Search results
Results From The WOW.Com Content Network
The necessity for primitive notions is illustrated in several axiomatic foundations in mathematics: Set theory: The concept of the set is an example of a primitive notion. As Mary Tiles writes: [6] [The] 'definition' of 'set' is less a definition than an attempt at explication of something which is being given the status of a primitive ...
The primitive notions of his theory were function and argument. Using these notions, he defined class and set. [1] Paul Bernays reformulated von Neumann's theory by taking class and set as primitive notions. [2] Kurt Gödel simplified Bernays' theory for his relative consistency proof of the axiom of choice and the generalized continuum ...
The simplest way to introduce cardinals is to add a primitive notion, Card(), and an axiom of cardinality to ZF set theory (without axiom of choice). [2] Axiom of cardinality: The sets A and B are equinumerous if and only if Card(A) = Card(B)
A bijection between two topological spaces is a homeomorphism if and only if the derived set of the image (in the second space) of any subset of the first space is the image of the derived set of that subset. [7] A space is a T 1 space if every subset consisting of a single point is closed. [8]
In fact the length of all of Tarski's axioms together is not much more than just one of Pieri's 24 axioms. It was the first system of Euclidean geometry that was simple enough for all axioms to be expressed in terms of the primitive notions only, without the help of defined notions. Of even greater importance, for the first time a clear ...
In mathematics, pointless topology, also called point-free topology (or pointfree topology) and locale theory, is an approach to topology that avoids mentioning points, and in which the lattices of open sets are the primitive notions. [1] In this approach it becomes possible to construct topologically interesting spaces from purely algebraic ...
Hilbert's axiom system is constructed with six primitive notions: three primitive terms: [5] point; line; plane; and three primitive relations: [6] Betweenness, a ternary relation linking points; Lies on (Containment), three binary relations, one linking points and straight lines, one linking points and planes, and one linking straight lines ...
The metalanguage includes primitive notions, axioms, and rules absent from the object language, so that there are theorems provable in the metalanguage not provable in the object language. The undefinability theorem is conventionally attributed to Alfred Tarski. Gödel also discovered the undefinability theorem in 1930, while proving his ...