Rational numbers and leibniz law

Question

Leibniz law says $a = b \implies f(a) = f(b)$. Unfortunately this law seems to fail for rational numbers e.g. ${1 \over 2} = {2 \over 4}$ but $numerator({1 \over 2}) \neq numerator({2 \over 4})$. I know that you can say $1 \over 2$ is just representation of "true" rational number and equality we use is just equivalence relation not real equality but this representation is how we think about rational numbers and how we define them in abstract algebra.

Question is: Is there some logically precise (e.g. first order logic) and "true" definition of natural number that respects Leibniz law and makes ${1 \over 2}$ and ${2 \over 4}$ truly equal not just equivalent.

Edit:

To rephrase the question: how do I define rational numbers to avoid problems with representation. In other words is there some axiomatic definition of rationals. Up to this point I've only seen algebraic definitions with pairs and equivalence classes.

Answer 1

You are confusing the object with its various names.

As an object, $\frac12=\frac24$. But you can present it differently. More specifically, in order for the numerator function to be well-defined, you need to choose a representation for each rational first.

Similarly, $1+1=2$, but the length of these two expressions is different. Names are syntax, objects are semantics. Leibniz's law is about semantics, not syntax.

Answer 2

Your problem is that you are separating the number $\frac12$ into a pair of numbers $(1,2)$. Your law would then still be true, since $(1,2)\neq(2,4)$ and there would be no reason to suppose that $f((1,2))=f((2,4))$.

As numbers, $\frac12$ and $\frac24$ are precisely the same number. Your function $\operatorname{numerator}(x)$ as you use it is not well-defined since it depends on the particular representation of the (rational) number $x$.

Answer 3

The law you're quoting holds only when $f$ is a function.

What you're calling "$\mathit{numerator}(\cdots)$" is not a function, because it depends on something external to which number its input is -- namely on how you've chosen to represent that number.

(Arguably the law is not really a deep thing, but rather part of the definition of what it means to be a function).

---

Axiomatically, you can define the rational numbers as a field of characteristic 0 which has no proper subfields. You can prove that any two such fields are uniquely isomorphic (so it doesn't matter which of them you choose to call $\mathbb Q$), and the equivalence-classes-of-pairs construction from textbooks shows you that believing in set theory is sufficient to know that there are fields with this property.