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The main reason that it takes so long to get to 1 + 1 = 2 1 + 1 = 2 is that Principia Mathematica starts from almost nothing, and works its way up in very tiny, incremental steps. The work of G. Peano shows that it's not hard to produce a useful set of axioms that can prove 1+1=2 much more easily than Whitehead and Russell do.
If you define that the symbols $1+1=2$, then you implicitly wrote an axiom which connects the symbols, and proves that $1+1=2$ is a true sentence. Often, however, we use the word "theorem" for statements whose proofs are not trivial. In this case, if you define $2$ as $1+1$ then this is not a theorem, this is a definition.
A reasonable proof in ZFC would be to prove 1 + 1 = 2 for the corresponding ordinal numbers. The first few ordinal numbers in ZFC are 0:={}, 1:={0} and 2:={0, 1} with the order 0 < 1 on {0, 1}. The sum of two ordinal numbers is the disjunct union of the two well-ordered sets, with the concatenation of the well-orders as the well-order for the sum.
0. You can! Math is axiomatic, so the exact proof would depend on which axioms you choose (some might choose 1 + 1 = 2 1 + 1 = 2 as an axiom itself, instead of a theorem). Whitehead and Russell's Principia Mathematica was an attempt to prove all of math from first principles. They famously got to showing that 1 + 1 = 2 1 + 1 = 2 on page 379.
You have to define what system you are working in. In Peano Arithmetic, PA, 1 1 and 2 2 are not part of the language. They are abbreviations for S(0) S (0) and S(S(0)) S (S (0)), where S S is intended as the successor function, so you are being asked to prove S(0) + S(0) = S(S(0)) S (0) + S (0) = S (S (0)). You can follow the Wikipedia proof ...
In your proof the symbols $\rm\:x,y\:$ denote abstract numbers, so let's specialize them to concrete ...
So, we have a proof in some system of the statement 1 + 1 != 2. Philosophers in the subject of logic, and mathematicians, would look closely at the details of this proof. Since all formal systems that anyone is interested in prove the opposite of this statement, also proving this statement demonstrates that whatever system was used, is ...
The claim that Russell and Whitehead "spent 362 pages to prove 1+1=2" is misleading in another way, since it suggests not only that the proof on page 362 relies in a logical sense on everything that precedes it (which as I said appears to be false), but also that proving this claim is the goal of all the preceding developments.
A friend showed me this proof: Proof: 2 = 1. Let x = y L e t x = y. Multiply both sides by x: x2 = xy x 2 = x y. Subtract y2 y 2 from both sides: x2 −y2 = xy −y2 x 2 − y 2 = x y − y 2. Factor: (x + y)(x − y) = y(x − y) (x + y) (x − y) = y (x − y)
$\begingroup$ This question was popularised by Einstein .This was discussed while developing a formal logical language for computers by Bertrnd Russel ,Whitehead and others.The Truth table used in Computer logic -> [ 0+0 =0 ;1 +0 =1 ;0+1 =1 ; 1+1 =1 ] , uses 1 +1 =1 When a circuit is switched ON ,it has state ' one' --when it is switched OFF , it has a state ' zero ' .