Understanding Equality in Mathematics: A Comprehensive Guide

[Dansuer]

I'm particularly interested in the foundation of mathematics. I've read various ways of defining the integers and addition. But I never encountered a formal definition of equality. It's seems (at least for what i read) that the equality is treated as something fundamental that does not need to be introduced. The only definition I found is that = is a relation that satisfies reflexive, transitive and symmetric properties.

I suspect that defining equality might be a lot harder than what I first thought. I'm asking for directions. Where should I look ? Does a definition even exist ? Could somebody suggest me a book that can clarify my ideas ?

Thank a lot !

 

[SteveL27]

I'm particularly interested in the foundation of mathematics. I've read various ways of defining the integers and addition. But I never encountered a formal definition of equality. It's seems (at least for what i read) that the equality is treated as something fundamental that does not need to be introduced. The only definition I found is that = is a relation that satisfies reflexive, transitive and symmetric properties.

I suspect that defining equality might be a lot harder than what I first thought. I'm asking for directions. Where should I look ? Does a definition even exist ? Could somebody suggest me a book that can clarify my ideas ?

Thank a lot !

This is a deep question!

The starting point is equality of sets. Two sets are equal if they have exactly the same elements. That definition goes a long way. But not all the way.

For example when we construct the natural numbers 1, 2, 3, ... we then use those to construct the real numbers. The real numbers contain a copy of the natural numbers, but the real number 3 is not the same set as the natural number 3. In this case we have to extend our notion of equality to the idea of isomorphism.

A mathematician named Barry Mazur wrote an essay about all this, well worth reading.

http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf

 

[chiro]

Hey Dansuer.

As you have hinted in your post, the definition is an equivalence relation:

http://en.wikipedia.org/wiki/Equivalence_relation

This relation will force a constraint on sets and what they can actually be for two things to have an equivalence relation.

 

[Dansuer]

This is a deep question!

The starting point is equality of sets. Two sets are equal if they have exactly the same elements. That definition goes a long way. But not all the way.

For example when we construct the natural numbers 1, 2, 3, ... we then use those to construct the real numbers. The real numbers contain a copy of the natural numbers, but the real number 3 is not the same set as the natural number 3. In this case we have to extend our notion of equality to the idea of isomorphism.

A mathematician named Barry Mazur wrote an essay about all this, well worth reading.

http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf

That was a very interesting reading. Although I'm looking for a more set theoretic answer.

Hey Dansuer.

As you have hinted in your post, the definition is an equivalence relation:

http://en.wikipedia.org/wiki/Equivalence_relation

This relation will force a constraint on sets and what they can actually be for two things to have an equivalence relation.

= is an equivalence relation, but which one ?
could be that = is the equivalence relation that partitions the set into singletons ?

for example = for the natural numbers would be the equivalence relation that partitions the natural numbers into {1},{2},{3},...

does that makes sense ?

 

[chiro]

It makes sense, but can a horse have no hairs at all?

 

[Dansuer]

what do you mean ?

 

[chiro]

I was just wondering if its possible for a horse to have no hair.

 

[Dansuer]

I guess it can.

 

[chiro]

Well it looks like you will have a tonne of equivalence classes that correspond to all the natural numbers plus 0.

 

[Dansuer]

oooh i see. all right.

 

[haael]

I like to think of equality as something different than identity.

Consider fractions. We have numerator and denominator. All fractions that are identical are also equal. But we also claim that some non-identical fractions are equal.

$$1/2 = 2/4$$

These fractions are not identical, but are equal. From set-theoretical perspective, a fraction can be identified with a pair of numbers. Then a fraction is a particular set. Equality relation is another set. Namely, set of pairs of all "equal" fractions.

Equality defines the theory. Consider a theory of, say, 5-tuples. Then consider two equivalence relations: one that says that the 5-tuples are equal if they have the same elements in the same order. The second one, that they have the same elements in any order. You get 2 different theories, despite the models of them are very similar.

Equality is not something fundamental. Identity is. Equality is defined by axioms using the notion of identity.