# 2.1 Truth vs Provability

## 2.1 Truth vs Provability

Mathematical statements have two distinct statuses: they can be true, and they can be provable. These notions are related but not identical.

A statement is **true** when it holds in a given interpretation or structure. This is a semantic notion. It depends on meaning.

A statement is **provable** when it can be derived from axioms using inference rules. This is a syntactic notion. It depends on formal manipulation of symbols.

| Notion      | Domain    | Definition            |
| ----------- | --------- | --------------------- |
| Truth       | Semantics | Holds in a structure  |
| Provability | Syntax    | Derivable from axioms |

To make this precise, we fix a language and a structure.

Let `L` be a formal language. Let `M` be a structure interpreting that language. For a statement `φ`:

```text
M ⊨ φ      (φ is true in M)
⊢ φ        (φ is provable in the system)
```

These symbols separate meaning from derivation.

A basic example comes from arithmetic. Consider the statement:

```text
∀x (x + 0 = x)
```

In the standard structure of natural numbers, this statement is true. It is also provable from the usual axioms of arithmetic. In this case, truth and provability align.

This alignment does not always hold.

A system is **sound** if every provable statement is true in its intended interpretation.

```text
⊢ φ  ⇒  M ⊨ φ
```

Soundness ensures that proofs do not produce false statements.

A system is **complete** if every true statement is provable.

```text
M ⊨ φ  ⇒  ⊢ φ
```

Completeness ensures that all truths are accessible by proof.

These properties depend on the system and the domain.

| System                    | Sound | Complete |
| ------------------------- | ----- | -------- |
| Propositional logic       | Yes   | Yes      |
| First-order logic         | Yes   | Yes      |
| Arithmetic (rich systems) | Yes   | No       |

For propositional and first-order logic, completeness holds in a precise technical sense. For sufficiently expressive systems such as arithmetic, completeness fails. There exist true statements that cannot be proved within the system.

This limitation follows from results such as the incompleteness phenomena associated with Kurt Gödel. These results show that any consistent system capable of expressing basic arithmetic has statements that are true but unprovable within that system.

The gap between truth and provability can be illustrated by independence.

A statement `φ` is **independent** of a system if neither `φ` nor its negation can be proved:

```text
⊬ φ   and   ⊬ ¬φ
```

In such cases, the system does not decide the statement. Different extensions of the system may assign different truth values.

Examples appear in set theory. Certain propositions cannot be settled using the standard axioms alone. Additional axioms lead to different consistent theories.

The distinction between truth and provability also depends on interpretation. A statement may be true in one structure and false in another.

| Statement          | Structure           | Status |
| ------------------ | ------------------- | ------ |
| `x^2 + 1 = 0`      | Real numbers        | False  |
| `x^2 + 1 = 0`      | Complex numbers     | True   |
| Parallel postulate | Euclidean geometry  | True   |
| Parallel postulate | Hyperbolic geometry | False  |

Provability, by contrast, depends only on the chosen axioms and rules, not on a specific structure.

This leads to two complementary viewpoints.

| View      | Focus              |
| --------- | ------------------ |
| Semantic  | Meaning and models |
| Syntactic | Formal derivations |

A proof operates syntactically. It transforms statements according to rules. Truth is evaluated semantically. It checks whether the statement holds in a structure.

A well-designed system aligns these views as much as possible. Soundness ensures that proofs preserve truth. Completeness ensures that truth is reachable by proof. When completeness fails, one must distinguish carefully between what is true and what can be proved.

In practice, mathematicians often work inside a standard interpretation. For arithmetic, this is the natural numbers. For analysis, this is the real numbers. In these settings, truth has an intended meaning, even if formal systems cannot capture all of it.

A working guideline:

| Question                   | Interpretation              |
| -------------------------- | --------------------------- |
| Is the statement provable? | Depends on axioms and rules |
| Is the statement true?     | Depends on the structure    |
| Can the system decide it?  | Depends on completeness     |

When reading a theorem, it is useful to identify both aspects. A theorem usually asserts provability. Its significance often lies in the truth it expresses about a structure.

Mathematics progresses by refining both sides. It develops stronger systems to prove more statements, and richer structures to interpret them. The boundary between truth and provability marks the limits of formal reasoning.