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In the usual approach to ordinals and cardinals (namely, von Neumann's approach), every cardinal is also an ordinal, and every ordinal is also a set. So if $\kappa$ is a cardinal, then the expression $2^\kappa$ has at least three meanings; so its meaning has to be specified in the surrounding text, or inferred from the context.

Now arguably, this isn't an issue of any real substance. That's a perfectly legitimate position; I don't happen to share it. For that reason, I'd like to experiment with the idea that set, ordinal and cardinal are mutually exclusive concepts. That way, if $\kappa$ is a cardinal, then expressions like $2^\kappa$ are unambiguous - e.g. in this case, its definitely cardinal exponentiation we're doing.

To give another example, there's good reason to distinguish between Conway's implementation of the surreal numbers, and the surreal numbers themselves. Otherwise, statements like $x=y$ are ambiguous - do we mean they're equal as numbers, or that they're equal as sets?

So we now have two reasons to distinguish between entities and the sets that represent them. The first was the ambiguity of expressions like $2^\kappa$. The second is the ambiguity of statements like $x=y$.

Lets focus our attention on the surreal numbers as an example. Suppose for a moment that we're sold on the above arguments. In this hypothetical universe, we want to distinguish between Conway's implementation of the surreal numbers, and the surreal numbers themselves.

Okay, one way of doing this would be to conservatively extend our language with new sorts. So we start off with a one sorted language whose elements are called 'sets.' Then we define a proper class $C$ whose elements are precisely the sets that Conway calls 'surreal numbers,' as well as an equivalence relation $\sim$ on $C$ that specifies which elements of $C$ are identical as numbers.

Now here comes the unconventional part. Instead of stopping there, we go one step further by extending our language with a new sort $S$ whose elements are called 'surreal numbers', and a canonical function $C \rightarrow S$ such that if $f$ is the aforementioned function, then $x \sim y \Leftrightarrow f(x)=f(y)$ for all $x,y \in C$.

It seems great in theory, but there's a problem: we can't collect surreal numbers into sets. After all, $\in$ is a relationship between two sets, so if $\alpha$ is a surreal number, then '$\alpha \in X$' isn't even a well-formed expression! Clearly, this isn't convenient.

Question. Can the approach using sorts be salvaged? If so, how?

I hope its salvageable, but lets assume for the moment that it isn't, or that no one knows how. Then we need another approach.

Enter the atom (or urelement, if you prefer). We begin the same: define $C$ and $\sim$. But, instead of adding a new sort $S$, we just collect a bunch of atoms together into a class, call it $S$ for 'surreal numbers.' Since atoms can belong to sets, concepts like $\alpha \in X$ make sense, even if $\alpha$ is surreal. Yay! We've solved the problem. That's the theory, but I've never actually seen this done in practice. What does it actually look like? Thus, I have a request.

Request. Can someone write up an example of how this would be done, in practice? I get the general gist of the idea, but I'm unsure about the details.

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    $\begingroup$ The point in having cardinals and ordinals and sets is that we don't need more than $\in$ and a few axioms in order to express everything. It's fine to define new types, but that'd be missing the idea of set theory. It's not about being nitpicky and pedantic, it's about rejecting set theory and accepting type theory. That's perfectly reasonable. $\endgroup$ – Asaf Karagila Jun 20 '13 at 7:58
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    $\begingroup$ I downvoted because the first paragraph is somewhat condescending by telling me what to read or not to read. If you would edit it, I'll be happy to remove my downvote. $\endgroup$ – Asaf Karagila Jun 20 '13 at 7:59
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    $\begingroup$ The other option, if I can be so bold, would be to typographically differentiate between the different meanings. Something like $\kappa^\lambda$ for cardinal exponentiation, $\kappa^{\cdot \lambda}$ for ordinal exponentiation, and ${^\lambda} \kappa$ for the set of all functions $\lambda \to \kappa$. $\endgroup$ – user642796 Jun 20 '13 at 8:03
  • $\begingroup$ @Asaf, i didnt mean to be condescending. Im not at my computer anymore so it would be a real pain to edit it at the moment, but I'll change it when i get the chance. $\endgroup$ – goblin Jun 20 '13 at 8:04
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    $\begingroup$ Also, note that this sort of trade off appears everywhere in mathematics. You can't remove it, because people are like that. People prefer the "shortcuts" that everyone in a given field easily fill in the gaps, rather than tediously work and make sure that everyone outside their field can fully understand everything. If Shelah were to write for the layman (or even the lay-set theorist), he wouldn't have had that much work done... $\endgroup$ – Asaf Karagila Jun 20 '13 at 8:06
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Well, I completely disagree with this approach, but let me give my opinion on how it should be done.

