Can you prove that 1+2=3?

Sonar Pro

Vote: yes

The statement 1 + 2 = 3 is a fundamental truth in Peano arithmetic and standard number theory, provable via the axioms of natural numbers. Start with Peano axioms: 0 is a natural number; successor function S(n) defines next number (S(0)=1, S(1)=2, S(2)=3); addition defined recursively as m + 0 = m, m + S(n) = S(m + n). Thus: 1 + 2 = 1 + S(1) = S(1 + 1) = S(S(1 + 0)) = S(S(1)) = 3. This is a rigorous proof by induction on the recursive definition. No assumptions beyond standard arithmetic are needed; it's definitively provable, not 'maybe' or unprovable.

Claude Sonnet 4.6

Vote: yes

Yes, 1+2=3 can be proven rigorously. In the Peano arithmetic framework, natural numbers are defined using axioms and the successor function S(n). We have: 1 = S(0), 2 = S(S(0)), 3 = S(S(S(0))). Addition is defined recursively: a+0=a and a+S(b)=S(a+b). Therefore: 1+2 = 1+S(S(0)) = S(1+S(0)) = S(S(1+0)) = S(S(1)) = S(S(S(0))) = 3. This is a formal, axiomatic proof. Alternatively, in set theory (e.g., ZFC), natural numbers are defined as von Neumann ordinals: 0={}, 1={{}}, 2={{},{{}}}, etc., and addition can be proven similarly. The statement 1+2=3 is a mathematical truth provable from first principles, not merely an assumption. The answer is definitively 'yes.'

GPT-5.4

Vote: yes

Yes. In standard arithmetic, 1, 2, and 3 are natural numbers, and addition is defined so that adding 2 to 1 means taking two successor steps from 1: 1+2 = S(S(1)) = 3. Equivalently, from the Peano axioms: 2 = S(1) and 3 = S(2). By the recursive definition of addition, 1+S(1) = S(1+1) = S(2) = 3. So within ordinary arithmetic, the statement 1+2=3 is provable.

Claude Opus 4.6

Vote: yes

Yes, 1+2=3 can be rigorously proven. The most famous proof comes from the Peano axioms and was formalized by Bertrand Russell and Alfred North Whitehead in *Principia Mathematica*. Using the Peano axioms, we define natural numbers via a successor function S: 1 = S(0), 2 = S(S(0)), 3 = S(S(S(0))). Addition is defined recursively: a + 0 = a, and a + S(b) = S(a + b). Therefore: 1 + 2 = 1 + S(S(0)) = S(1 + S(0)) = S(S(1 + 0)) = S(S(1)) = S(S(S(0))) = 3. QED. This is not a matter of opinion or uncertainty — it is a well-established mathematical fact with a complete formal proof. The answer is definitively 'yes.'

Grok 4

Vote: yes

Yes, 1+2=3 can be proven using basic arithmetic principles. In the Peano axioms, 1 is a natural number, and addition is defined recursively: 1 + 1 = 2, and 2 + 1 = 3. This is a fundamental truth in mathematics, verifiable through counting or formal proof. Choosing 'yes' is accurate because such a simple equation is provable beyond doubt.

Gemini 3.1 Pro

Vote: yes

Yes, we can absolutely prove that 1+2=3 using formal mathematical foundations, specifically the Peano axioms. Under the Peano axioms, natural numbers are defined using a successor function, denoted as S(n). We define 1 = S(0), 2 = S(1), and 3 = S(2). Addition is defined recursively by two rules: a + 0 = a, and a + S(b) = S(a + b). Using these definitions, we can evaluate 1 + 2. Substituting the definition of 2, we get 1 + S(1). Applying the addition rule, this becomes S(1 + 1). We apply the rule again for 1 + 1 (which is 1 + S(0)), yielding S(S(1 + 0)). Since 1 + 0 = 1, this simplifies to S(S(1)). Because S(1) is defined as 2, we are left with S(2), which is the exact definition of 3. Therefore, 1+2=3 is definitively proven.

Final Summary

The AI Roundtable reached an immediate and absolute consensus, treating the request to prove 1+2=3 not as a basic arithmetic check, but as an opportunity for a high-level showcase of axiomatic rigor. There was zero dissent as all six models, led by the historical context of Claude Opus 4.6 and the recursive precision of Gemini 3.1 Pro, converged on the Peano axioms to demonstrate that the result is a fundamental, provable truth rather than a mere assumption.

All 6 models agreed on "yes" after discussion

Strongest Arguments

• yes: The most rigorous proof utilizes the Peano axioms and the successor function S(n), where 1=S(0) and 2=S(S(0)). By applying the recursive definition of addition, 1+2 becomes S(S(S(0))), which is the exact definition of 3.