ancestral relation

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Giuseppe Peano (1858—1932)


Gottlob Frege (1848—1925) German philosopher and mathematician, founder of modern logic



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Intuitively, a relation that stands to another as ‘ancestor of’ stands to ‘parent of’: an ancestor is a parent, or parent of a parent, and so on. The formal definition of the ancestral is due to Frege. Suppose, to simplify, we allow that y is one of its own ancestors. Then the ancestors of y form a set that fulfils two conditions: the initial condition that y is a member, and the closure condition that all parents of members are members. So x is an ancestor of y if x belongs to all sets satisfying those two conditions. Formally this may be put (∀z)(y ϵ z) & (∀u)(∀w)(u ϵ z & Pwuw ϵ z) → x ϵ z). This says that all classes satisfy this condition: if y belongs, and if, for anything at all, if it belongs then its parents do, then x belongs. It is notable that this definition can only be given by quantifying over classes. With ancestor defined in this way, the ancestral of a relation is that relation that stands to it as ancestor does to parent. The class of numbers greater than a given number is the ancestral of the successor relation. See also Peano's postulates.

Subjects: Philosophy.

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