The axiom schema of replacement states that if F is a definable class function, as above, and A is any set, then the image F is also a set. This can be seen as a principle of smallness: the axiom states that if A is small enough to be a set, then F is also small enough to be a set. It is implied by the stronger axiom of limitation of size.
Because it is impossible to quantify over definable functions in first-order logic, one instance of the schema is included for each formula φ in the language of set theory with free variables among w1, ..., wn, A, x, y; but B is not free in φ. In the formal language of set theory, the axiom schema is:
Read more about Axiom Schema Of Replacement: Axiom Schema of Collection, Example Applications, History and Philosophy, Relation To The Axiom Schema of Specification
Famous quotes containing the words axiom and/or replacement:
““You are bothered, I suppose, by the idea that you can’t possibly believe in miracles and mysteries, and therefore can’t make a good wife for Hazard. You might just as well make yourself unhappy by doubting whether you would make a good wife to me because you can’t believe the first axiom in Euclid. There is no science which does not begin by requiring you to believe the incredible.””
—Henry Brooks Adams (1838–1918)
“Not even the visionary or mystical experience ever lasts very long. It is for art to capture that experience, to offer it to, in the case of literature, its readers; to be, for a secular, materialist culture, some sort of replacement for what the love of god offers in the world of faith.”
—Salman Rushdie (b. 1947)