Axiom Systems and Formal Systems

Solution 1:

Euclid's Elements satisfies the criteria for being an axiomatic system. It does not, however, satisfy the criteria for being a formal system; the reason being that, from the point of view of formalism, certain steps in Euclid's proofs are left implied or tacit. In other words: all formal systems are axiomatic, but not all axiomatic systems are formal.

In nearly all practicality - except when studying formal systems - one would rarely do work in a bona fide formal system, but always in an axiomatic, or one is not doing math.

Solution 2:

A useful suggestion is Richard Kaye, The Mathematics of Logic, (Cambridge U.P., 2007).

In Chapter 3 : Formal systems, he describes formal systems as :

kinds of mathematical games with strings of symbols and precise rules.

Rules are of two basic kind :

rules of formation : how to generate well formed (i.e.admissible) strings
rules of transformation : how to produce new (well formed) strings from existing ones.

The following chapters deal with the typical formal systems of Math Log : Propositional Logic and First-Order Logic.

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