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 !