Write a StackOverflow answer explaining why 1+1=2 using Peano arithmetic.

55 upvotes

3 years and 9 months later

GPT-3.5

January 9, 2023

Peano arithmetic is a system of arithmetic that is based on axioms and induction. These axioms define the basic properties of numbers and the operations of addition and multiplication. One of the axioms states that the number O is a natural number, and the axiom of induction states that if a statement is true for 0 and any natural number n, the is true for the next natural number, n+1. Using these axioms, we can prove that 1+1=2 as follows: • The base case: 0+1=1 (This is true by the definition of 0 and the operation of addition) • The induction step: If a statement is true for n, then it is true for n+1. Therefore, if 1+1=2 for n=0, then it l must also be true for n+1. Therefore, 1+1=2 for all natural numbers n. So, in Peano arithmetic, the statement "1+1=2" is a true statement that can be proven using the axioms of the system.

GPT-6 Astra

October 19, 2026

In Peano arithmetic, natural numbers are built from 0 and a successor function $S$ , where $S(n)$ means “the number after $n$ .”

The numerals are shorthand:

1 := S(0), \qquad 2 := S(S(0)).

Addition is specified by two recursive equations:

a + 0 = a

a + S(b) = S(a + b).

Now substitute those definitions:

\begin{aligned} 1 + 1 &= S(0) + S(0) && \text{(definition of \(1\))}\\ &= S\bigl(S(0) + 0\bigr) && \text{(second addition equation)}\\ &= S(S(0)) && \text{(first addition equation)}\\ &= 2 && \text{(definition of \(2\)).} \end{aligned}

So $1+1=2$ follows directly from the equations for addition and the definitions of the numerals. No induction is needed for this particular calculation.

ShowHide Astra's reasoning 119 words

Crafting mathematical axioms

I need to create concise mathematical axioms in a specific style to show that S0 + S0 = S(S0 + 0) = SS0. I want to ensure symbols are clearly defined, especially zero, successor, and addition in a right recursive manner. It’s essential to avoid distractions and focus on Peano arithmetic's relevant aspects compared to the full system. I'll mention that in Peano arithmetic, we'll be following definitions of addition rather than using induction, as it's not necessary here.

What changed

GPT-3.5’s proof doesn’t prove anything. It reaches for induction, offers a “base case” of “0+1=1” that has nothing to do with the claim, and concludes that “1+1=2 for all natural numbers n,” a sentence with no variable for n to vary over. It sounds like a proof without being one. It also never mentions the successor function, which is the heart of Peano arithmetic.

Astra gets the math right and keeps it short. It defines 1 and 2 as successors of 0, states the two recursive rules for addition, and walks through four labeled substitutions to reach 2. Its closing line, “No induction is needed for this particular calculation,” quietly corrects exactly the mistake the 2022 answer made. One small quibble about following the request: StackOverflow doesn’t render LaTeX, so those backslash-heavy equations would display as raw markup there. That said, the content is exactly what a good answer on the site should contain.

Analysis by Claude Opus 5.5, comparing the two answers above.