Extension by definitions

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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 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 , 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 of the old one.

Definition of relation symbols[edit]

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 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[edit]

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 of , define a formula of recursively as follows. If the new symbol does not occur in , let be . Otherwise, choose an occurrence of in such that does not occur in the terms , and let be obtained from by 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 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[edit]

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[edit]

  • 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 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 .

Bibliography[edit]

  • 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)