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Surprisingly, the Wikipedia article on addition doesn't contain the answer. I looked elsewhere online for it, but didn't find it.

Intuitively, the cardinal of the union of two sets seemed appealing. But that approach doesn't work, given that the union of 2 and 2 is 2. I wondered whether ordered sets might be essential to the answer.

So, how does ZFC describe addition?

Hal
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    Addition of what kinds of objects? – Cameron Buie Feb 12 '14 at 15:35
  • Natural numbers – Hal Feb 12 '14 at 15:37
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    he probably means ordinal addition – Kaa1el Feb 12 '14 at 15:37
  • I have answered this very question at least once on the site. This is not your first duplicate, either. Please try and search the site before posting questions (and not just Wikipedia!). – Asaf Karagila Feb 12 '14 at 15:39
  • Yeah, I always look through the first two pages of search results. I didn't see it. I also don't know many terms yet, so I may have missed a question that used different words to ask what I asked. – Hal Feb 12 '14 at 15:42
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    See at list the post on Set theory and addition – Mauro ALLEGRANZA Feb 12 '14 at 15:51
  • Hal, maybe it'd be better if you sit with a book, which has a structure leading from one theorem to another, rather than just "try to come up with the next thing that you don't understand, and ask Wikipedia/here". Doing so is counterproductive for actually generating knowledge or actual understanding. – Asaf Karagila Feb 12 '14 at 15:58
  • I'm not sure if I understand your request. How can one study foundation of mathematics without knowing some sufficient amount of mathematics? Do you know any good books on quantum mechanics which assume that the reader doesn't know any physics? – Asaf Karagila Feb 12 '14 at 16:24
  • @AsafKaragila I am trying to learn math from the axioms up. Most people learn it through the standard curriculum in high-school, and then university - where ZFC isn't taught until the upper years. Accordingly, the set theory books I dipped into relied on analogies to abstract algebra and advanced calculus. But ZFC builds up to those concepts, so they can't be necessary to understand ZFC. – Hal Feb 12 '14 at 16:27
  • Yes. Because mathematics shouldn't be taught from the axioms up. It'd be missing the whole points both of the mathematics and the foundational beauty in constructing mathematics using the axioms. This is like deciding that you want to understand human behavior by understanding quantum mechanics. Sure, ultimately one can find some relation, but it would be missing the entire point. – Asaf Karagila Feb 12 '14 at 16:31
  • @AsafKaragila what is the point? – Hal Feb 12 '14 at 16:33
  • A good book to start with ? Kenneth Kunen, The Foundations of Mathematics (2009); from Introduction : "It is assumed that the reader knows basic undergraduate mathematics. Specifically: You should feel comfortable thinking about abstract mathematical structures such as groups and fields. You should also know the basics of calculus [...]. You should also know the basics of logic [...]." You can supplement it with Béla Bajnok, An Invitation to Abstract Mathematics (2013), and John Stillwell, The Real Numbers An Introduction to Set Theory and Analysis (2013). – Mauro ALLEGRANZA Feb 12 '14 at 16:39
  • If you are interested also to an historical perspective on "foundations", see : Morris Kline, Mathematics The Loss of Certainty (1980), Marcus Giaquinto, The Search for Certainty: a philosophical account of foundations of mathematics (2002). – Mauro ALLEGRANZA Feb 12 '14 at 16:46
  • Hal, fine. Here's a different analogy. Learning to appreciate and criticize art by studying the physics of photons and wavelengths and their relation to color. The point is that in order to explain foundation, you need to understand the thing you are trying to express in set theory. At least a little bit. – Asaf Karagila Feb 12 '14 at 16:58
  • @MauroALLEGRANZA thank you for the recommendations. I appreciate them. – Hal Feb 12 '14 at 17:18
  • @AsafKaragila Right I completely agree. But for now and some time to come, I will need to understand only relations, functions, and arithmetic. Learning calculus and abstract algebra seems like a major detour to get to that end. Also, there's the very practical matter that I learn best when I know how things fit together and how we arrive at them. Everything in my mind is organized in a tree, which branches from a set of basic assumptions (that's not just for math). So I say that to say, I don't think working from the foundations is an undesirable way to understand what I want to understand. – Hal Feb 12 '14 at 17:22

3 Answers3

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Define $n++$ by $n++=n\cup\{n\}$, called the successor of $n$.

Define recursively,

$n+0=n$

$n+(m+1)=(n+m)++$

Kaa1el
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    While n++ works great for some programming language this is one of the ugliest, most horrifying, and un-mathematical ways to write the successor function. – Asaf Karagila Feb 12 '14 at 15:46
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There are a few ways we can go about it, which ultimately amount to the same operation.

One is as the cardinal of the disjoint union of sets, which avoids the problem of union that you mentioned--and is simply cardinal addition, restricted to finite cardinals. Another is as the order type of the disjoint union, where the elements of the first set are supposed to be less than all the elements of the second, and the sets themselves keep the same orders--this is the non-recursive definition of ordinal addition. Yet another is to proceed recursively--for any ordinal $\alpha,$ we have $S(\alpha):=\alpha\cup\{\alpha\},$ and for any natural numbers $n,m,$ we define:

  • $n+0:=n$
  • $n+S(m):=S(n+m)$

Regardless, we end up with an addition operation having all the usual properties that we are used to.

Cameron Buie
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It is defined recursively by $$ n + 0 = n\\ n + (a + 1) = (n + a) + 1 $$ so to figure out $2+3$ (where $2 = 1 + 1$ and $3 = 2 + 1$), you do: $$ 2 + 3 = 2 + (2 + 1) = (2 + 2) + 1 = (2 + (1 + 1)) + 1 = \\((2 + 1) + 1) + 1 = (3 + 1) + 1 = 4 + 1 = 5 $$

Arthur
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