definitional extension
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In
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...
, more specifically in the
proof theory Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Barwise (1978) consists of four corresponding part ...
of
first-order theories In first-order logic, a first-order theory is given by a Set (mathematics), set of axioms in some language. This entry lists some of the more common examples used in model theory and some of their properties. Preliminaries For every natural mat ...
, 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 is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
to introduce a symbol \emptyset for the
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
which has no member. In the formal setting of first-order theories, this can be done by adding to the theory a new constant \emptyset and the new
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
\forall x(x\notin\emptyset), meaning "for all ''x'', ''x'' is not a member of \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.


Definition of relation symbols

''Let'' T be a
first-order theory First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quan ...
and \phi(x_1,\dots,x_n) a
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...
of T such that x_1, ..., x_n are distinct and include the variables free in \phi(x_1,\dots,x_n). Form a new first-order theory T' from T by adding a new n-ary relation symbol R, the logical axioms featuring the symbol R and the new axiom :\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 R. If \psi is a formula of T', let \psi^\ast be the formula of T obtained from \psi by replacing any occurrence of R(t_1,\dots,t_n) by \phi(t_1,\dots,t_n) (changing the bound variables in \phi if necessary so that the variables occurring in the t_i are not bound in \phi(t_1,\dots,t_n)). Then the following hold: # \psi\leftrightarrow\psi^\ast is provable in T', and # T' is a conservative extension of T. The fact that T' is a conservative extension of T shows that the defining axiom of R cannot be used to prove new theorems. The formula \psi^\ast is called a ''translation'' of \psi into T. Semantically, the formula \psi^\ast has the same meaning as \psi, but the defined symbol R has been eliminated.


Definition of function symbols

Let T be a first-order theory ( with equality) and \phi(y,x_1,\dots,x_n) a formula of T such that y, x_1, ..., x_n are distinct and include the variables free in \phi(y,x_1,\dots,x_n). Assume that we can prove :\forall x_1\dots\forall x_n\exists !y\phi(y,x_1,\dots,x_n) in T, i.e. for all x_1, ..., x_n, there exists a unique ''y'' such that \phi(y,x_1,\dots,x_n). Form a new first-order theory T' from T by adding a new n-ary function symbol f, the logical axioms featuring the symbol f and the new axiom :\forall x_1\dots\forall x_n\phi(f(x_1,\dots,x_n),x_1,\dots,x_n), called the ''defining axiom'' of f. Let \psi be any atomic formula of T'. We define formula \psi^\ast of T recursively as follows. If the new symbol f does not occur in \psi, let \psi^\ast be \psi. Otherwise, choose an occurrence of f(t_1,\dots,t_n) in \psi such that f does not occur in the terms t_i, and let \chi be obtained from \psi by replacing that occurrence by a new variable z. Then since f occurs in \chi one less time than in \psi, the formula \chi^\ast has already been defined, and we let \psi^\ast be : \forall z(\phi(z,t_1,\dots,t_n)\rightarrow\chi^\ast) (changing the bound variables in \phi if necessary so that the variables occurring in the t_i are not bound in \phi(z,t_1,\dots,t_n)). For a general formula \psi, the formula \psi^\ast is formed by replacing every occurrence of an atomic subformula \chi by \chi^\ast. Then the following hold: # \psi\leftrightarrow\psi^\ast is provable in T', and # T' is a conservative extension of T. The formula \psi^\ast is called a ''translation'' of \psi into T. As in the case of relation symbols, the formula \psi^\ast has the same meaning as \psi, but the new symbol 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 T' obtained from T by successive introductions of relation symbols and function symbols as above is called an extension by definitions of T. Then T' is a conservative extension of T, and for any formula \psi of T' we can form a formula \psi^\ast of T, called a ''translation'' of \psi into T, such that \psi\leftrightarrow\psi^\ast is provable in 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' of ''T'' is not distinguished from the original theory ''T''. In fact, the formulas of 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 = (equality) and \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 \subseteq, the constant \emptyset, the unary function symbol ''P'' (the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
operation), etc. All of these symbols belong in fact to extensions by definitions of ZF. * Let T be a first-order theory for
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
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 ::\forall x(x \times e = x\texte \times x = x), :and what we obtain is an extension by definitions T' of T. Then in 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 T'' obtained from T' by adding a unary function symbol f and the axiom ::\forall x(x \times f(x)=e\textf(x) \times x=e) :is an extension by definitions of T. Usually, f(x) is denoted x^.


See also

* Conservative extension *
Extension by new constant and function names In mathematical logic, a theory can be extended with new constants or function names under certain conditions with assurance that the extension will introduce no contradiction. Extension by definitions is perhaps the best-known approach, but it r ...


Bibliography

* S.C. Kleene (1952), ''Introduction to Metamathematics'', D. Van Nostrand * E. Mendelson (1997). ''Introduction to Mathematical Logic'' (4th ed.), Chapman & Hall. * J.R. Shoenfield (1967). ''Mathematical Logic'', Addison-Wesley Publishing Company (reprinted in 2001 by AK Peters) {{Mathematical logic Mathematical logic Proof theory