1
$\begingroup$

In the Wikipedia article on "Peano axioms" I read this (source):

For all $a$ and $b$, if $a$ is a natural number and $a = b$, then $b$ is also a natural number. That is, the natural numbers are closed under equality.

Seems legit (to a computer scientist such as me, anyway, maybe not to a mathematician). But then in the Wikipedia article on "Closure" I read this (source):

A set has closure under an operation if performance of that operation on members of the set always produces a member of the same set.

I think the two above statements contradict each other. The first statement basically says: if an element of a set is operated on by a certain operator, the operator will not allow for elements of a different set to be the other operand, this set is considered closed under that operation. But the second statement says that the result of that operation is supposed to be of the same type (for lack of a better term). So one natural number added to another produces another natural number, so the natural numbers are closed under addition. But the equality operator produces a Boolean/truth value, not a natural number.

Question: What's the deal with equality and closure? I take it that the natural numbers are closed under equality, but how exactly? Which of the above statements is flawed?

  • 0
    I didn't find your statement "if a is a natural number and a=b, then b is a natural number" anywhere in the wiki article. Maybe you could copy the whole area around where you found the statement, so someone could find it. In my opinion, such a statement should *not* appear (as you have quoted it) in any serious article about Peano arithmetic. The statement is really redundant!2012-10-23
  • 0
    weird, it's the fifth numbered item under the paragraph "The axioms".2012-10-23
  • 0
    The axioms you are looking at were probably from Peano's original paper in Latin. The axiom in question seems redundant to us today. The most commonly used modern version of Peano's axioms does not given the axioms of equality. They define only 0 (or 1), the successor function and sometimes addition and multiplication. The axioms of equality are now subsumed in the rules and axioms of logic.2015-06-02

2 Answers 2