Question

which phrase best describes a theorem in an axiomatic system?
a. the accepted meaning of a term
b. an assumption made with little supporting evidence
c. an accepted fact that cannot be proven
d. a statement proven to be true using logic
e. a list of geometric facts

Explanation:

To determine the correct answer, we analyze each option:

• Option A: The accepted meaning of a term is a "definition", not a theorem. So A is incorrect.
• Option B: An assumption with little evidence is a "conjecture" or "hypothesis", not a theorem. Theorems require proof. So B is incorrect.
• Option C: An accepted fact that cannot be proven is an "axiom" (or postulate), not a theorem. Theorems are proven. So C is incorrect.
• Option D: In an axiomatic system, a theorem is a statement that is proven to be true using logical reasoning (based on axioms, definitions, and previously proven theorems). This matches the definition of a theorem.
• Option E: A list of geometric facts is not a theorem. A theorem is a single proven statement. So E is incorrect.

Answer:

D. a statement proven to be true using logic