ω-consistent theory

In mathematical logic, a theory is called ω-consistent (or omega-consistent) if it cannot prove an existence statement, if it can refute all concrete instances of this statement.

definition

Let T be a theory that interprets arithmetic, which means that every natural number n can be assigned a term of the language, which in the following is denoted by. T is called ω-consistent if there is no formula such that both   and for every natural number n can be proved. Formally: ${\displaystyle {\dot {n}}}$ ${\displaystyle \phi (x)}$ ${\displaystyle \exists x\phi (x)}$ ${\displaystyle \neg \phi ({\dot {n}})}$

${\displaystyle {\text{T ist }}\omega {\text{-konsistent }}\leftrightarrow {\text{ Es gibt keine Formel }}\phi (x){\text{, so dass T}}\vdash \exists x\phi (x){\text{ und für jede natürliche Zahl n: T}}\vdash \neg \phi ({\dot {n}})}$

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

Relationship to other principles of consistency

If a theory T can be axiomatized recursively , then, according to a result by C. Smoryński, the ω-consistency can be characterized as follows:

T is ω-consistent if and only if is consistent. ${\displaystyle T+\mathrm {RFN} _{T}+\mathrm {Th} _{\Pi _{2}^{0}}(\mathbb {N} )}$

Here denotes the set of all Π ⁰ ₂ sentences which are valid in the standard model of arithmetic . is the uniform reflection principle for T, which is derived from the axioms ${\displaystyle \mathrm {Th} _{\Pi _{2}^{0}}(\mathbb {N} )}$ ${\displaystyle \mathrm {RFN} _{T}}$

${\displaystyle \forall x\,(\mathrm {Prov} _{T}(\varphi ({\dot {x}}))\to \varphi (x))}$

for every formula with a free variable. ${\displaystyle \varphi }$

In particular, a finite axiomatizable theory T in the language of arithmetic is ω-consistent if and only if T + PA -correct. ${\displaystyle \Sigma _{2}^{0}}$

example

Let PA denote the theory of Peano arithmetic and Con (PA) be the arithmetic statement which formalizes the assertion PA is consistently . Usually Con (PA) will be of the following form:

For any natural number n: n is not the Gödel number of a proof of 0 = 1 in PA (i.e. there is no proof of the contradiction 0 = 1)

On the basis of Gödel's incompleteness theorem we know that if PA is consistent, PA + ¬Con (PA) must also be consistent. However, PA + ¬Con (PA) is not ω-consistent for the following reason: For every natural number n, PA already proves that n is not the Gödel number of a proof of 0 = 1, so PA + ¬Con (PA) certainly proves this too. However, ¬Con (PA) also proves that there is a natural number m, so that m is the Godel number of a proof of 0 = 1 (which is precisely the statement ¬Con (PA) itself).

Individual evidence

1. ↑ Craig Smoryński: Self-reference and modal logic , in: The Journal of Symbolic Logic, 53: 1 (1988), pp. 306-309. Springer, Berlin 1985.