# Using definitions instead of axioms.

Lets take (classical) first-order logic for granted, including an equality symbol and its associated axioms.

Given all this, a rigorous work of mathematics will typically begin with a signature - that is, a collection of symbols. For example, the symbols $\in$ for set theory, or the symbols $0,1,+,\times,<$ for arithmetic.

Lets just stick with set theory as our motivating example, so our starting symbol is $\in$.

The next thing is to list axioms which will be assumed true for the remainder of the work. For example, in set theory a common axiom is:

Axiom of Extensionality. For all $X$ and all $Y$, it holds that

• (if) for all $o$ it holds that $o \in X$ precisely when $o \in Y,$
• (then) $X=Y.$

The writer then proceeds from these axioms to prove theorems.

For instance, using the axioms of ZFC we can prove the existence of a unique empty set. This might be Theorem 1.

Theorem 1. There exists unique $X$ such that for all $o$ it holds that not $o \in X$.

What I'm looking for is a slightly different approach to rigor. Rather than asserting some axioms at the get-go, I'd like a work that merely defines axioms.

Definition 1. Extensionality means that for all $X$ and $Y$, it holds that

• (if) for all $o$ it holds that $o \in X$ precisely when $o \in Y,$
• (then) $X=Y.$

Where we would normally state a theorem, we instead define another axiom, and then write the theorem after.

Definition 5. Unique empty set means there exists unique $X$ such that for all $o$ it holds that not $o \in X$.

Theorem 1. If Extensionality and Empty set, then Unique empty set.

Is there a work of mathematics structured in this fashion?

• Definition is just a word. Coincidentally, axiom is just a word as well. Definitions happen to be a list of axioms for an object to satisfy. You can define $\in$ to be an extensional relation, and then prove that if $R$ is an extensional relation then bla bla bla; or assume that $R$ satisfies the axiom of extensionality and then prove bla bla bla. Why do you expect to find any difference between the two? – Asaf Karagila Apr 20 '13 at 12:18
• @AsafKaragila, I don't get what you're saying..... from my point of view, the Axiom of extensionality says that $\in$ is an extensional relation. But it doesn't make sense to say that "$R$ satisfies the the Axiom of extensionality," but only that "$R$ is extensional." Anyway, can you rephrase your question? – goblin Apr 20 '13 at 12:27
• My question is why do you think the two approaches are different? – Asaf Karagila Apr 20 '13 at 12:30
• Oh. They aren't - or at least, the difference is only a difference in emphasis. – goblin Apr 20 '13 at 12:30