How do we define equality in real numbers? How do we define equality in real numbers? I know in logic we define equality by Leibniz's law.
   $$ \forall x \forall y[x=y \rightarrow \forall P(Px \leftrightarrow Py)] $$
But how do we define the set of all $P$'s for a system. For example for fractions we say $1/2$ and $2/4$ are not equal but in rational numbers they are. But it seems to me the set of all propositions that make sense about fractions and that of rational numbers are the same.

This not a response as much as it is an observation regarding equality and it's meaning in mathematics that I feel is worth sharing
Consider a function described for an expression $\frac{a}{b}$ that returns $a$
Such a function would clearly differentiate between the two fractions you have presented.
Equality as we know really means "equivalent with respect to" and then a suitable definitions.
When examining an object such as a number we can ask questions about the numerical value represented or we can abstract and discuss about how this value itself is represented etc...
Just because two things are numerically equal does not mean they are textually equal.
And perfect equality is useless to us because it makes substation impossible. For example in solving $2+2-3$ the ability to say "2+2" and "4" are different yet "equivalent" in numerical value is important to us. 
