# 1+1=2

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Addition is an axiom of mathematics, or am I wrong?

I heard that the Peano Axioms can prove 1+1=2, but I don't really know why or how. Can any of you more advanced students dechiper this?

(I'm only a freshman in high school, remember, and it's frustrating to not be educated higher in math.)

So, if anyone can relate this to me, I'd be grateful.

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please let me also know if you can

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2 is, by definition, the result of adding 1 to 1 when we're dealing with numbers as abstract objects.

Peano's axioms give a way of 'concretely' realizing numbers as a collection of objects labelled 1,2,3,... and an operation, 'addition', such that adding two objects labelled 1 gives the object labelled 2, and further that all the other properties of addition hold. Let me use [n] to mean the object labelled by n, then you need to check that in the Peano system that doing ([n]+[m])+[p] is the same as doing [n]+([m]+[p]), and that [n]+[m]=[m]+[n]. I will say that [n] is a collection of sets of sets (of sets of sets) and addition is something like taking the union, so you do need to check that these constructions behave properly. I don't propose to give it here since it is notationally heavy, I'll only get it wrong, and there are plenty of other places where it is explained properly.

Thus we're saying that there is something relatively 'concrete' that fits our abstract set of rules for the natural numbers. Some people find this to be reassuring.

There are results about the natural numbers as an abstract object that cannot be deduced from the Peano Axioms alone (try googling mathworld peano to find out more on this).

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Does this, in short term, prove that 1 + 1 = 2, then?

S(x) + S(x) = S(S(x))

S = Successor ; x = Natural Number.

x is 1 ; S of 1 is 2.

Thus, 1 + 1 = 2 because S(x) + S(x) = S(S(x))?

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Define the plus of S(x)+S(x)

But I would say no, since what you've written is, with the 'best interpretation' saying that 2+2=3.

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Or in general you appear to be saying that x+1+x+1=x+2

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• 2 months later...

you guys just confuse me, why do people want to know really why 1 + 1 = 2?

I think, because if you have one thing have another one, now you have two of the one thing, or basically two thing.

1 + 1 = 2 (1) = 2

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And what is a 'thing'? how do you knowthat if you put two things together you always get two?

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Have a look at this - it may add to what has already been stated here

Cheers,

Ryan Jones

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Addition is an axiom of mathematics' date=' or am I wrong?

I heard that the Peano Axioms can prove 1+1=2, but I don't really know why or how. Can any of you more advanced students dechiper this?

(I'm only a freshman in high school, remember, and it's frustrating to not be educated higher in math.)

So, if anyone can relate this to me, I'd be grateful.[/quote']

Hello, God here. Most of you don't know me. I'm that omnipotent entity to whom many of you attribute the creation of the Universe. There have been a lot of misconceptions regarding me and my work and I'm here today to set the record straight once and for all.

Before I created the universe we didn't have numbers like you do today. When nothing existed nowhere, the only figure that came to mind was "Ø", but after my seven day project, I had some serious counting to do. I decided to call each instance of existence "one" when considered alone, but when a pair of them was considered together; I thought it would be nice to call them "two". Of course numbers are a lot like potato chips, once you start, well - you know - so when a pair and a singleton were considered together, I called them "three" - and so on. Now remember, I made an infinity of them, so it took me quite some time to create names for them all - and I'm afraid I became somewhat repetitive. And then, of course, I had to have a negative number for every positive one so that the entire set of numbers added together still equaled nothing. When I finally got through, there was no way you could count the entire set. Of course I could because I'm omnipotent. And I tried to once - just for grins. I must have counted for what seemed to be eternity before I got distracted and lost my place. That's where I got the concept of Hell from . . . but I digress.

1+1=2

Well Done !!!

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And what is a 'thing'? how do you knowthat if you put two things together you always get two?

try it yourself....

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Ok, I have a drop of water, I add another drop of water, and oh look they coalesce and I have one drop of water.... perhaps 1+1 is not 2 after all..... never mind the other far more complex philsophical issues at stake here either. (Try reading Russell.)

And how do you prove that even if you define 'thing' properly that addition *always* behaves properly? Remember 'well, it is cos it is' is not a *proof* in mathematics. You either need to accept something as an axiom, or try to prove it if at all possible in your model, or disprove it, or whatever you wish. Whilst no one would argue that 'of course 1+1=2' that is not a proof. And some people feel the need to prove it from simpler axioms. Look at Peano Arithmetic for a firm axiomatic description of the natural number system.

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