N. Bourbaki. Th'eorie des Ensembles, Livre I, Chapitre III, Exercice 6. Hermann, Ed. 1956  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... Topos Theoretic Fix Point Theorem Thierry Coquand Chalmers University Preliminary version, June 1995 Introduction We present a fix point theorem that can be seen as an intuitionistic alternative to Bourbaki s lemma used in his presentation of Zorn s lemma [3]. Here however intuitionistic is taken in a generalized sense since our proof uses impredicativity. We present then an application of this lemma to the semantics of Frege Structure and of Type Theory, and compare it to S. Allen s justification of this semantics. 1 A Fix Point Theorem Theorem: ....

N. Bourbaki. Th'eorie des Ensembles, Livre I, Chapitre III, Exercice 6. Hermann, Ed. 1956

....related to it were already deduced (i.e. written down) These string manipulation rules are called rules of inference . We describe in this section the axioms and the rules of inference that we will accept into our logical calculus. These are chosen so that a logic equivalent to that in [Bou, En] results. The characterizing feature will be that all primary rules of inference are propositional . This will entail a very simple version of the Deduction Theorem that is applicable without constraints. We have the choice of taking all tautologies as schemata in Ax1 below, or restricting the ....

Bourbaki, N. Elemens de Mathematique; Theorie des Ensembles, Ch. 1, Paris: Hermann, 1966.

....because of the inherent complexity of the semantic of the SML module system. We also believe our system may help in designing a safe and powerful module system for Elf. In the same direction, it would be interesting to compare our module system with Bourbaki s mathematical notion of theory [Bou70] which is for instance implemented in the IMPS [FGT95] prover. ....

Nicolas Bourbaki. El'ements de Math'ematique; Th'eorie des Ensembles, chapter IV. Hermann, Paris, 1970.

....for two reasons: for the avoidance of error, and for the systematization and collation of mathematical knowledge. Indeed in recent times, formalizability in some system such as ZF set theory has become widely accepted as the ultimate arbiter of correctness for mathematical reasoning. Bourbaki [3] clearly says that the correctness of a mathematical text is verified by comparing it, more or less explicitly, with the rules of a formalized language , and Mac Lane [20] says almost the same thing (p. 377) As to precision, we have now stated an absolute standard of rigor: A Mathematical ....

Nicolas Bourbaki, Theory of sets, Elements of mathematics, AddisonWesley, 1968. Translated from French `Theorie des ensembles' in the series `Elements de mathematique', originally published by Hermann in 1968.

....difficult to predict whether a change in a proof will break proofs depending on it, since there is no clear notion of the specification exported by a given file. Some theorem provers already address some of these issue. Thus IMPS [FGT95] implements Bourbaki s notion of structures and theories [Bou70] allowing to instantiate a general theory on a given structure at once, getting every instantiations of theorems. Unfortunately, this notion is well suited in a set theoretic framework but less in a type theoretic one. The Standard ML programming language has a very powerful module system ....

Nicolas Bourbaki. El'ements de Math'ematique; Th'eorie des Ensembles, chapter IV. Hermann, Paris, 1970.

.... for instance lambda calculus like systems for classical logic such as the calculus of Parigot [13] 2 The graphical representation of terms We deal with a graph representation of terms which unifies in a single shot the usual representation (referred to as the Bourbarki representation, [3]) linking the bound variables to their lambda, the dynamic graph [5, 14, 6] and the sharing graph representation as defined in [1] Our graphs are unoriented but have a natural orientation defined on the figures below. The edges are labeled by some weight belonging to the dynamic algebra (see ....

N. Bourbaki. Th'eorie des ensembles. Hermann & C. Editeurs, 1954.

....of M , and k i with the variables bound by f (the latters will be called bound ports of the partition and have a positive polarity) Observe that bound variables correspond to (bound) ports of the form f. Thus our graphs are cyclic. Indeed, already Bourbaki used this notation for predicate logic [6]: this is the reason for calling Bourbaki representations our graphical representations of expressions. The unique port which does not belong to a partition is called the output port of f. All the forms, have a principal port, which is drawn with an outgoing arrow (the arrow is omitted when it is ....

N. Bourbaki. Th'eorie des ensembles. Hermann & C. Editeurs, 1954.

....difficult to predict whether a change in a proof will break proofs depending on it, since there is no clear notion of the specification exported by a given file. Some theorem provers already address some of these issues. Thus IMPS [FGT95] implements Bourbaki s notion of structures and theories [Bou70] allowing to instantiate a general theory on a given structure at once, getting every instantiations of theorems. Unfortunately, this notion is well suited in a set theoretic framework but less in a type theoretic one. The Standard ML programming language has a very powerful module system ....

Nicolas Bourbaki. Elements de Mathematique; Theorie des Ensembles, chapter IV. Hermann, Paris, 1970.

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