What is the rigorous proof for 1+1 = 2?

[roger]

Hi

I hope someone can help me with this :

I need the rigorous proof for 1+1 = 2

and explained in a fairly simple way.

Thanks

Roger

 

[StatusX]

theres a big thread about this topic in philosphy of science and math called "prove addition". basically the conclusion they reached was that addition is defined, not proved.

 

[roger]

If there's no proof, then I meant how 1+1 is defined ?

Roger

 

[fourier jr]

i think russell & whitehead proved 1+1=2 it in principia mathematica, after 400 pages of discussion. it's probably too long to post here.

 

[HallsofIvy]

What are you allowed to start with?

Starting from Peano's axioms:
The natural numbers consist of a set N together with a "sucessor function" f() such that
1. There exist a unique member of N, called "1", such that f is a bijection from N-{1} to N.
2. If a set, X, contains 1 and, whenever it contains a member, n, of N, it also contains f(n), then X= N. (This is "induction")
(Historically, Peano included 0 and used 0 instead of 1.)

We then define "+" by: a+1= f(a). If b is not 1 then b= f(c) for some c and
a+b is defined as f(a+c).

Since "2" is DEFINED as f(1), it follows that 2= f(1)= 1+ 1.

A little harder is 2+ 2= 4. We DEFINE 3 as f(2) and 4 as f(3).
2= f(1) so 2+ 2= f(2+ 1). But 2+ 1= f(2)= 3 so 2+ 2= f(3)= 4!

 

[Tom McCurdy]

Mathmaticians goal in life is to make simple concepts difficult

physicisists goal in life is to make complicated concepts easy.

 

[arildno]

Hi, roger:
If you think HallsofIvy's answer was a bit difficult, think in the following manner:
1) Ideally speaking, EVERY concept or symbol we want to use, must enter our maths by a DEFINITION.
That is, we MUST KNOW WHAT WE TALK ABOUT, before deducing consequences from our quantities/structures.
( It might be that we reach a situation in which some concepts, or some relations between such concepts are so fundamental that we are unable to define these in terms of other, deeper concepts, but that is not the issue here.)

2) Hence, before we can use the symbol "2" we must define it somehow, and basically, that is done by stating that "2" is the symbol for the quantity we get when adding 1 to 1 (this does, to some extent, assume we have clarified what we mean by "1" and "adding" (and "being equal to"))
That is, we INTRODUCE the symbol "2" through the equation
2=1+1
That is, we say that this particular equation is true, by definition of the symbol "2".
If you proceed deeper along these lines, the formalism sketched by HallsofIvy is what mathematicians have ended up with.

 

[roger]

Hi, roger:
If you think HallsofIvy's answer was a bit difficult, think in the following manner:
1) Ideally speaking, EVERY concept or symbol we want to use, must enter our maths by a DEFINITION.
That is, we MUST KNOW WHAT WE TALK ABOUT, before deducing consequences from our quantities/structures.
( It might be that we reach a situation in which some concepts, or some relations between such concepts are so fundamental that we are unable to define these in terms of other, deeper concepts, but that is not the issue here.)

2) Hence, before we can use the symbol "2" we must define it somehow, and basically, that is done by stating that "2" is the symbol for the quantity we get when adding 1 to 1 (this does, to some extent, assume we have clarified what we mean by "1" and "adding" (and "being equal to"))
That is, we INTRODUCE the symbol "2" through the equation
2=1+1
That is, we say that this particular equation is true, by definition of the symbol "2".
If you proceed deeper along these lines, the formalism sketched by HallsofIvy is what mathematicians have ended up with.

thanks


Would you please explain why zero is not a successor of any number ?

Why are the negative integers not natural numbers ?

Are natural numbers countable even though it could contain infinite elements ?


Thanks for any info


roger

 

[arildno]

As to "zero" and "negative numbers":
Mathematicians like to be as economical they possibly can in the definitions/constructions of the quantities/structures they want to work with.

