Gödel's Theorem by A. W. Moore

Author:A. W. Moore [Moore, A. W.]
Language: eng
Format: epub
ISBN: 9780192663580
Publisher: OUP Oxford
Published: 2022-09-12T00:00:00+00:00

An axiomatization can involve infinitely many axioms because, although the set of axioms has to be decidable, decidability doesn’t entail finitude.

Decidability doesn’t entail finitude. It nevertheless involves finitude—in as much as, when a set of statements is decidable, there is a finite account of what it takes for a statement to be a member of the set; or, equivalently, there is a finite description of the relevant algorithm. And indeed this is why a decidable set of axioms always corresponds to a single basic principle or to a finite set of basic principles, as illustrated in the example above. (This is the subtlety to which I referred in Chapter 1. It is because there is a corresponding basic principle or finite set of basic principles that the statements in a decidable set are fit to serve as axioms at all.)

Something else illustrated in the example above is the idea of a property that is expressible in a language for arithmetic. This leads to the next definition.

Arithmetical predicate: An arithmetical predicate is defined as an expression in a formal language that expresses a property of natural numbers.

To amplify. An arithmetical predicate Π in a formal language ℒ must contain, as well as symbols from ℒ, one or more occurrences of the ellipsis ‘_’. When each of these occurrences of ‘_’ is replaced by an expression e in ℒ that stands for a natural number, what results is a statement s in ℒ that has e as its grammatical subject. (Note that if there is more than one occurrence of ‘_’ in Π, then each of them is to be replaced by the same expression: the resultant statement s will then have a grammatical subject that occurs more than once in it.) Π can be thought of as picking out some property that any given natural number may or may not possess. It applies to any natural number that does possess the property; it fails to apply to any that does not. If it applies to the natural number that e stands for, then the statement s that results when each occurrence of ‘_’ in Π is replaced by e is true; if it fails to do so, then s is false.

Examples of arithmetical predicates (again, couched here not in their pristine form, but in a mixture of formal vocabulary and English, together with some familiar abbreviations) are:

• _ is prime;

• 10 < _ < 15;

• _ is the sum of four squares;

• _ < 2 _

(this was the example considered above);

• _ is odd and _ is divisible by 4;

• .

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