What is the Gödel sentence G, constructed in the proof of the First Incompleteness Theorem?
AA sentence that is both true and false in the standard model
BA sentence that encodes 'this sentence is not provable in T'
CA sentence whose truth requires the axiom of choice
DA sentence that is true in some models of PA and false in others
The Gödel sentence G is constructed via the diagonal lemma to assert its own unprovability: G says 'I am not provable in T.' If T proved G, then G would be false (it claimed to be unprovable), making T inconsistent. If T proved ¬G, then T would assert G is provable, but then G is provable — contradiction. In a consistent T, G is undecidable. Moreover, G is true in the standard model since T genuinely cannot prove it.
Question 2 True / False
Gödel's incompleteness theorems demonstrate that mathematics is fundamentally inconsistent.
TTrue
FFalse
Answer: False
The incompleteness theorems assume consistency — the First Theorem says *if* T is consistent, then there exists an undecidable sentence; the Second says *if* T is consistent, T cannot prove Con(T). They show no single consistent formal system can prove all arithmetic truths, not that mathematics is contradictory. Mathematics is not identified with any one formal system, and informal mathematical reasoning may exceed what any particular formal theory can capture.
Question 3 Short Answer
What does the Second Incompleteness Theorem say, and why did it collapse Hilbert's program?
Think about your answer, then reveal below.
Model answer: The Second Incompleteness Theorem states that a consistent system T extending PA cannot prove its own consistency (the sentence Con(T) is unprovable in T). This collapsed Hilbert's program because Hilbert sought to secure mathematics by proving the consistency of formal systems from within those systems — which the theorem shows is impossible.
Hilbert's program aimed to find a complete, consistent formal system that could verify its own consistency from the inside. The Second Incompleteness Theorem shows the consistency verification requirement is unachievable: any system strong enough to represent basic arithmetic cannot prove its own consistency without importing assumptions from a stronger theory. This doesn't make mathematics unreliable — it means foundational certainty cannot be fully bootstrapped from within.