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The general axiom scheme for building up complex sets like the ordinals is called replacement. It is an infinite list of axioms. These axioms could be defined by a single finite expression, but they are usually defined as an easily generated sequence.

The axiom of replacement scheme describes how new sets can be defined from exiting sets using any relationship that defines as a function of . Recall that a function maps any element in its range (any input value) to a unique result or output value. The axiom of replacement scheme asserts that for any set and any function defined on all sets, one can construct a new set which consists of the sets obtained by applying to each element of .

The following notation simplifies the formal expression. says there exists one and only one set such that is true. The replacement axioms schema is as follows.

This first part says if defines uniquely as a function of then the for all there exists such that is true. The second part defines as equivalent to if and only if there exists an such that is true. is the set defined by applying the function defined by to . Since is not defined in the form of a function one has to use this somewhat convoluted definition.

This axiom schema came about because previous attempts to
formalize mathematics were too general and led to contradictions
like the Barber Paradox^{6.1}. By
restricting new sets to those obtained by applying well defined
functions to the elements of existing sets it was felt that one
could avoid such contradictions. Sets are explicitly built up from
sets defined in safe axioms. Sets cannot be defined as the
*universe* of all objects satisfying some relationship. One
cannot construct the set of all sets which inevitably leads to
paradox.

We now turn our attention to developing the ordinals.

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