How do you prove that one plus one equals two?

Answer 1

Because if you have one apple and then take another and put it with that one apple you would then have two apples. Therefore 1 + 1 = 2 ANSWER 2- Well you could test it by going 2 - 1 = 1, but somethings in math you just have to go along with, it just works ANSWER NUMBER 3: Russell's & Whitehead's proof uses pure symbolic logic. It's pretty arcane, though, because first you have to come to a definition of what "1" means, what "2" means, and even what "+" & "=" means! It's a lot harder than you think without using circular definitions (eg "1" is when there is only one of something - this won't do as a definition).

The proof goes something like this (remember, it depends on set theory):

We start off by defining the natural numbers (ie positive integers) in terms of sets.

For any set S, define a "successor" function f as

f(S) = {S, {S}}

ie, the set containing: S and the set containing S.

Then we define the natural integers as such:

Define the number 0 to be the empty set, which I'll write here as O.

Then each successive integer (ie "n+1") is just the preceding integer put through the successor function, ie

1 = {O, {O}}

2 = {O, {O, {O

3 = {O, {O, {O, {O}

and so on.

Then 1+1=f(1)={O, {O, {O=2. QED.

By the way, from this definition of natural integers we can work out the entire system of arithmetic, including rational numbers, irrational & transcendental numbers, multiplication, division etc...