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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..?)

hwe
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    Under what context are we defining $1$, $2$, and $+$? – yearning4pi Feb 18 '13 at 08:51
  • See this answer and this other answer. – Asaf Karagila Feb 18 '13 at 08:57
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    I find the combination of your questions quite strange. First you ask about relative constructibility and admissible ordinals. Then you ask about how to prove that $1+1=2$, I'd imagine that the answer to this would be apparent to anyone who solved the first exercise about defining addition for ordinals (which in turn tells you how to prove this in PA). Can you provide some context as for where these questions are coming from? – Asaf Karagila Feb 18 '13 at 09:01

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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:

  1. How do I convince someone that $1+1=2$ may not necessarily be true?
  2. How would one be able to prove mathematically that $1+1 = 2$?
Community
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Asaf Karagila
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Here's an example of such proof: http://tachyos.org/godel/1+1=2.html

Ludolila
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