Completed second draft of this book

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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 says if there is a relationship $A(x,y)$ that defines $y$ as a function of $x$ 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. $\exists\,!\,y\,g(y)$ 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 the elements of $u$. 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 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