1
$\begingroup$

What are the conditions for a set $X$ to be considered a set of integers? I know there are isomorphisms and there is only one set of integers but what are the conditions? Equal with rational numbers.

  • 0
    I might be revealing my ignorance here, but wouldn't it be enough to say 1 is an integer, and any number that can be obtained by repeatedly adding or subtracting 1 starting from 1 is also an integer? – 2017-02-05

1 Answers 1

1

The set of integers is simply defined as the set of natural numbers along with their additive inverses. The set of natural numbers is defined by the following axioms described by Peano. Let S(x) be the successor function.

  1. For all natural numbers $0 \neq S(x)$
  2. If $S(x) = S(y)$ then $x = y$.
  3. $x + 0 = x$
  4. $x + S(y) = S(x + y)$
  5. $x*0 = 0$
  6. $x*S(y) = x*y + x$
  7. The axiom of induction which is kind of complicated to write.

The rationals are simply any number that can be written in the form of $\frac{p}{q}$ where $p,q$ are integers.