What are the 'equality axioms'?

Say there's a relation ~ between the two objects a and be such that a ~ b. We call ~ an equivalence relation if: i) a ~ a. Ii) If a ~ b.

Then be ~ a. Iii) If a ~ be and be ~ c, then a ~ c. Where c is another object.

The three properties above are called the reflexive, symmetric, and transitive properties. Were those three properties all that was needed to define the equality relation, we could safely call them axioms. However, one more property is needed first.

To show you why, I'll give an example. Consider the relation, "is parallel to," represented by . We'll check the properties above to see if is an equivalence relation.

I) a a. Believe it or not, whether this statement is true is an ongoing debate. Many people feel that the parallel relation isn't defined for just one line, because it's a comparison.

Well, if that were true, then you would have to say the same thing for every binary equivalence relation; e.g., a triangle couldn't be similar to itself, or, even more preposterously, the statement a = a would have to be tossed out the window too. But, just to be formal, we'll use the following definition for parallel lines: Two lines are not parallel if they have exactly one point in common; otherwise they are parallel. So, with that definition in hand, i) holds for .

Ii) If a b, then be a. True. Iii) If a be and be c, then a c.

True. Thus, the relation is an equivalence relation, but two parallel lines certainly don't have to be equal! So, we need an additional property to describe an equality relation: iv) If a ~ be and be ~ a, then a = b.

Let's check iv) and see if this works for our relation : If a be and be a, then a = b. False. But, does it hold for the equality relation?

If a = be and be = a then a = b. True. This is what's known as the antisymmetric property, and is what distinguishes equality from equivalence.

But wait, we have a problem. We used the relation = in one of our "axioms" of equality. That doesn't work, because equality wasn't part of the signature of the formal language we're using here.

By the way, the signature of the formal language that we are using is ~. So, any other non-logical symbol we use has to either be defined, or derived from axioms. Well, we have three possible ways out of this.

We can either: 1) Figure out a way to axiomize the = relation through the use of the ~ relation. 2) Define the = relation. 3) Add = to our language's signature.

Well, 1) is not possible without the use of sets, and since the existence of sets isn't part of our signature either, we'd have to define a set, or add it to our language. This isn't very hard to do, but I'm not going to bother, because the result is what we're going to obtain from 2). Anyways, speaking of 2), let's define =.

For all predicates (also called properties) P, and for all a and b, P(a) if and only if P(b) implies that a = b. In other words, for a to be equal to b, any property that either of them have must also be a property of the other. In this case, the term property means exactly what you think it means; e.g. red, even, tall, Hungarian, etc. So, the million dollar question is, by defining =, are our properties now officially axioms?

For three of the properties, the answer is no. In fact, because we just defined =, we've turned properties ii), iii), and iv) from above into theorems, not axioms. Why?

Because, property iv) still has that = relation in it, which we had to define. So, iv) is a true statement, but we had to use another statement to prove it. That's the definition of a theorem!

And, since the qualifier for iv)'s truth was that a ~ be and be ~ a, we can now freely replace be with a in ii), giving us "If a ~ a, then a ~ a." Well, now ii)'s proven as well, but we had to use iv) to do it. Thus, both ii) and iv) are now theorems.

Finally, iii) can be proven in a similar was as ii) was, so it, too, is a theorem. However, our definition of = only related a to b, it never related a to itself. Thus, we need to include i), from above, as an axiom.

Just for kicks, let's try plan 3) too. The idea here is to make = a part of our signature, which means now we don't need to define it. In fact, we can't define it if we put it in our signature; because by placing it there, we're assuming that it's understood without definition.

Therefore, iv) must now be assumed to be true, because we have no means to prove it; that sounds like an axiom to me! However, just like before, we can prove both ii) and iii) through the use of iv), so they get relegated back to the land of theorems and properties. Interestingly though, iv) makes no mention of reflexivity, and since our formal definition of = is gone, we have no way to prove i).

Once again, we have to assume that it's true. Thus i) is an axiom as well. So, to paraphrase our two separate situations: In order for the relation ~ to be considered an equality relation between the objects a and b, one axiom must be satisfied if we define =: 1) For all a, a ~ a, as well as three theorems: 1) If a ~ b, then be ~ a 2) If a ~ be and be ~ c, then a ~ c, where c is another object 3) For all a and b, if a ~ be and be ~ a, then a = b.

Additionally, In order for the relation ~ to be considered an equality relation between the objects a and b, two axioms must be satisfied if we put = into our signature: 1) For all a, a ~ a 2) For all a and b, if a ~ be and be ~ a, then a = b, as well as two properties: 1) If a ~ b, then be ~ a 2) If a ~ be and be ~ c, then a ~ c, where c is another object. What the one right above did is include "=" into our formal language, but "=" is equality, so he actually came up with a fairly well axiom before he finishes with the circular looking one. His axiom: We say ~ is an equality relation means whenever x ~ y, for any condition P, P(x) iff P(y) The axiom is the bolded part.

After discussion with my Math prof. This morning, that axiom becomes a properties follows from this more formal definition. It does not need to include any more things then what we already have for the formal language.

We say ~ is an equality relation on a set A if (a set is something that satisfies the set axioms) For any element in A, a ~ a. If follows that P(a) is true and y ~ a, then P(y) is also true, vice versa. Because in this case, y has to be a for it to work.

You might argue well the definition for an equivalence relation have this statement in it too, does that mean equivalence IS equality? No! It's the other way around, equality is equivalence.

Equality is the most special case for any relation, say *, where a * a. Take an equivalence relation, say isomorphisms for instance (don't know what that word mean? Google or as it on this website), we know any linear transformation T is isomorphic to T, in particular this isomorphism IS equality.

Of course it would be boring if isomorphism is JUST equality, so it's MORE. The other axioms in a definition of a relation are to differ THEM from equality, because equality is the most basic. Equality must always be assumed, it always exist, any other relation is built upon it.

It is the most powerful relation, because ALL relations have it. (I mean all relations, say *, such that a * a for all a must at least be equality).

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