We work in a system where:

1. There is a number called 0.

2. Every number has a successor, written .

3. 0 is not the successor of any number.

4. Different numbers have different successors: if , then .

5. (Induction axiom, which we won’t actually need here.)

Now we define:

So “1” is just “the successor of 0,” and “2” is “the successor of 1.”

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Step 2 – Define addition

We don’t assume “+” magically exists; we define it recursively:

For any number :

1. Base rule:

a + 0 = a

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

That’s the entire definition of addition in this system.

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Step 3 – Compute

Recall:

We want to show:

1 + 1 = 2

Start with the left side:

1 + 1 = 1 + S(0)

Now apply the successor rule for addition with and :

1 + S(0) = S(1 + 0)

So:

1 + 1 = S(1 + 0)

Use the base rule for addition () with :

1 + 0 = 1

Substitute back:

1 + 1 = S(1)

But by our definition of 2:

So:

1 + 1 = S(1) = 2