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Axiom scheme of replacement

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 $A(x,y)$ that defines $y$ as a function of $x$. 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 $x$ and any function $f$ defined on all sets, one can construct a new set which consists of the sets obtained by applying $f$ to each element of $x$.

The following notation simplifies the formal expression. $\exists\,!\,y\,g(y)$ says there exists one and only one set $y$ such that $g(y)$ is true. The replacement axioms schema is as follows.

$$[ \forall x \exists\,!\,y A_n(x,y) ]\rightarrow \forall u \exists v (B(u,v))$$

$$B(u,v) \equiv [ \forall r (r \in v \equiv \exists s [ s \in u \wedge A_n(s,r)])]$$

This first part says if $A_n$ defines $y$ uniquely as a function of $x$ then the for all $u$ there exists $v$ such that $B(u.v)$ is true. The second part defines $B(u,v)$ as equivalent to $r \in v$ if and only if there exists an $s \in u$ such that $A_n(s,r)$ is true. $v$ is the set defined by applying the function defined by $A_n$ to $u$. Since $A_n$ 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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