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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 says if there is a relationship that defines as a function of then you can apply this function to the elements of any set to create a new set.

To simplify the formal expression we introduce a new notation. 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 the elements of . 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
axiomatize mathematics were too general and led to contradictions
like the Barber Paradox^{4.1}. 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.

Completed
second draft of this book

PDF version
of this book

**Next:** Ordinal numbers **Up:** Creative mathematics
**Previous:** Ordinal induction **Contents**

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