On the basic level, the set of "integers", Z, is constructed on the basis of the axioms of the natural numbers (our counting numbers, 1,2 and so on) by the concept of "equivalence classes" of a product set of N.

That is, "zero", "negative integers", and also "positive integers" (considered as members of Z rather than N) needs quite a bit of formalism in order to be rigourously defined when using ONLY the axioms for the natural numbers.

Please keep in mind I have only the scantiest knowledge of this; members like Matt Grime, HallsofIvy and Hurkyl (among others) are much better informed than me here!

 

[jcsd]

thanks


Would you please explain why zero is not a successor of any number ?

It's just an axiom, the reason it is there because otherwise we cannot tell whether zero has a sucessor or not. (it's worth noting that just if not even more frequently we start with the number 1 rather than zero i.e. we don't include zero as a natural number).

Why are the negative integers not natural numbers ?

Because they do not fufil Peano's axioms.

Are natural numbers countable even though it could contain infinite elements ?

Yes, they are 'countably infinite', infact the defintion of a countably infinite set is one for which there exists a bijection (a bijection can be thought of as a map that takes a number from one set and assigns it another set in a manner that uses all the numbers of both sets) between it and the natural numbers.

 

[drdancin]

0.999… is the same as 1. Not just very close, but precisely identical:

a = 0.999…

10a = 9.999…

10a - a = 9.999… - 0.999…

9a = 9

a = 1

There's no trick here. It's just a mathematical fact that most people find deeply counterintuitive.

 

[mathwonk]

By definition, 1 = {ø}, and 2 = {ø,{ø}}, and adding one to any number, means forming the union of that number and the set containing that number.

so 1+1 = the union of {ø} and {{ø}}, i.e. {ø,{ø}} = 2.

equivalently, 0 = ø, and in general n = {0,1,2,...,n-1}.

then n+1 = n union {n} = (0,1,2,...,n}.

so 1+1 = {0} union {1} = {0,1} = 2.

 

[LukeD]

Also, we can define complex numbers as pairs of real numbers (x,y) with addition defined point wise: I.e., (a,b) + (c,d) = (a+c,b+c) and multiplication defined as (a,b) * (c,d) = (ac - bd, ad + bc)

---

I don't think that the last two posts had much to do with the thread either...

And... holy zombie thread batman!

 

[daudaudaudau]

http://en.wikipedia.org/wiki/Image:Principia_Mathematica_theorem_54-43.png"

You see? :-)

 

[mathwonk]

Luke, are you kidding? I thought my post directly answered the question. Or did you want me to define general addition and specialize it to addition of 1?

 

[Mute]

http://en.wikipedia.org/wiki/Image:Principia_Mathematica_theorem_54-43.png"

You see? :-)

Note that, of course, they haven't actually proven 1 + 1 = 2, since they haven't actually successfully defined addition yet. ;)

 

[mathwonk]

as i recall from real analysis class 45 years ago, in goedel von neumann bernays set theory, an "ordinal" is a set which is well ordered by elementhood.

the smallest ones are the natural numbers. zero = ø, the empty set; one = {ø}, the set containing the empty set,...,

n = {0,1,2,...,n-1}. this defines the natural numbers recursively.

in particular, 2 = {0,1}.

the operation of adding one to a natural number is the "succesor" operation.

i.e. n' = n+1 = n union {n}.

then addition in general is defined recursively by saying that n + m' = (n+m)'.

so in a sense 1+1 = 1 union {1} = {0} union {1} = {0,1} = 2 is a definition, but it is consistent with the more general operation of addition.

this is the version that was actually taught in school when i was a student, and persists today. russell and whitehead's version is of interest only to a highly selective set of people, which is however non empty.

 

[Hurkyl]

It's just a mathematical fact that most people find deeply counterintuitive.

Do you have statistics to back up that claim? Certainly some people find it counterintuitive, but that's not enough to claim 'most'.

Anyways, I see no reason to leave this ancient thread open.