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I think I’m taking this thread way to seriously, but here I go anyway…
First, we need to clarify what we actually mean by “1” and “2.” These are numbers denoted in the decimal numeral system. Here’s a quick summary of how it works for natural numbers:
– The natural number refered to as “1” in the Peano axioms is denoted as “1”.
– To get the notation of the natural number x’ (the successor of x, in some sources S(x)), in the notation of x replace the rightmost digit by the next digit. (All digits in order: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) If the digit is a “9”, replace it by a “0” and also replace the digit to the left of the digit just replaced in the way just described (assuming a “0” where there are no more digits).
With this definition, we now know that the number denoted as “2” is actually 1’ (and obvioulsy, “1” refers to 1).
Now we just have to find out what 1 + 1 means. “+” is defined for natural numbers like this:
(1) For all natural numbers x, x + 1 = x’.
(2) For all natural numbers x and y, (x + y)’ = x + (y’).
It has to be and can be shown that this indeed defines a unique operation.
Using (1) from the definition of “+”, we get 1 + 1 = 1’. And with the knowledge from before that 1’ can be denoted as 2, we come to the conclusion that indeed 1 + 1 = 2.
