Ω-consistent theory

In mathematical logic, a theory is referred to as ω - consistent ( or omega- consistent) if it can not prove the existence theorem if they can refute all concrete instances of this statement.

Definition

T is a theory which interprets the arithmetic, this means that any natural number n, a term of the language to be recognized, will be referred to with the. T is called ω - consistent if there is no formula, so that both n and for every natural number is provable. Formal:

An ω - consistent theory is consistent automatically, conversely, there are consistent theories that are not ω - consistent, see example.

Relationship to other consistency principles

Is a recursively axiomatizable theory T, then one can characterize the ω - consistency according to a result of C. Smoryński as follows:

Here, the set of all Π02 sentences, which are valid in the standard model of arithmetic. is the uniform reflection principle for T, which consists of the axioms

Exists for every formula with a free variable.

In particular, a finitely axiomatizable theory T in the language of arithmetic if and only if T is ω - consistent PA correctly.

Example

Denote by PA the theory of Peano arithmetic and Con (PA ) is the one arithmetic statement that is consistent formalizes the assertion PA. Most Con (PA ) will be of the following form:

On the basis of Gödel's incompleteness theorem, we know that if PA is consistent, also PA ¬ Con (PA ) must be consistent. However, PA ¬ Con (PA ) is not ω - consistent because of the following: For every natural number n already proves PA that n is the Gödel number of a proof of 0 = 1 is not, thus proving PA ¬ Con (PA ) this safe as well. However ¬ Con proves (PA ) also that there exists a natural number m such that m is the Gödel number of a proof of 0 = 1 ( which is in fact just the statement ¬ Con (PA ) itself ).