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Extension by definitions

In mathematical logic, more specifically in the proof theory of first-order theories, extensions by definitions formalize the introduction of new symbols by means of a definition. For example, it is common in naive set theory to introduce a symbol ${\displaystyle \emptyset }$ for the set that has no member. In the formal setting of first-order theories, this can be done by adding to the theory a new constant ${\displaystyle \emptyset }$ and the new axiom ${\displaystyle \forall x(x\notin \emptyset )}$, meaning "for all x, x is not a member of ${\displaystyle \emptyset }$". It can then be proved that doing so adds essentially nothing to the old theory, as should be expected from a definition. More precisely, the new theory is a conservative extension of the old one.

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Definition of relation symbols

Let ${\displaystyle T}$ be a first-order theory and ${\displaystyle \phi (x_{1},\dots ,x_{n})}$ a formula of ${\displaystyle T}$ such that ${\displaystyle x_{1}}$, ..., ${\displaystyle x_{n}}$ are distinct and include the variables free in ${\displaystyle \phi (x_{1},\dots ,x_{n})}$. Form a new first-order theory ${\displaystyle T'}$ from ${\displaystyle T}$ by adding a new ${\displaystyle n}$-ary relation symbol ${\displaystyle R}$, the logical axioms featuring the symbol ${\displaystyle R}$ and the new axiom

${\displaystyle \forall x_{1}\dots \forall x_{n}(R(x_{1},\dots ,x_{n})\leftrightarrow \phi (x_{1},\dots ,x_{n}))}$,

called the defining axiom of ${\displaystyle R}$.

If ${\displaystyle \psi }$ is a formula of ${\displaystyle T'}$, let ${\displaystyle \psi ^{\ast }}$ be the formula of ${\displaystyle T}$ obtained from ${\displaystyle \psi }$ by replacing any occurrence of ${\displaystyle R(t_{1},\dots ,t_{n})}$ by ${\displaystyle \phi (t_{1},\dots ,t_{n})}$ (changing the bound variables in ${\displaystyle \phi }$ if necessary so that the variables occurring in the ${\displaystyle t_{i}}$ are not bound in ${\displaystyle \phi (t_{1},\dots ,t_{n})}$). Then the following hold:

1. ${\displaystyle \psi \leftrightarrow \psi ^{\ast }}$ is provable in ${\displaystyle T'}$, and
2. ${\displaystyle T'}$ is a conservative extension of ${\displaystyle T}$.

The fact that ${\displaystyle T'}$ is a conservative extension of ${\displaystyle T}$ shows that the defining axiom of ${\displaystyle R}$ cannot be used to prove new theorems. The formula ${\displaystyle \psi ^{\ast }}$ is called a translation of ${\displaystyle \psi }$ into ${\displaystyle T}$. Semantically, the formula ${\displaystyle \psi ^{\ast }}$ has the same meaning as ${\displaystyle \psi }$, but the defined symbol ${\displaystyle R}$ has been eliminated.

Definition of function symbols

Let ${\displaystyle T}$ be a first-order theory (with equality) and ${\displaystyle \phi (y,x_{1},\dots ,x_{n})}$ a formula of ${\displaystyle T}$ such that ${\displaystyle y}$, ${\displaystyle x_{1}}$, ..., ${\displaystyle x_{n}}$ are distinct and include the variables free in ${\displaystyle \phi (y,x_{1},\dots ,x_{n})}$. Assume that we can prove

${\displaystyle \forall x_{1}\dots \forall x_{n}\exists !y\phi (y,x_{1},\dots ,x_{n})}$

in ${\displaystyle T}$, i.e. for all ${\displaystyle x_{1}}$, ..., ${\displaystyle x_{n}}$, there exists a unique y such that ${\displaystyle \phi (y,x_{1},\dots ,x_{n})}$. Form a new first-order theory ${\displaystyle T'}$ from ${\displaystyle T}$ by adding a new ${\displaystyle n}$-ary function symbol ${\displaystyle f}$, the logical axioms featuring the symbol ${\displaystyle f}$ and the new axiom

${\displaystyle \forall x_{1}\dots \forall x_{n}\phi (f(x_{1},\dots ,x_{n}),x_{1},\dots ,x_{n})}$,

called the defining axiom of ${\displaystyle f}$.

