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Everyone knows that 1+1=2, but how would one mathematically prove that this equation is true? Or can you?

  • A naïve answer would be to take the sum and the RHS modulo $1$; however, a "number" $\mathfrak{a}$ modulo any number $n$ is in fact an equivalence class $$\mathfrak{a}=[a]_{n}={b: a\sim_n b},$$ where $\sim_n$ is the equivalence relation defined by $a\sim_n b\iff a=b+kn$ for some $k\in \Bbb Z$. Then $$[\color{blue}{1}]_1\color{green}{+_1}[\color{red}{1}]_1\color{green}{:=}[\color{blue}{1}+\color{red}{1}]_1=[0]_1=[3]_1$$ because $1+1=0+(2\times 1)$ and $3=0+(3\times 1)$. –  Jan 22 '20 at 03:56
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    It all depends on what axioms you start with. The Peano axioms are a fairly standard way to axiomatize addition for natural numbers, and with those you can prove that 1+1=2 fairly easily. – Bram28 Jan 22 '20 at 03:56
  • @Bram28 Ok, but how? –  Jan 22 '20 at 03:56
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    Welcome to Mathematics Stack Exchange. Somewhat infamously, several hundred pages of *Principia Mathematica" precede the proof of the validity of the proposition $1+1=2$. – J. W. Tanner Jan 22 '20 at 03:56
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    This has been discussed here and here, you may be interested in reading what's in those links. – dxdydz Jan 22 '20 at 03:57
  • @everyone My account is scheduled for deletion –  Jan 22 '20 at 03:59
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    I don't understand the downvotes here. This is a perfectly good topic, and shows someone who is asking the right questions. – Michael Biro Jan 22 '20 at 04:01
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    Among the suggested reasons to downvote: "This question does not show any research effort; it is unclear or not useful." Arguably, this question fits all of those criteria. Had the op searched, they would have found several posts on this site and elsewhere already, including the (in)famous book alluded to elsewhere. It is unclear because the OP has not explained what they already know on the topic and what axiomatic system they are using or how they define the symbols $1,+,2$ and $=$, most sensible definitions of which lead to an immediate answer – JMoravitz Jan 22 '20 at 04:04
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    In my mind, this is simply the definition of the number $2$. – littleO Jan 22 '20 at 04:05
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    @JMoravitz have you tried searching yourself? I searched "1 + 1 = 2" and there was nothing relevant on the first page. "Proof 1 + 1 = 2" doesn't have it either. "Prove 1 + 1 = 2" has a some results, but none of the titles are quite analagous to "How do we prove 1 + 1 = 2?" – Michael Biro Jan 22 '20 at 04:11
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    As a user continues typing a question, the system analyzes the text and will populate a "Related" list, the top choice among them currently being How would one be able to prove mathematically that $1+1=2$. Looking at the list before pressing the submit button would have prevented this question from being posted as it is clearly a duplicate, both in spirit and content. – JMoravitz Jan 22 '20 at 04:14
  • As for searching, typing in 1+1=2 proof into google, the entire first page is relevant to the question being asked, including some links to stackexchange duplicates as well as a link to the wikipedia page on the aforementioned book – JMoravitz Jan 22 '20 at 04:17
  • @JulianTiemann You can find a proof based on the Peano axioms here: https://math.stackexchange.com/questions/2364039/is-11-2-a-logical-truth/2364060#2364060] – Bram28 Jan 22 '20 at 21:13

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You can! Math is axiomatic, so the exact proof would depend on which axioms you choose (some might choose $1 + 1 = 2$ 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 $1 + 1 = 2$ on page 379.

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Michael Biro
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  • People always mention Principia Mathematica, and it always sounds horrifying. But I have heard that nowadays there are much more efficient ways to develop the foundations. So perhaps we shouldn't make too much of the fact that the proof of $1+1=2$ appeared on p. 379. – littleO Jan 22 '20 at 04:08