Thread: 1+1

In university, if the question ask:
Prove that 1+1=2.

How to prove?

Originally Posted by SengNee

In university, if the question ask: Prove that 1+1=2.
How to prove?

It depends upon your axiom set.

Originally Posted by SengNee

In university, if the question ask:
Prove that 1+1=2.

How to prove?

You need to first define "integer". That is what Plato said.

Having seen this before, if you were not given a set of axioms, then Peano's axioms may be helpful for this. They 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.

Originally Posted by galactus

Having seen this before, if you were not given a set of axioms, then Peano's axioms may be helpful for this. They are:

1. Let S be a set such that for each element x of S there exists a
unique element x' of S.

what are the properties of this element x'? how does it relate to x?

The way I have seen it done is thus:

As a matter of notation, we write 1' = 2, 2' = 3, etc. We define
addition in S as follows:





The element x + y is called the sum of x and y.

Now to prove that 1 + 1 = 2.

From 1: 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.

Originally Posted by SengNee

In university, if the question ask:
Prove that 1+1=2.

How to prove?

Alfred North Whitehead and Bertrand Russell wrote Principia Mathematica and published it in three volumes in the years 1910-1913. In it they laid the foundation of modern mathematics. On page 362, they finally got around to proving that 1 + 1 = 2.

On the other hand, 1 + 1 = 3 for 1 sufficiently large

Originally Posted by mr fantastic

Alfred North Whitehead and Bertrand Russell wrote Principia Mathematica and published it in three volumes in the years 1910-1913. In it they laid the foundation of modern mathematics. On page 362, they finally got around to proving that 1 + 1 = 2.

I remember reading that too. Most people do not realize how complicated simple addition is.

I still confuse. Anywhere, thanks.

Originally Posted by SengNee

I still confuse. Anywhere, thanks.

what axioms are you using? you have to tell us what you have at your disposal for us to properly help you

I only a student after SPM(only available in Malaysia, its standard is same as "O" level), waiting for result.
I post this because I hope to learn somethings extra.

Originally Posted by SengNee

I still confuse. Anywhere, thanks.

In the Peano system 1+1 is another name for the successor of 1 which
by convention is named 2, so 1+1=2 is saying nomore than 1+1=1+1.

RonL

Originally Posted by ThePerfectHacker

I remember reading that too. Most people do not realize how complicated simple addition is.

Most people don't, but they are assuming the basic axioms of mathematics.

Originally Posted by SengNee

I only a student after SPM(only available in Malaysia, its standard is same as "O" level), waiting for result.
I post this because I hope to learn somethings extra.

well, so far i think galactus' approach is the most rigorous, so you may want to note that. however, do give careful considerations to any other answer that was given

Thank you everyone.