# Closed under equality

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?

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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! – coffeemath Oct 23 '12 at 12:34
weird, it's the fifth numbered item under the paragraph "The axioms". – pancake Oct 23 '12 at 12:47
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. – Dan Christensen Jun 2 '15 at 17:34

There are two ways to approach understanding this.

1. Closure refers to two different things: the idea that elements in a set are only related to other elements of the set (for relations), or are only mapped to other elements of the set (for functions of one or more arguments). Using the same word for different concepts is not too harmful if the meaning is clear from context. In fact, there are more meanings from topology and algebra: a set is closed under limits if all convergent sequences in the set have their limit also contained in the set (although "taking limits" isn't an operation or a relationship on elements of a set); and a field is algebraically closed if all polynomials over that field factor as a product of linear polynomials (which could be formulated as a relation from polynomials to sets of polynomials, but isn't usually). We think these ideas have something in common — a sort of family resemblance — so we give them similar names.

2. In the case of closure under an operation or under a relation, we can unify the concepts somewhat by recognising that functions are relations. A function $F: X \to Y$ is often described as a relation $F \subseteq X \times Y$ such that $$\forall x \in X \, \exists! y \in Y : (x,y) \in F$$ (where $\exists!$ denotes unique existence). We may say that a set $Y$ is closed under a relation $R$ between $X$ and $Y$, where $\mathcal P(Y) \subseteq X$ or $Y^n \subseteq X$ forsome $n \geqslant 0$, if every element of $X \cap \mathcal P(Y)$ or $X \cap Y^n$ as appropriate is related only to elements in $Y$. In the special case of binary relations such as equality, and binary operations such as addition — or considering the set of points in a converging sequence, for that matter — we then recover the usual meanings. (The "family resemblance" of the definitions might be capturable in a single, more general definition.)

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What is said about equality, is a (or in fact the) general property of identity: If $a=b$ then $\Phi(a)\iff\Phi(b)$ holds for any predicate (if one rejects strange things like "... is the first letter of the alphabet").

The term closure refers to operations, not equality. With natural numbers, you have the successor function, addition, multiplication, under which the set of naturals is closed (as opposed to subtraction).

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this is known as the identity of indiscernibles – Theta33 Mar 3 '13 at 1:39
If you're working in a logic without $=$, then such silly kinds of axioms can be necessary. – dfeuer Aug 19 '13 at 21:34