Our school teacher just asked us to write the best possible proof of $1+1=2$. First thought he was joking but he was very serious about this question.
I could not even try an attempt.
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2You can't prove a definition. – MathematicsStudent1122 Oct 13 '16 at 14:07
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See Principia Mathematica (where Russell and Whitehead proved $1+1=2$ in an axiomatic set-theoretical context). – Parcly Taxel Oct 13 '16 at 14:09
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Arguably the Principia Mathematica has the most thorough proof of 1+1=2. – Matt Oct 13 '16 at 14:10
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What is the solid ground you're allowed to assume? How are $1$ and $2$ defined? I mean, assume that $1+1=2$ does not hold and show that your favorite arithmetic theorem doesn't either. – aphorisme Oct 13 '16 at 14:10
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Thnx Matt and Parcly just saw that and will show the same to my teacher. – Sanmukh Oct 13 '16 at 14:13
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3You need definitions first: how is defined '$1$'...? how is defined '$2$'...? How is defined '$+$'...? and how is defined '$=$'? Possibly the proof will follow directly from those definitions. – CiaPan Oct 13 '16 at 14:13
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The notorious version is Theorem $*54\cdot 43$ of Whitehead & Russell's Principia Mathematica p. $360$ which then says "From this proposition it will follow, when arithmetical addition has been defined, that $1+1=2$." In modern notation, you can find the proof at http://us.metamath.org/mpegif/pm54.43.html – Henry Oct 13 '16 at 14:44
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2See this question (and maybe some of the posts linked there and perhaps also this question. – Martin Sleziak Oct 14 '16 at 05:10
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You have to define what system you are working in. In Peano Arithmetic, PA, $1$ and $2$ are not part of the language. They are abbreviations for $S(0)$ and $S(S(0))$, where $S$ is intended as the successor function, so you are being asked to prove $S(0)+S(0)=S(S(0))$. You can follow the Wikipedia proof, which you may need to update a bit depending on how your version of the axioms are written. Basically this should look a lot like your axioms that define addition.
Ross Millikan
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