Godel and the limits of proof

A consistent, effectively axiomatized theory rich enough for arithmetic cannot settle every arithmetic statement.

The first incompleteness theorem establishes undecidable sentences for sufficiently strong formal theories under the relevant consistency assumptions. The second, under standard technical conditions, limits a theory's ability to prove its own consistency using its own resources.

For example: Adding an undecidable sentence as an axiom can expand a theory, but a new theory satisfying the same conditions still faces incompleteness.

Keep in mind: These theorems concern specified formal systems. They do not show that nothing is true, that every system is inconsistent, or that humans are automatically beyond computation.

Think about it

Where are you confusing 'not provable from these rules' with 'false' or 'unknowable by any method'?

Connected ideas

Infinity comes in sizes · Falsifiability and its limits · The map is not the territory

Sources and further reading

• Stanford Encyclopedia of Philosophy: Godel's Incompleteness Theorems

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