Let ${\displaystyle \psi }$ be any atomic formula of ${\displaystyle T'}$. We define formula ${\displaystyle \psi ^{\ast }}$ of ${\displaystyle T}$ recursively as follows. If the new symbol ${\displaystyle f}$ does not occur in ${\displaystyle \psi }$, let ${\displaystyle \psi ^{\ast }}$ be ${\displaystyle \psi }$. Otherwise, choose an occurrence of ${\displaystyle f(t_{1},\dots ,t_{n})}$ in ${\displaystyle \psi }$ such that ${\displaystyle f}$ does not occur in the terms ${\displaystyle t_{i}}$, and let ${\displaystyle \chi }$ be obtained from ${\displaystyle \psi }$ by replacing that occurrence by a new variable ${\displaystyle z}$. Then since ${\displaystyle f}$ occurs in ${\displaystyle \chi }$ one less time than in ${\displaystyle \psi }$, the formula ${\displaystyle \chi ^{\ast }}$ has already been defined, and we let ${\displaystyle \psi ^{\ast }}$ be

${\displaystyle \forall z(\phi (z,t_{1},\dots ,t_{n})\rightarrow \chi ^{\ast })}$

(changing the bound variables in ${\displaystyle \phi }$ if necessary so that the variables occurring in the ${\displaystyle t_{i}}$ are not bound in ${\displaystyle \phi (z,t_{1},\dots ,t_{n})}$). For a general formula ${\displaystyle \psi }$, the formula ${\displaystyle \psi ^{\ast }}$ is formed by replacing every occurrence of an atomic subformula ${\displaystyle \chi }$ by ${\displaystyle \chi ^{\ast }}$. Then the following hold:

1. ${\displaystyle \psi \leftrightarrow \psi ^{\ast }}$ is provable in ${\displaystyle T'}$, and
2. ${\displaystyle T'}$ is a conservative extension of ${\displaystyle T}$.

The formula ${\displaystyle \psi ^{\ast }}$ is called a translation of ${\displaystyle \psi }$ into ${\displaystyle T}$. As in the case of relation symbols, the formula ${\displaystyle \psi ^{\ast }}$ has the same meaning as ${\displaystyle \psi }$, but the new symbol ${\displaystyle f}$ has been eliminated.

The construction of this paragraph also works for constants, which can be viewed as 0-ary function symbols.

Extensions by definitions

A first-order theory ${\displaystyle T'}$ obtained from ${\displaystyle T}$ by successive introductions of relation symbols and function symbols as above is called an extension by definitions of ${\displaystyle T}$. Then ${\displaystyle T'}$ is a conservative extension of ${\displaystyle T}$, and for any formula ${\displaystyle \psi }$ of ${\displaystyle T'}$ we can form a formula ${\displaystyle \psi ^{\ast }}$ of ${\displaystyle T}$, called a translation of ${\displaystyle \psi }$ into ${\displaystyle T}$, such that ${\displaystyle \psi \leftrightarrow \psi ^{\ast }}$ is provable in ${\displaystyle T'}$. Such a formula is not unique, but any two of them can be proved to be equivalent in T.

In practice, an extension by definitions ${\displaystyle T'}$ of T is not distinguished from the original theory T. In fact, the formulas of ${\displaystyle T'}$ can be thought of as abbreviating their translations into T. The manipulation of these abbreviations as actual formulas is then justified by the fact that extensions by definitions are conservative.

Examples

• Traditionally, the first-order set theory ZF has ${\displaystyle =}$ (equality) and ${\displaystyle \in }$ (membership) as its only primitive relation symbols, and no function symbols. In everyday mathematics, however, many other symbols are used such as the binary relation symbol ${\displaystyle \subseteq }$, the constant ${\displaystyle \emptyset }$, the unary function symbol P (the power set operation), etc. All of these symbols belong in fact to extensions by definitions of ZF.
• Let ${\displaystyle T}$ be a first-order theory for groups in which the only primitive symbol is the binary product ×. In T, we can prove that there exists a unique element y such that x×y = y×x = x for every x. Therefore we can add to T a new constant e and the axiom
${\displaystyle \forall x(x\times e=x{\text{ and }}e\times x=x)}$,
and what we obtain is an extension by definitions ${\displaystyle T'}$ of ${\displaystyle T}$. Then in ${\displaystyle T'}$ we can prove that for every x, there exists a unique y such that x×y=y×x=e. Consequently, the first-order theory ${\displaystyle T''}$ obtained from ${\displaystyle T'}$ by adding a unary function symbol ${\displaystyle f}$ and the axiom
${\displaystyle \forall x(x\times f(x)=e{\text{ and }}f(x)\times x=e)}$
is an extension by definitions of ${\displaystyle T}$. Usually, ${\displaystyle f(x)}$ is denoted ${\displaystyle x^{-1}}$.