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I recently stumbled upon this interesting definition of mathematics:

Math is the study of things that can be described as sets.

I am aware that the integers and the real numbers can be defined in terms of sets, but how can we define the operations of addition and subtraction?

hb20007
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  • Addition and subtraction where? – Git Gud Nov 09 '14 at 18:16
  • Natural numbers are finite ordinals and for all ordinals we define addition as the order type of the set that we get when we take the disjoint union of the sets and order all elements in each component by the old order and everything in the left summand is less than anything in the right summand. This gives the usual addition on the natural numbers. To give but one example. – Henno Brandsma Nov 09 '14 at 18:16
  • @GitGud: Well, since I mentioned them, addition and subtraction in the real and integer number systems. – hb20007 Nov 09 '14 at 18:18
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    http://math.stackexchange.com/a/1009721/119110 – George Chen Nov 11 '14 at 09:51
  • @GeorgeChen Thanks, ur answer to that question in the link is relevant and helpful – hb20007 Nov 11 '14 at 09:59
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    https://archive.org/details/settheorystruct00hami – Floridus Floridi Nov 26 '20 at 23:01

2 Answers2

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You can use cardinals of sets to construct the natural numbers and prove the Peano axioms. Also you can use the ordinals defining $0:=\emptyset$ and $n^+:=n\cup \{n\}$, and so prove the Peano axioms again.

To construct the integers from natural numbers, you can use the cartesian product to define integers $(a,b)$ and $(c,d)$ such that two integers are equal if $a+d=c+b$.

The topic is technical so you need to read a book about set theory abd foundations. The Wikipedia has good references.

Cristhian Gz
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Just to give an idea ( but I'm far from being an authority).

You may have a look at Peterseon, Theory Of Arithmetic.

  • One may say that : $0$ is ( by definition) the same object as $\emptyset$.

  • Define $S(n)$ ( the successor of number $n$) as the set : $n \cup \{n\}$.

So, for example, $S(0) = S(\emptyset)= \emptyset\cup\{\emptyset\}=\{\emptyset\}$.

Call this number: " number 1". Hence : $1 =\{\emptyset\} =\{0\}$ by definition.

Note : continuing the process , you will get this ( at first sight) " strange" result that every natural number is the set of all its predecessors.

  • Then state two rules in order to define addition as a function taking as input any couple of natural numbers ( that is, any element of the cartesian product $N\times N$ ) and sending back as output an element of $N$ (meaning that addition is an operation on $N$):

(1) $n+O=n$ ( $n$ being any natural number)

(2) $n+ S(m) = S (n+m)$ ( $n$ and $m$ being any natural numbers).

Note : the rules mean that, $a+b= a$ if $a = O$ and that $a+b= S(a+ $ predecessor of $b$) if $b\neq 0$.

  • So for axample:

$3+2 = 3+S(1) = S(3+1) = S (3+S(0))= S S( 3+0) = SS(3) =S(4)= 5$.

( 5 being " the successor of 4" by definition).

  • As to substraction, onece you have defined the integers ( using an equivalence relation on the set of couples of natural numbers) , you can say that :

$n-m = n+(-m)$.

Floridus Floridi
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