How would you prove that 1 plus 1 equals 2?

Answer 1

If you don't want to use complex, dazzling higher math theories, you should go about it like this:

-Show a single finger of your choice on one of your hands.

-Ask the inquiring person: 'How many is this?'. The answer should be 'one'.

-Show a single finger of your choice on your other hand, and ask the same question, receiving the same answer.

-Now show two hands and two finger simultaneously. Ask: 'How many is this?'. Now, the answer should be 'two'.

-Congratulate yourself on crushing your opponent with this flawless logic.

The above isn't a proof of course, but it illustrates that we consider the idea of addition to be self evident. Mathematicians prefer to define the integers and addition from a set-theoretic construction. I'll outline it for you.

First a few definitions:

Informally, a set is a collection of things called elements. Elements in a set can be almost anything you like, including other sets. On restriction is that sets cannot include themselves.

There is a set with no elements, called the null set. The null set has a special symbol of a 0 with a slash through it. Since I can't represent that properly here I'll use 0 .

We need the idea of a mapping between (non-null) sets. Informally a mapping is just a way of saying which elements of one set go with an element of another set. If you draw a picture of two sets and draw lines from every element of the first set to an element of the second, then the set of lines is the mapping--it just answer the question of what in the first set goes with what in the second. Note that every element of the first set is connected one element of the second set, but it's possible that not every element of the second set is connected to an element of the second set.

For example, "owns" is a mapping from the set of toaster owners (a sub-set of the set of people) to the set of toasters. Some toaster enthusiasts may own multiple toasters (such a mapping is called one-to-many), and some toasters may be owned by multiple people (such a mapping is many-to-one).

If every toaster is owned, this mapping is called onto. Technically a mapping from one set to another is onto if every element it the second set is the mapping of at leastone element from the first set.

A mapping is called a function if every element of the first set is mapped to exactly one element of the second set.

A function is a bijection if it is also a one-to-one and onto mapping.

Some examples:

1. Let UPPER be the mapping from the lowercase letters to the uppercase letters.
2. 1. UPPER is one-to-one since every uppercase letter is mapped by at most one lowercase letter.
   2. UPPER is onto since every uppercase letter is mapped by at least one lowercase letter.
   3. UPPER is a function since lowercase letter maps to exactly one uppercase letter.
   4. The function is a bijection since it is one-to-one and onto.
   1. IS_VOWEL is a function, since the mapping can be made for every letter and gives exactly one answer for each letter. (Note that sometimes is a distinct answer from true and false.)
   2. IS_VOWEL is onto since there is at least one letter that maps to each element in the answer set.
   3. IS_VOWEL is not one-to-one since some letters map to the same answer.
   4. IS_VOWEL is not a bijection since it is not one-to-one.
3. Let IS_VOWEL be a mapping from the letters (ignoring case for this example) to the answer set {true, false, sometimes}. A, E, I, O and U, map to true, Y maps to sometimes, and the rest map to false.
   1. Squaring is a function, since every integer has a square.
   2. Squaring is NOT one-to-one, since, for example, -22 = 4, and 22 = 4.
   3. Squaring is NOT onto, since, for example there is no integer whose square is 3.
4. Integer squaring is a mapping from the set of integers (positive and negative) into the set of non-negative integers.

Now we can develop a notion of "size" of a set. We can say that two sets have the same cardinality if there is a bijection from one set to the other. For example, the UPPER bijection given above says that the cardinality of the lowercase letters and uppercase letters is the same.

As notation, we can use |A| to mean the cardinality of set A.

Now we can define the integers in terms of cardinality:

0 = | 0 | (cardinality of the null set)

1 = |{ 0 }| (cardinality of the set containing the null set)

2 = |{ 0 , { 0 }}|

3 = |{ 0 , { 0 }, { 0 , { 0 |

...

As notation, we'll name these sets N0, N1, N2, N3, ....

Notice that (informally) the 0 set has no elements, the 1 set has 1 element, the 2 set has 2, elements, etc.

Each set is constructed as the successor of the previous set using

Ni+1 = Ni U {Ni}

where U is the traditional union operator. This says that the successor of Ni is constructed by taking all the elements of Ni and adding Ni itself. So |Ni+1| > |Ni|, in the sense that Ni+1 contains everything in Ni and something else.

Now that we've defined the integers, we just need to define addition.

Let Ni* be the set constructed by taking each element of Ni and paring it with the * symbol. So for example, N2* = {{ 0 ,*}, {{ 0 },*}}.

Clearly |Ni*| = |Ni|, by the bijection n -> {n,*}. However, no element of any of the N*s is in any of the Ns.

So we come to the crux of the matter:

For integers i and j, define

i + j ## |Ni U Nj*|

Here, "##" means "is defined as" since the typical symbol of 3 horizontal bars is unavailable.

So to show, for example, that 2 + 3 = 5, we need to show that |N2 U N3*| = |N5| we need to find a bijection between N2 U N3* = N5.

N2 ={

0 ,

{ 0 }

}

N3* = {

{ 0 ,*},

{{ 0 },*},

{{ 0 , { 0 }},*}

}

N5 = {

0 ,

{ 0 },

{ 0 , { 0 }},

{ 0 , { 0 }, { 0 , { 0 ,

{ 0 ,{ 0 },{ 0 , { 0 }}, { 0 , { 0 }, { 0 , { 0 },

}

The bijection is

0 -> 0

{ 0 } -> { 0 }

{ 0 , { 0 }}, -> { 0 ,*}

{ 0 , { 0 }, { 0 , { 0 -> {{ 0 },*}

{ 0 ,{ 0 },{ 0 , { 0 }}, { 0 , { 0 }, { 0 , { 0 } -> {{ 0 , { 0 }},*}

This can be generalized by an appropriate transformation of the elements in the N*, so that an appropriate bijection is created for arbitrary integers, but that's more than I'm willing to do, and probably more than you're willing to read!

Finally, to answer the original question:

1 + 1 = |N1 U N1*| = |N2| = 2

With the bijection

0 -> 0

{ 0 , *} -> { 0 }

This is how integer arithmetic can be construction purely from set theory.