Element of ordinal a subset of the same ordinal

Question

I've got a very short question in set theory. I am currently reading P. T. Johnstone's book Notes on Logic and set theory, and the proof of the fact that Every element of an ordinal is a subset of that ordinal goes as follows:

$\alpha\in\beta$ implies $\alpha\subseteq\beta$, since $\beta$ is transitive.

I assume that he meant the following: If $\gamma\in\alpha$, then by transitivity $\gamma\in\beta$, so $\alpha\subseteq\beta$ follows. $\Box$

Now: Isn't that a tautology? $\beta$ is transitive was defined as $$\forall(x,y,z\in\beta)(((x\in y)\wedge(y\in z))\implies(x\in z)),$$ already assuming that (in the above proof) $\gamma \in \beta$.

Any help? How would you prove that Every element of an ordinal is a subset of that ordinal?

Cheers!

Answer 1

The fine point here is that the term "transitivity" is applied twice to the definition of an ordinal.

1. It is a transitive set. In the context of set theory we say that $x$ is a transitive set if whenever $y\in x$, we have $y\subseteq x$; or in other words whenever $z\in y$ and $y\in x$, we have that $z\in x$.

2. It is well-ordered by $\in$, and in particular this means $\in$ is a partial order on the ordinal, so it is a transitive relation. Here transitive means the same thing is usually means for relations, i.e. for every three elements of the ordinal, $x,y,z$ if $x\in y$ and $y\in z$ then $x\in z$.

Why do we need both? The latter one is fairly obvious, because we want $(\beta,\in)$ to be a well-ordered set, so we need $\in$ to be a partial order. But why the former? Well, the reason is that this guarantees the ordinals are canonical, meaning there is exactly one ordinal of each order type.

For example, consider any singleton $\{x\}$, it is a well-ordered set by $\in$, so which singleton will represent the well-order which only has one element? Or for example $\{x,y\}$ where $x\in y$ is a well-ordered set with two elements, regardless to the values of $x$ and $y$ (as long as $x$ is an element of $y$). How do we decide which is going to be the ordinal?

Well, it turns out that if we require that ordinals are transitive sets then this uniqueness follows, and we get a bunch of other very nice properties. For example, now an ordinal is itself a set of smaller ordinals. But note from the two examples, that without requiring transitivity (in the sense of a transitive set) then it won't be true anymore.

Answer 2

That's exactly what is meant. It is simply a rephrasing of the transitivity definition.

I note that the definition of "$\beta$ is transitive" that you gave is incorrect. Rather, it should read $$\forall x,y\Bigl(\bigl((x\in y)\wedge(y\in\beta)\bigr)\implies(x\in\beta)\Bigr).$$ The definition you gave says that $\in$ is a transitive binary relation on the set $\beta.$ This is yet another thing that we want to be true, especially if one defines an ordinal to be a transitive set that is (well-)ordered by $\in.$