Thread: How do you prove these really basic statements?

Hi. Out of curiousity, I was wondering how one goes about proving simple statements such as 1+1=2. I don't know a lot about the set theoretic formulation of natural numbers; only, that 0=ø, eccetera. Could you please tell me how addition is defined using this formulation of N?

Are you asking how one proves that "addition" works, or that the integer 2 can be represented by 1+1?

I would check out this entry on Addition; it is one of those "basic" ideas you just accept I suppose: when we "add" something we are combining one set of objects with another (I also like the "length" argument).

If your intent is to prove that 2 is equal to 1+1, you can just note that 2 is an even number, and all even numbers can be represented as 2k, where k is some integer. In this case your integer is 1 (1+1 => 2(1)).

Hopefully that answers, a little bit, your question - assuming I understood the question.

No, that dosen't work because you've pretty much assumed what you wanted to prove. IE 1+1=2(1), which is what you wanted to show in the first place. My question is about how you prove these simple statements, using set theory. Your solution requires a definition of addition, and a definition of multiplication.

In set theory, the natural numbers are constructed thusly:

0=ø ,1={0) , 2={0, 1} ,... I don't know how addition is defined using this construction.

Originally Posted by Chris11

No, that dosen't work because you've pretty much assumed what you wanted to prove. IE 1+1=2(1), which is what you wanted to show in the first place. My question is about how you prove these simple statements, using set theory. Your solution requires a definition of addition, and a definition of multiplication.

In set theory, the natural numbers are constructed thusly:

0=ø ,1={0) , 2={0, 1} ,... I don't know how addition is defined using this construction.

How intense are you trying to get into this? Look up Principia Mathematica

Originally Posted by Chris11

No, that dosen't work because you've pretty much assumed what you wanted to prove. IE 1+1=2(1), which is what you wanted to show in the first place. My question is about how you prove these simple statements, using set theory. Your solution requires a definition of addition, and a definition of multiplication.

In set theory, the natural numbers are constructed thusly:

0=ø ,1={0) , 2={0, 1} ,... I don't know how addition is defined using this construction.

I have not reviewed such proofs, but this proposition in Principia Mathematica by Russell and Whitehead is pretty famous.

Originally Posted by Chris11

No, that dosen't work because you've pretty much assumed what you wanted to prove. IE 1+1=2(1), which is what you wanted to show in the first place. My question is about how you prove these simple statements, using set theory. Your solution requires a definition of addition, and a definition of multiplication.
In set theory, the natural numbers are constructed thusly:
0=ø ,1={0) , 2={0, 1} ,... I don't know how addition is defined using this construction.

Here is a suggestion of a book.
An Outline of Set Theory by James M Henle.
It is written for a Moore style class, thus making it ideal for self-study.
He does a beautiful job of answering your questions.

Thanks Plato. I'll see if it's available at my school's library.

For this question, I don't think it is very helpful to refer to Principia Mathematica, whose (arguably, over) elaborate system (even unclear in certain respects) has been supplanted in common mathematical use by more streamlined first order systems such as Z set theory.

The question of proving "1+1=2" requires answering "Prove in what sense?" In a formal sense? Then in what particular system? First order Peano Arithmetic (PA)? Z set theory?

As for PA, '+' is not defined, but rather is primitive with axioms ('S' is a primitive 1-place function symbol, intuitively understood as 'successor of'):

n + 0 = n
n + S(k) = S(n + k)

From the axioms (and the definitions of '1' and '2'), a proof is trivial :

1 =df S(0)
2 =df S(1)

1+1 = S(0) + S(0)
= S(S(0) + 0)
= S(S(0))
= 2

As for Z set theory, usually the addition function is proven as a set by a definition by recursion theorem, which provides as theorems the set theoretic versions of the PA axioms, along with 'S' defined by:

S(n) = n union {n}

and in which case we get a proof essentially as in PA.

0 =df the empty set
1 =df S(0) = 0 union {0} = {0}
2 =df S(1) = 1 union {1} = {0 1}

So (looking at the sets themselves):

1+1 = {0} + {0}
= S({0} + 0)
= S({0})
= {0} union {{0}}
= {0 1}
= 2

P.S. I recommend for set theory texts:

'Elements Of Set Theory' - Enderton
'Axiomatic Set Theory' - Suppes

Those are both authoritative, in common use (which makes them suitable if one wishes to communicate readily with other people about the subject), well written, systematic, and comprehensive per a beginning level.

In combination (while making a few easy tweaks to the Suppes text), they're an excellent introduction to the subject.

Thanks for your reply MoeBlee. I'll see if my school has those books in its library. Also, if you could answer this quesiton: do you know if set theory is still an active research area?

Originally Posted by Chris11

do you know if set theory is still an active research area?

Indeed, it is active. Some of the old veterans are still at work along with new researchers in the subject. In fact, about a couple of years back was published is a tome 'Handbook Of Set Theory' that discusses some of the relatively recent results in the subject.