It seems obvious that 2 + 2 ≠ 5…

A rigorous proof that 2 + 2 ≠ 5 is not particularly trivial. In order to be concise, we need to begin with some definitions:

Which is seems more complicated than it is.

  • “∀” means “for all”.
  • “∈” means “element of” “belongs to” or “in”.
  • “ℕ” is a symbol for the natural numbers (0, 1, 2, 3 … etc.).
  • “¬” means “not”.
  • “⇔” means “if and only if”.

The only other part here that hasn’t been described is “S()”, the successor function. The successor of a number is the number after itself, for example, S(0) = 1, and S(3) = 4.

The axioms above in English for reference are as follows:

  1. 0 is a natural number
  2. For every natural number x, the number after x is also a natural number.
  3. For any two natural numbers x and y, if x = y, then S(x) = S(y) and if S(x) = S(y) then x = y
  4. 0 does not come after any natural number.
  5. Any natural number plus zero is itself
  6. For any two natural numbers x and y, x + S(y) = S(x + y)

This is sufficient to prove 2 + 2 ≠ 5. For brevity we have left out some axioms of equality.

We define 2 as S(S(0)) and 5 as S(S(S(S(S(0))))).

To prove that 2 + 2 ≠ 5, we need to prove ¬(2 + 2 = 5). ¬(2 + 2 = 5) is equivalent to 2 + 2 = 5 ⇒ ⊥ where “⇒” means “implies” and “⊥” means “falsehood” or “contradiction”. So if we can prove that 2 + 2 = 5 implies a contradiction or falsehood, then we have our proof.

We start with our definition of 2 and 5:

Then we apply our axioms of addition:

We use axiom 3 to reduce our problem by removing one layer of successor functions each time.

For which we already know by axiom 4 that ¬(S(0) = 0), or as we have seen before, S(0) = 0 ⇒ ⊥.

Therefore:

or:

Qed.

Originally published at andymac-2.github.io.

The stories I write are a part of a learning journey through life, logic and programming. Share this journey with me.

The stories I write are a part of a learning journey through life, logic and programming. Share this journey with me.