I saw how Russell and Whitehead come to "prove" (may be not, depending on views) that 1+1 = 2. So how does modern logic/set theory prove that 1+1=2? (Is it just that we derive it from Peano axioms, and just say that it is true by axiom..?)
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$\begingroup$Since I have answered this question in the PA context many times before (many=twice), as the links below can reveal, let me focus on set theory this time. First we need to be clear as to what $1$ and $2$ mean in modern set theory.
In the common von Neumann interpretation, $0=\varnothing; n+1=n\cup\{n\}$. So we have that $1=\{\varnothing\}$ and $2=\{\varnothing,\{\varnothing\}\}$.
Now what does $n+k$ equal? It equals the unique finite number $m$ such that there is a bijection between $m$ and the disjoint union of $n$ and $k$.
In this case it is trivial to see that $1+1=\{\varnothing\}\coprod\{\varnothing\}$ has two elements so we can write the bijection with $2$.
See also:
- How do I convince someone that $1+1=2$ may not necessarily be true?
- How would one be able to prove mathematically that $1+1 = 2$?
Here's an example of such proof:
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