Proof that 1 + 1 = 2 Using Peano's PostulatesDate: 09/12/2002 at 11:38:08 From: Janrik Öberg Subject: How to prove that 1+1=2. How do I prove that 1 + 1 = 2? Could you please explain this, or is it too difficult? (For me to understand or for you to explain.) I've read a paper proving it, but unfortunately I only understand about half of it. -nb5 Date: 09/12/2002 at 12:25:33 From: Doctor Jerry Subject: Re: How to prove that 1+1=2. Hi Janrik, Peano's Postulates are: 1. Let S be a set such that for each element x of S there exists a unique element x' of S. 2. There is an element in S, we shall call it 1, such that for every element x of S, 1 is not equal to x'. 3. If x and y are elements of S such that x' = y', then x = y. 4. If M is any subset of S such that 1 is an element of M, and for every element x of M, the element x' is also an element of M, then M = S. Just as a matter of notation, we write 1' = 2, 2' = 3, etc. We define addition in S as follows: (a1) x + 1 = x' (a2) x + y' = (x + y)' The element x + y is called the sum of x and y. Now to prove that 1 + 1 = 2. From (a1), with x = 1, we see that 1 + 1 = 1' = 2. Standard properties of addition - for example, x + y = y + x for all x and y in S - can be proved by induction (which is based on Peano's Postulate #4. If the above proof seems too easy, we can try to show that 2 + 2 = 4. 2 + 2 = 2 + 1' = 3' = 4. - Doctor Jerry, The Math Forum http://mathforum.org/dr.math/ |
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