Was your math professor a Discrete Mathematics professor?
I will attempt to prove that 1+1=2 with math induction. I didn’t invent this, some other crazy math whiz did.
The problem is, depending on how complex the number system assigned, the proof gets more complicated. That is why with real numbers it cannot be proven effectively.
You also have to assume that the addition procedure works the way it does, the proper way to prove the addition procedure would be to prove it equal to some other easily defined term (such as binary addition, an even simpler form of addition) and then show it equal in all cases, which I think is possible but is beyond my scope and knowledge of math. You asked me to prove 1 + 1 = 2, not the addition procedure so…
Anyways, Let us assume that 1+1=2, in the universe of discourse of all whole numbers, 1 to infinite is true.
To prove this right, I need to prove this for all possible cases. The problem is, there is an infinite number of possible cases where 1 thing gets added to one of another. The logical contrapositive equivalent is ~2= ~(1 + 1)
This is easier to prove, not by much, since it’s still infinite, but it can be done.
If any number but two will not be 1 + 1
Base Case:
1 = 1 + 0, which is not 1 + 1, check
3 = 1 + 1 + 1, which is not 1 + 1 check.
Let N be a number greater than 2 within the universe of discourse.
Let N’s base be the number 3
I’ve already shown the base case to be true,
so N = 1 + 1 + 1
So we know from the addition procedure that 3 = 1 + 1 + 1.
Now take the statement N + 1 = 1 + 1 + 1 + 1
If we subtract 1 from both sides we get N = 1 + 1 + 1, our base step.
In other words, If N is true, N + 1 is ALSO true. This is the inductive step.
Thus any whole number from 3 all the way to infinite will NOT be 1 + 1.
The contrapositive is always true, thus the statement 1 + 1 is always true.
Something like that. The whole proof would be some long drawn out thing…like this wasn’t drawn out enough.