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10041 | Impredicative Definitions refer to the totality to which the object itself belongs |
Full Idea: Impredicative Definitions are definitions of an object by reference to the totality to which the object itself (and perhaps also things definable only in terms of that object) belong. | |||
From: Kurt Gödel (Russell's Mathematical Logic [1944], n 13) |
21716 | In simple type theory the axiom of Separation is better than Reducibility |
Full Idea: In the superior realist and simple theory of types, the place of the axiom of reducibility is not taken by the axiom of classes, Zermelo's Aussonderungsaxiom. | |||
From: report of Kurt Gödel (Russell's Mathematical Logic [1944], p.140-1) by Bernard Linsky - Russell's Metaphysical Logic 6.1 n3 | |||
A reaction: This is Zermelo's Axiom of Separation, but that too is not an axiom of standard ZFC. |
10035 | Mathematical Logic is a non-numerical branch of mathematics, and the supreme science |
Full Idea: 'Mathematical Logic' is a precise and complete formulation of formal logic, and is both a section of mathematics covering classes, relations, symbols etc, and also a science prior to all others, with ideas and principles underlying all sciences. | |||
From: Kurt Gödel (Russell's Mathematical Logic [1944], p.447) | |||
A reaction: He cites Leibniz as the ancestor. In this database it is referred to as 'theory of logic', as 'mathematical' seems to be simply misleading. The principles of the subject are standardly applied to mathematical themes. |
10042 | Reference to a totality need not refer to a conjunction of all its elements |
Full Idea: One may, on good grounds, deny that reference to a totality necessarily implies reference to all single elements of it or, in other words, that 'all' means the same as an infinite logical conjunction. | |||
From: Kurt Gödel (Russell's Mathematical Logic [1944], p.455) |
10038 | A logical system needs a syntactical survey of all possible expressions |
Full Idea: In order to be sure that new expression can be translated into expressions not containing them, it is necessary to have a survey of all possible expressions, and this can be furnished only by syntactical considerations. | |||
From: Kurt Gödel (Russell's Mathematical Logic [1944], p.448) | |||
A reaction: [compressed] |
10046 | The generalized Continuum Hypothesis asserts a discontinuity in cardinal numbers |
Full Idea: The generalized Continuum Hypothesis says that there exists no cardinal number between the power of any arbitrary set and the power of the set of its subsets. | |||
From: Kurt Gödel (Russell's Mathematical Logic [1944], p.464) |
10039 | Some arithmetical problems require assumptions which transcend arithmetic |
Full Idea: It has turned out that the solution of certain arithmetical problems requires the use of assumptions essentially transcending arithmetic. | |||
From: Kurt Gödel (Russell's Mathematical Logic [1944], p.449) | |||
A reaction: A nice statement of the famous result, from the great man himself, in the plainest possible English. |
10043 | Mathematical objects are as essential as physical objects are for perception |
Full Idea: Classes and concepts may be conceived of as real objects, ..and are as necessary to obtain a satisfactory system of mathematics as physical bodies are necessary for a satisfactory theory of our sense perceptions, with neither case being about 'data'. | |||
From: Kurt Gödel (Russell's Mathematical Logic [1944], p.456) | |||
A reaction: Note that while he thinks real objects are essential for mathematics, be may not be claiming the same thing for our knowledge of logic. If logic contains no objects, then how could mathematics be reduced to it, as in logicism? |
10045 | Impredicative definitions are admitted into ordinary mathematics |
Full Idea: Impredicative definitions are admitted into ordinary mathematics. | |||
From: Kurt Gödel (Russell's Mathematical Logic [1944], p.464) | |||
A reaction: The issue is at what point in building an account of the foundations of mathematics (if there be such, see Putnam) these impure definitions should be ruled out. |