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 ∀ ∀ --> x ( x ∉ ∉ --> ∅ ∅ --> ) {\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.
Let T {\displaystyle T} be a first-order theory and ϕ ϕ --> ( x 1 , … … --> , x n ) {\displaystyle \phi (x_{1},\dots ,x_{n})} a formula of T {\displaystyle T} such that x 1 {\displaystyle x_{1}} , ..., x n {\displaystyle x_{n}} are distinct and include the variables free in ϕ ϕ --> ( x 1 , … … --> , x n ) {\displaystyle \phi (x_{1},\dots ,x_{n})} . Form a new first-order theory T ′ {\displaystyle T'} from T {\displaystyle T} by adding a new n {\displaystyle n} -ary relation symbol R {\displaystyle R} , the logical axioms featuring the symbol R {\displaystyle R} and the new axiom
called the defining axiom of R {\displaystyle R} .
If ψ ψ --> {\displaystyle \psi } is a formula of T ′ {\displaystyle T'} , let ψ ψ --> ∗ ∗ --> {\displaystyle \psi ^{\ast }} be the formula of T {\displaystyle T} obtained from ψ ψ --> {\displaystyle \psi } by replacing any occurrence of R ( t 1 , … … --> , t n ) {\displaystyle R(t_{1},\dots ,t_{n})} by ϕ ϕ --> ( t 1 , … … --> , t n ) {\displaystyle \phi (t_{1},\dots ,t_{n})} (changing the bound variables in ϕ ϕ --> {\displaystyle \phi } if necessary so that the variables occurring in the t i {\displaystyle t_{i}} are not bound in ϕ ϕ --> ( t 1 , … … --> , t n ) {\displaystyle \phi (t_{1},\dots ,t_{n})} ). Then the following hold:
The fact that T ′ {\displaystyle T'} is a conservative extension of T {\displaystyle T} shows that the defining axiom of R {\displaystyle R} cannot be used to prove new theorems. The formula ψ ψ --> ∗ ∗ --> {\displaystyle \psi ^{\ast }} is called a translation of ψ ψ --> {\displaystyle \psi } into T {\displaystyle T} . Semantically, the formula ψ ψ --> ∗ ∗ --> {\displaystyle \psi ^{\ast }} has the same meaning as ψ ψ --> {\displaystyle \psi } , but the defined symbol R {\displaystyle R} has been eliminated.
Let T {\displaystyle T} be a first-order theory (with equality) and ϕ ϕ --> ( y , x 1 , … … --> , x n ) {\displaystyle \phi (y,x_{1},\dots ,x_{n})} a formula of T {\displaystyle T} such that y {\displaystyle y} , x 1 {\displaystyle x_{1}} , ..., x n {\displaystyle x_{n}} are distinct and include the variables free in ϕ ϕ --> ( y , x 1 , … … --> , x n ) {\displaystyle \phi (y,x_{1},\dots ,x_{n})} . Assume that we can prove
in T {\displaystyle T} , i.e. for all x 1 {\displaystyle x_{1}} , ..., x n {\displaystyle x_{n}} , there exists a unique y such that ϕ ϕ --> ( y , x 1 , … … --> , x n ) {\displaystyle \phi (y,x_{1},\dots ,x_{n})} . Form a new first-order theory T ′ {\displaystyle T'} from T {\displaystyle T} by adding a new n {\displaystyle n} -ary function symbol f {\displaystyle f} , the logical axioms featuring the symbol f {\displaystyle f} and the new axiom
called the defining axiom of f {\displaystyle f} .
Let ψ ψ --> {\displaystyle \psi } be any atomic formula of T ′ {\displaystyle T'} . We define formula ψ ψ --> ∗ ∗ --> {\displaystyle \psi ^{\ast }} of T {\displaystyle T} recursively as follows. If the new symbol f {\displaystyle f} does not occur in ψ ψ --> {\displaystyle \psi } , let ψ ψ --> ∗ ∗ --> {\displaystyle \psi ^{\ast }} be ψ ψ --> {\displaystyle \psi } . Otherwise, choose an occurrence of f ( t 1 , … … --> , t n ) {\displaystyle f(t_{1},\dots ,t_{n})} in ψ ψ --> {\displaystyle \psi } such that f {\displaystyle f} does not occur in the terms t i {\displaystyle t_{i}} , and let χ χ --> {\displaystyle \chi } be obtained from ψ ψ --> {\displaystyle \psi } by replacing that occurrence by a new variable z {\displaystyle z} . Then since f {\displaystyle f} occurs in χ χ --> {\displaystyle \chi } one less time than in ψ ψ --> {\displaystyle \psi } , the formula χ χ --> ∗ ∗ --> {\displaystyle \chi ^{\ast }} has already been defined, and we let ψ ψ --> ∗ ∗ --> {\displaystyle \psi ^{\ast }} be
(changing the bound variables in ϕ ϕ --> {\displaystyle \phi } if necessary so that the variables occurring in the t i {\displaystyle t_{i}} are not bound in ϕ ϕ --> ( z , t 1 , … … --> , t n ) {\displaystyle \phi (z,t_{1},\dots ,t_{n})} ). For a general formula ψ ψ --> {\displaystyle \psi } , the formula ψ ψ --> ∗ ∗ --> {\displaystyle \psi ^{\ast }} is formed by replacing every occurrence of an atomic subformula χ χ --> {\displaystyle \chi } by χ χ --> ∗ ∗ --> {\displaystyle \chi ^{\ast }} . Then the following hold:
The formula ψ ψ --> ∗ ∗ --> {\displaystyle \psi ^{\ast }} is called a translation of ψ ψ --> {\displaystyle \psi } into T {\displaystyle T} . As in the case of relation symbols, the formula ψ ψ --> ∗ ∗ --> {\displaystyle \psi ^{\ast }} has the same meaning as ψ ψ --> {\displaystyle \psi } , but the new symbol f {\displaystyle f} has been eliminated.
The construction of this paragraph also works for constants, which can be viewed as 0-ary function symbols.
A first-order theory T ′ {\displaystyle T'} obtained from T {\displaystyle T} by successive introductions of relation symbols and function symbols as above is called an extension by definitions of T {\displaystyle T} . Then T ′ {\displaystyle T'} is a conservative extension of T {\displaystyle T} , and for any formula ψ ψ --> {\displaystyle \psi } of T ′ {\displaystyle T'} we can form a formula ψ ψ --> ∗ ∗ --> {\displaystyle \psi ^{\ast }} of T {\displaystyle T} , called a translation of ψ ψ --> {\displaystyle \psi } into T {\displaystyle T} , such that ψ ψ --> ↔ ↔ --> ψ ψ --> ∗ ∗ --> {\displaystyle \psi \leftrightarrow \psi ^{\ast }} is provable in T ′ {\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 T ′ {\displaystyle T'} of T is not distinguished from the original theory T. In fact, the formulas of T ′ {\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.
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