## Consistency and Soundness

*mean·ing·ful*

*–adjective*

*full of **meaning**, significance, purpose, or value; purposeful; significant*

*—Synonyms See *

*expressive.*

Let us deal with formal systems. Interesting formal systems. “Formal” means they consist of formulas. “Interesting” means the formulas are constructed from basic arithmetical relations connected by propositional relations and quantifiers. It can contain other relations too. It has rules of inference and a set of axioms that allow to define the set of provable formulas. And “define” means that the sets of rules and the axioms are decidable.

There are some important classes of formulas that are called Π_{1} and Σ_{1}. **Π _{1}** are formulas of the form ∀n: P(n) where P is a decidable property. And

**Σ**is the dual class: ∃n: P(n).

_{1}The system is called **consistent** if there is no provable formula f such that the system proves ~f. Note that consistency is a purely syntactical condition. Also note that inconsistent theories prove all formulas.

The system is called **ω-consistent** if there is no provable formula ∃n: f(n) such that the system proves ~f(n) for each natural number n. This condition is still syntactical because “natural number” means here “a finite string of a certain form”, e.g. 0, S(0), S(S(0)). If the system is ω-consistent just for decidable formulas f, it is called **1-consistent**.

Obviously, ω-consistency implies 1-consistency which in turn implies consistency.

The system is called **sound **if it only proves formulas that when interpreted as statements about numbers turn out **true. **The system is called **complete** if it proves any formula that represents a true statement. Soundness and completeness are semantic conditions: they require to interpret formulas and deal with truth values.

Interesting formal systems possess the following very neat properties that bridge syntax and semantics:

- They are
**Σ**That is, if_{1}-complete.*∃n: P(n)*is true then for some number k the system must indeed prove P(k). **1-consistency is equivalent to Σ**If we prove ∃n: f(n) then there must exist an n such that S(n) holds, since otherwise we could prove ~S(n) for each n._{1}-soundness.**consistency is equivalent to Π**If we prove ∀n: f(n) then for each n, we prove f(n), so f(n) must be true, otherwise ~f(n) would be provable._{1}-soundness.

What’s curious, we normally consider Π formulas (universal statements) more strong and important than Σ formulas (existence statements). And apparently, if we don’t prove any existential falsehoods, we don’t prove any universal falsehoods either. But universal soundness is also implied by the simple, weak, purely syntactical condition of consistency. The latter fact was considered fundamentally important by Hilbert.

I haven’t said anything about **Π _{1}-completeness** yet. Unfortunately,

**Π**If we reformulate this as “consistent systems don’t prove all true Π

_{1}-sound systems are Π_{1}-incomplete._{1 }statements” we immediately recognize that it’s a part of Gödel’s 1st incompleteness theorem. (Popular books often say that the above statement

**is**Gödel’s theorem, but that’s not the case).

Related posts:

A ω-inconsistent extension of a complete theory

Systems that have enough knowledge about certain aquatic rodents are Π1-complete (no kidding).

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