My background: I have been studying set theory for a couple of months, via Paul Halmos's book «Naive Set Theory» (which, as it turns out, does not present Cantor's naïve and inconsistent set theory, but rather a non-formalized version of ZFC, lacking one or two axioms).
I have also read a famous paper by Paul Benacerraf entitled «What Numbers Could Not Be», in which he claims there is an indefinite amount of ways one can ground arithmetic (and perhaps the rest of mathematics) in set theory, each of these ways corresponding to a different set-theoretic definition/construction of numbers/ordinals. He gave the example of Zermelo's numbers and of von Neumann's numbers, highlighting that they are mathematically equivalent, but not set-theoretically equivalent.
(His chief objective was advancing a structuralist conception of the metaphysics of mathematics, arguing against those who identify numbers with certain sets. But this is of no consequence here.)
My question: Can one really make sense of the whole theory of ordinals using Zermelo's construction, as well as indefinitely many others? — I ask this because I am quite naïve in set theory so far, but I have yet to think of a way to satisfy all the properties of the ordinals using the Zermelo numbers.
Perhaps my puzzlement arises because some of the properties I've been trying to meet are idiosyncratic to Halmos's construction of the ordinals (which uses von Neumann's numbers), which is the only construction I know of. The properties which I found puzzling are following: (i) ordinals are transitive sets, (ii) each ordinal is the set of its predecessors, (iii) all the elements of an ordinal are ordinals (joined with ii, this becomes puzzling), (iv) and each element of an ordinal number is also contained in it.
Thanks for the attention. Cheers, Dan.
The only requirements that are necessary to the concept of ordinals are that
Any model that satisfies these three conditions captures the entire behavior of ordinals as order-types.
Now von Neumann's construction adds a number of additional properties that are very convenient for dealing with ordinals, most particularly that $\alpha = \{\beta \mid \beta < \alpha\}$. Because they are so handy, most mathematicians prefer to just equate "ordinal" with von Neumann's construction. However, these are only conveniences: the theory can be phrased to not make use of them.
Concerning your 4 properties, (i) and (iii) are simple consequences of (ii): If for all ordinals $\alpha$, $\alpha = \{\beta \mid \beta < \alpha\}$, then, yes, every member of that set is an ordinal, so all elements of an ordinal are also ordinals. And if $\beta \in \alpha$ and $\gamma \in \beta$, then $\gamma < \beta$ and $\beta < \alpha$, so $\gamma < \alpha$ and therefore $\gamma \in \alpha$.
And (iv) is a property of any set. In fact, it is just saying the same relationship in two different ways: "$a$ is an element of $b$" and "$a$ is contained in $b$" mean the exact same thing.