Relationship between axioms postulates and theorems

Euclid's geometry is built in layers, each resting on the one below it.

• Undefined terms — point, line, plane. Their meaning is taken as intuitively understood; Euclid does not (and cannot) define them using anything more basic.
• Definitions — descriptions of further objects built from the undefined terms, such as "line segment," "angle," or "circle."
• Axioms / postulates — statements assumed true without proof, the starting rules of the system. Euclid's general self-evident truths were his common notions (axioms); his geometry-specific starting assumptions were his five postulates. Both play the identical logical role: accepted, unproved starting points.
• Theorems — statements proved by a chain of logical deduction, using only definitions, axioms/postulates, and theorems already established earlier in the chain.

The relationship between them is one-directional: every theorem's proof must ultimately trace back to axioms and postulates. No step in a proof can be justified by "it looks true in the diagram" — it must be justified by an axiom, a postulate, a definition, or a theorem proved earlier. This is exactly what separates a mathematical proof from a plausible-looking picture. The dependency runs one way only, and never backwards:

Two standard statements from this chapter make the distinction concrete:

• Axiom (unproved): "Given two distinct points, there is one and only one line which passes through them."
• Theorem (must be proved): "Two distinct lines cannot have more than one point in common."

Notice the theorem can be proved using the axiom — this is the exact relationship the syllabus wants you to see and reproduce.

There is no formula to derive here — the "relationship" is a rule about logical structure, and it holds because of how proof itself is defined in an axiomatic system. A proof is a finite chain of statements, each one justified by something already accepted: a definition, an axiom/postulate, or a theorem proved in an earlier link of the chain. If you trace any single step of any valid proof backwards, it must land on one of these three sources — there is no fourth option. Diagrams may suggest a result, but a diagram is not on the list of allowed justifications, because a picture can mislead (a special-case drawing can hide a general-case exception) in a way a logical statement cannot. Follow that backward trace far enough, through however many earlier theorems it passes, and it must eventually bottom out at the axioms and postulates — because those are the only statements in the system accepted without a proof of their own. If it never bottomed out, the "proof" would be an infinite regress, never actually establishing anything; if it looped back on itself, it would be circular reasoning, not a proof. So the structure has to be: axioms/postulates sit at the base, unproved by design, and every theorem's chain of justification is finite and terminates there. This is precisely why axioms cannot themselves be proved within the system — proving one would require another accepted starting point, pushing the question back a level rather than answering it. The buck has to stop somewhere, and Euclid chose to stop it at a short, explicit list of common notions and postulates that felt self-evident.

Question: Prove the theorem "Two distinct lines cannot have more than one point in common," using the axiom "Given two distinct points, there is one and only one line through them."

Step 1. State what is given: two distinct lines, $\ell$ and $m$.

Step 2. Suppose, for contradiction, that $\ell$ and $m$ have two points in common, say $P$ and $Q$, with $P \neq Q$.

Step 3. By the axiom, since $P$ and $Q$ are distinct points, there is one and only one line passing through both of them.

Step 4. But both $\ell$ and $m$ pass through $P$ and $Q$ by the assumption in Step 2. Since the line through $P$ and $Q$ is unique, $\ell$ and $m$ must be the same line: $\ell = m$.

Step 5. This contradicts the given fact that $\ell$ and $m$ are distinct. So the assumption made in Step 2 must be false.

Answer: Two distinct lines cannot have more than one point in common — every step traces to the axiom, not to a picture.

Competitive stretch

Question: Classify each statement as an axiom/postulate (accepted without proof) or a theorem (needs proof), with a one-line reason: (a) "A circle can be drawn with any given centre and any given radius." (b) "If two lines intersect, the vertically opposite angles are equal."

Step 1. Statement (a) is exactly Euclid's Postulate 3, stated as a basic starting assumption about what constructions geometry allows — nothing more basic backs it up. Classification: postulate.

Step 2. Statement (b) feels visually obvious, but the syllabus (Lines and Angles) proves it using the linear-pair axiom (angles on a straight line sum to $180°$) plus algebra — it is derived, not assumed. Classification: theorem.

Answer: the test is never how "obvious" a claim looks; it is whether the statement is used as an unproved starting rule or derived through a logical chain from other accepted facts.

• Judging axiom versus theorem by how "obvious" a statement looks — obviousness is irrelevant; the only test is whether a logical proof is required.
• Thinking a theorem can be established by a diagram or by measurement alone — Euclid's method demands proof from definitions, axioms/postulates, and earlier theorems, never visual inspection.
• Believing axioms themselves need to be, or even can be, proved inside the system — they are the accepted base and are not derived from anything else within it.
• Forgetting that a theorem's proof may lean on previously proved theorems, not only on axioms directly — the justification chain can run several links long before it bottoms out.

Ask the tutor: Want to practise classifying a fresh batch of geometric statements as axioms versus theorems, or walk through another short proof-by-contradiction like the one above?