Using the Von Neumann representation of the non-negative integers, where the empty set corresponds to zero, and the successor function is defined as the function on a set that returns the union of the set with the set that contains the set, how do you define addition, subtraction, multiplication, and division?
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What's downvote-worthy about this question? – hmakholm left over Monica Feb 05 '15 at 19:58
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Just to share my motivation for asking the question, I've been reading "Structure and Interpretation of Computer Programs". The book mentions that it is possible to perform arithmetic with nothing more than function calls without ever declaring a number. I wanted to understand how the arithmetic works so that I could program a proof of concept. – Vivian River Feb 06 '15 at 02:57
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2Then you should probably look into Church numerals in the lambda calculus instead. That's a much more computationally flavored setting than set theory. – hmakholm left over Monica Feb 06 '15 at 11:38
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That's a great suggestion, Henning! I've already looked at Church numerals as well as Zermelo numerals, as well, so I know of at least three ways to "computer without numbers". I figured out yesterday that all you really need to do this is some definition of zero, an increment operation, and a check for equality, and I wouldn't be surprised if it turns out I can do with even less. – Vivian River Feb 06 '15 at 14:08
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Addition and multiplication are defined by recursion: if by $S(n)$ we denote successor of $n$, then:
$$a+0=a,a+S(b)=S(a+b)$$ $$a\cdot 0=0,a\cdot S(b)=a\cdot b+a$$
Note that in definition of multiplication we use earlier defined addition.
For subtraction, we define $a-b$ as a unique $c$ such that $a=c+b$ (if one exists). For division, $a/b$ is a unique $d$ such that $a=d\cdot b$.
It's worth noting that in terms of von Neumann natural numbers some subtractions and divisions are undefined, like $1-2,2/5,0/0$ (the last one because we require $d$ to be unique).
Wojowu
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I'm not following the definition you're given: a + 0 = a, a + S(b) = S(a + b). How would you use this definition to add 3 + 2? – Vivian River Feb 05 '15 at 20:28
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@DanielAllenLangdon You have to note that 2 is actually defined as S(1), and 1 is S(0). So you have 3+2=3+S(1)=S(3+1)=S(3+S(0))=S(S(3+0))=S(S(3))=S(4)=5. – Wojowu Feb 05 '15 at 20:44
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If I'm not mistaken, your definition implies the existence of a predecessor function as well. It is clear that 3 + 2 = 3 + S(1), but it stating this, I've implicitly said that 1 is the predecessor of 2. – Vivian River Feb 05 '15 at 22:08
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I know this is really a separate question, but in doing theoretical arithmetic with Von Neumann numbers, do we typically assume that we can compare two numbers for equality? – Vivian River Feb 05 '15 at 22:15
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@Daniel: Set theory is usually done in the context of classical logic, in which $x=y\lor x\ne y$ is a theorem for arbitrary mathematical objects. Thus in proofs we are indeed allowed to do case analysis on whether two sets are the same or not, and in particular this holds for the sets that represent numbers. But even if you do set theory in the more restricted setting of constructive logic, it will usually be possible to prove $x=y\lor x\ne y$ for all $x,y\in\omega$. – hmakholm left over Monica Feb 05 '15 at 23:22
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There are two main approaches.
In the first one, you start by proving that $\omega$ satisfies the (second order) Peano axioms, which allows you to prove that primitive recursive function definitions work. Then you can define addition and multiplication by the standard recursion equations from Peano Arithmetic:
$$ \begin{align} a+0 &= a \\ a+(b^+) &= (a+b)^+ \\ a\cdot 0 &= 0 \\ a\cdot(b^+) &= (a\cdot b)+a \end{align} $$
(Later on these definitions can be extended to full ordinal addition and multiplication by adding cases for limit ordinals).
The second approach is to develop a general theory of ordinals and well-orders, and then define ordinal arithmetic in general by saying:
$\alpha+\beta$ is the order type of $(\{0\}\times\alpha) \cup (\{1\}\times\beta)$ with the lexicographical order.
$\alpha\cdot\beta$ is the order type of $\beta\times\alpha$ with the lexicographical order.
(where $\times$ is the set-theoretic Cartesian products in both definitions).
You would then need to prove that these operations are closed on the finite ordinals -- that is, if $\alpha$ and $\beta$ are finite, then the lexicographical orders of the definitions are order isomorphic to a finite ordinal. That is not terribly difficult, however: both operations preserve the property that every element of the well-order is either the initial one or has an immediate predecessor, which one can show is equivalent to being finite.
In either case you would go on to define subtraction and division as the (partial) inverse operations to addition and multiplication.
hmakholm left over Monica
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Another approach that has not been mentioned is to use cardinality. Given a set $S$ and a von Neumann natural number $n$, we say that $S$ has cardinality $n$ iff there is a bijection between $S$ and $n$. You can prove that for any $S$, there is at most one $n$ such that $S$ has cardinality $n$; if such an $n$ exists, we say $S$ is finite. You can then prove that a union or cartesian product of two finite sets is finite. Finally, you can defined the sum $m+n$ as the cardinality of the set $m\times\{0\}\cup n\times\{1\}$ (i.e., a union of copies of $m$ and $n$ modified so that they are disjoint sets) and the product $m\cdot n$ as the cardinality of the set $m\times n$.
As the other answers have mentioned, subtraction and division are not always defined, but when they are defined they can be defined in terms of addition and multiplication.
Eric Wofsey
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