The problem with atoms is that they are completely indiscernible from one another. If $a,b$ are atoms and $\varphi(a)$ is true then $\varphi(b)$ is true as well. At least in the language where all we have to express is $\in$ which is a relation pertaining sets and their elements.

This is why you ought to be adding two things. The class of atoms and an ordering on the atoms. Then you can add an axiom stating that the atoms and their ordering is a well-order class isomorphic to the usual class of ordinals. Then you can declare that from now on we shall refer to these atoms when we talk about ordinals.

Similarly we add a third sort for cardinals, and we can include the order or addition/multiplication into the language.

But the point is that there are no theorems which will be stated differently. Just whenever we refer to ordinals it will be clear that these ordinals have a different type than the rest of the sets. Of course we have to tweak a lot of definitions, for example a sequence of length $\alpha$ is no longer a function from the ordinal $\alpha$, but rather a function whose domain is $\{x\in{\sf Ord^*}\mid x<\alpha\}$. And so on and so forth. But beyond these sort of tweaks and changes there shouldn't be any change whatsoever. The only difference is that now when you say "Let $\alpha,\beta$ be two ordinals and $\gamma=\alpha+\beta$" we know that this cannot be addition as cardinals or anything because ordinals make a completely different type of object.

This is a particularly good time to bring up (again) the point that in set theory we really write meta-proofs rather than proofs. That is, we actually say "suppose that $\varphi(x,y,z)$ is the formula stating that $z$ is the ordered pair $\langle x,y\rangle$, then bla bla bla".

We don't really care about the formula. It's quite easy to take Kuratowski's definition for ordered pairs, but there are others. We just care about the fact that such definition exists and it's simple enough. Rarely we do care that the encoding is simple enough, but even then there are degrees of freedom.

Similarly we don't really care about the actual way of devising ordinals or cardinals, and all our proofs can be easily re-fitted to any definitions of these mathematical notions. But sometimes it makes things simpler that way, so it's more convenient to just use it like that. Of course when we have the von Neumann ordinals then often we can make nice shortcuts (such as the aforementioned about sequences), but it's not very difficult seeing that this is not a real mathematical issue. In fact, I doubt that you would find it confusing too.

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    $\begingroup$ A comment on your last sentence: its not about me! Its about pedagogy, and general accessibility. Anyway, thank you for your answer. $\endgroup$ – goblin Jun 21 '13 at 4:15
  • $\begingroup$ Sadly mathematics is not about accessibility for everyone. It is about mathematics. And it is very convenient to have these things, just as much as it is convenient to have small "holes" in other fields that would make them inaccessible. People who want to understand the results of set theory will study set theory. If you insist on making things readable you will inevitably complicate them to a point they are unreadable by anyone. At least now a set theorist can (theoretically) read most set theory texts, even if they do have some confusing parts. $\endgroup$ – Asaf Karagila Jun 21 '13 at 6:39
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    $\begingroup$ I'm not quite sure what you're saying, but I think you're suggesting that, "the most important thing is mathematical progress." That's probably true, but note that most fields do indeed organize their subject matter by defining functions between sets of objects whose internal structure remains undefined. This doesn't seem to hamper the people in those fields - so why should it hamper progress in set theory? $\endgroup$ – goblin Jun 21 '13 at 6:52
  • $\begingroup$ I'm saying even more than that. I'm saying that most of those people are uninterested in set theory to begin with. If they are uninterested, why should we bother to accommodate them? $\endgroup$ – Asaf Karagila Jun 21 '13 at 6:56
  • $\begingroup$ You shouldn't. But if they ARE interested, you should reward their interest by maximizing the return/effort ratio. Not at the price of undermining mathematical progress of course. Anyway, I didn't mean my original question to be specifically about set theory for the sake of set theory. It was meant to be about set theory for the sake of studying various other number systems. I chose the ordinals and cardinals because they're well-known and the issue of ambiguity actually comes up in practice, rather than just in theory, but the question could easily have been about the reals or surreals. $\endgroup$ – goblin Jun 21 '13 at 7:37

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