Extension by definitions

# Extension by definitions

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In mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

, more specifically in the proof theory
Proof theory
Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed...

of first-order theories
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...

, extensions by definitions formalize the introduction of new symbols by means of a definition. For example, it is common in naive set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

to introduce a symbol for the set which has no member. In the formal setting of first-order theories, this can be done by adding to the theory a new constant and the new axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...

, meaning 'for all x, x is not a member of '. 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
Conservative extension
In mathematical logic, a logical theory T_2 is a conservative extension of a theory T_1 if the language of T_2 extends the language of T_1; every theorem of T_1 is a theorem of T_2; and any theorem of T_2 which is in the language of T_1 is already a theorem of T_1.More generally, if Γ is a set of...

of the old one.

## Definition of relation symbols

Let be a first-order theory and a formula of such that , ..., are distinct and include the variables free in . Form a new first-order theory from by adding a new -ary relation symbol , the logical axioms featuring the symbol and the new axiom,
called the defining axiom of .

If is a formula of , let be the formula of obtained from by replacing any occurrence of by (changing the bound variables in if necessary so that the variables occurring in the are not bound in ). Then the following hold:
1. is provable in , and
2. is a conservative extension
Conservative extension
In mathematical logic, a logical theory T_2 is a conservative extension of a theory T_1 if the language of T_2 extends the language of T_1; every theorem of T_1 is a theorem of T_2; and any theorem of T_2 which is in the language of T_1 is already a theorem of T_1.More generally, if Γ is a set of...

of .

The fact that is a conservative extension of shows that the defining axiom of cannot be used to prove new theorems. The formula is called a translation of into . Semantically, the formula has the same meaning as , but the defined symbol has been eliminated.

## Definition of function symbols

Let be a first-order theory (with equality) and a formula of such that , , ..., are distinct and include the variables free in . Assume that we can prove
in , i.e. for all , ..., , there exists a unique y such that . Form a new first-order theory from by adding a new -ary function symbol , the logical axioms featuring the symbol and the new axiom,
called the defining axiom of .

If is an atomic formula
Atomic formula
In mathematical logic, an atomic formula is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic...

of , define a formula of recursively as follows. If the new symbol does not occur in , let be . Otherwise, choose an occurrence of in , and let be obtained from be replacing that occurrence by a new variable . Then since occurs in one less time than in , the formula has already been defined, and we let be

(changing the bound variables in if necessary so that the variables occurring in the are not bound in ). For a general formula , the formula is formed by replacing every occurrence of an atomic subformula by . Then the following hold:
1. is provable in , and
2. is a conservative extension
Conservative extension
In mathematical logic, a logical theory T_2 is a conservative extension of a theory T_1 if the language of T_2 extends the language of T_1; every theorem of T_1 is a theorem of T_2; and any theorem of T_2 which is in the language of T_1 is already a theorem of T_1.More generally, if Γ is a set of...

of .

The formula is called a translation of into . As in the case of relation symbols, the formula has the same meaning as , but the new symbol 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 obtained from by successive introductions of relation symbols and function symbols as above is called an extension by definitions of . Then is a conservative extension of , and for any formula of we can form a formula of , called a translation of into , such that is provable in . 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 of T is not distinguished from the original theory T. In fact, the formulas of 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 (equality) and (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 , the constant , the unary function symbol P (the power-set operation), etc. All of these symbols belong in fact to extensions by definitions of ZF.

• Let be a first-order theory for groups
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

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,

and what we obtain is an extension by definitions of T. Then in we can prove that for every x, there exists a unique y such that x.y=y.x=e. Consequently, the first-order theory obtained from by adding a unary function symbol and the axiom
is an extension by definitions of T. Usually, is denoted .