Formal Languages

Mathematics, as a human endeavor, is much more than a formal language processing. However modern mathematics would not be possible without the development of a special, relatively formal language. Mathematical expositions are written in a mixture of a common and that special mathematical language. The proportion of each depends on the level of complexity of the subject, the level of sophistication of the intended audience, and the taste of the writer. A presentation made exlusively in the mathematical language would be well nigh incomprehensible even to specialists, see, for example, a page from Russell and Whitehead's Principia Mathematica.

The Model Theory which is the foundation of the Non-standard Analysis is one of the branches of mathematics where employment of formal languages is unescapble. A (formal) language consists of an alphabet of symbols that combine into formulas (also called statements or sentences) and the rules of building the formulas. All languages of interest are extensions of a basic logical language. The alphabet of that language consists of

The first does not say much, except that, if need be, x can be used in lieu of y ∨ z. The second asserts reflexivity of equality; the third stipulates the transitivity of equality. The fourth asserts that there are at least three objects. In the first formula all the variables are free; in the rest, all the variables are bound.

In mathematics, one often extends the language with convenience symbols. For example, x ≠ y is a common replacement for ¬(x = y). Thus the fourth statement can be written as

• ∀x∀y∃z {(z ≠ x) ∧ (z ≠ y)}.

Languages, like the one defined above, underlie every attempt at formalization of mathematical theories, each getting associated with a language extension. Extensions may augment the alphabet with new symbols, of which some may be said to be constant. Extensions also add rules for formula construction.

The language of arithmetic extends the above language with two symbols "0" and "1" (constants), two operations "+" and "×" and a relation "<". The notion of formula is subsequently extended to include individual constants, operations, like x + 1, comparisons, like 0 < 1, and (implied) rules (a×(b + c) = a×b + a×c), or properties (∀x [x + 0 = x]), etc.

To define the language of Group theory one does not need that much. For multiplicative groups, we may only add symbol "1" and operation "×"; for additive groups, the addons are "0" and "+". In general, we need a constant for a special, unity element, say, e, and a symbol of operation, say, "•". In a group language we encounter formulas like

Both multiplicative and additive groups are models of Group theory.

The language, as defined above, falls into the category of first-order languages. It only allows binding of variables, as above, but not formulas (nor functions or predicates that were omitted altogether from the specifications.)

• Infinitesimals. Non-standard Analysis
• Formal Languages
• Theories and Proofs
  • Formal Systems - an Example
• Models and Metamathematics
• Hyperintegers and Hyperreal Numbers
• Structure of Hyperreal Numbers
• Transfer Principle
• Is .999... = 1?

Back to What Is Infinity?

References

1. D. Marker, Logic and Model Theory, in T. Gowers (ed.), The Princeton Companion to Mathematics, Princeton University Press, 2008, pp. 635-646
2. A. Robinson, Non-standard Analysis, Princeton University Press (Rev Sub edition), 1996
3. R. M. Smullyan, First-Order Logic, Dover, 1995