Mathematics. Springer-Verlag, 1969.

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