A good post.

Incidentally, there are topics in Natural Number and Associative Property in this context.

]]>We start by defining the sequence of natural numbers (1):

1, 2, 3, ... s,t, ...

Here, s, and t represent two consecutive natural numbers.

So far in our presentation, the natural numbers are only symbols.

We define their values by introducing the value equation, which defines uniquely the value of every natural number except for 1 in terms of the value of 1:

for every natural number s: t = s + 1

The natural number s should go through the natural numbers sequentially from 1 so that the right-hand side of the value equation is defined for every value of s.

To make the definition of the value equation complete, we should define the +-operator.

However, we choose instead to introduce an axiom of the natural numbers, the associativity axiom (2):

for every natural number a, b, and c: a + (b + c) = (a + b) + c

We are now in a position to prove:

Theorem:

2 + 2 = 4

Proof:

We operate on the left-hand side of the equation 2 + 2 = 4:

2 + 2

= 2 + (2) as a parenthesis can be put around a singular natural number

= 2 + (1 + 1) as 2 = 1 + 1 by the value equation

= (2 + 1) + 1 by the associativity axiom

= (3) + 1 as 3 = 2 + 1 by the value equation

= 3 + 1 as a parenthesis around a singular natural number can be removed

= 4 as 4 = 3 + 1 by the value equation

We see from this that we have proved that 2 + 2 = 4.

This is what we intended to prove, so the proof is complete.

References:

1 Natural number https://en.wikipedia.org/wiki/Natural_number

2 Associative Laws https://www.mathsisfun.com/associative-commutative-distributive.html