I'm interested in whether ordinal numbers can be described by a first-order theory without presupposing ZFC or any particular set theory. Such a theory might look like Peano arithmetic, but generalized to include all the infinite ordinals that appear in standard treatments. A similar question is asked here, but the top answer there assumes a "background" set theory, and it even explicitly claims that "unlike the Peano axioms, these axioms require a larger theory of sets."
I understand that the standard treatment of ordinals lives within set theory, where the set/class distinction is important. We might like to say something like, "for any set—but not class—of ordinals, there exists a supremum," but logical predicates extend to classes, not sets, so there are clearly some difficult issues that need to be handled carefully. But I would still be surprised if it were impossible to capture the structure of the class of all ordinals and their arithmetic as a standalone first-order theory.
Initial stab:
It seems natural to begin with the axioms of a strictly well-ordered class. We have a built-in binary relation $<$, and we can define $\leqslant,\nless,>,$ etc. in the usual ways.
- $\forall x:x\nless x$
- $\forall x,y,z:(x<y)\wedge(y<z)\to(x<z)$
- $\forall x,y:(x\leqslant y)\vee(y\leqslant x)$
- (Schema) $\exists x\varphi(x)\to\exists m\Big(\varphi(m)\wedge\big(\forall y\varphi(y)\to m\leqslant y\big)\Big)$
We should also have a constant $0$ and unary function $s$ obeying the same axioms as in Peano arithmetic, with the added axiom that $\forall x:x<s(x)$.
But this is not enough the guarantee the existence of $\omega$, and we will probably need a special axiom for this, as ZFC also needs an axiom for $\mathbb N$. We could include a unary relation in our language called $f$ (for "finite") and add the axioms $f(0)$ and $\forall x:f(x)\to f(s(x))$. Then we could add an "axiom of infinity": $\exists y\forall x:f(x)\to x<y,$ and from this we can prove the existence of a unique smallest element that is greater than all the finite elements.
But this is where I get stuck/confused. I think we need some analog of replacement to get to $\omega\cdot2$ and higher limit ordinals. But I can't even imagine what we need in order to get to $\omega_1$, since this will require the language having a way of comparing cardinalities.
I'm not an expert but I'm very interested in both ordinals and in doing math without ZFC so I'll have a go at it. I'll start by rewriting your axioms to use the same notation throughout and to only use $<$ (personal preference to not write the axioms with symbols that are not part of the signature).
Axioms 1 through 4 ensure that $<$ is a well ordering, but put no other constraints on the universe, which can even be empty. Rather than adding $0$ $s$ and the limit ordinal as constants to the signature of the language, I want to try forcing their existence via axioms.
Axiom 4 is a good candidate for identifying unique elements of our language because, once we can prove that $\varphi$ is satisfied by at least one element, the $m$ from the axiom can be proved to be unique. Now, we can add an axiom to make sure our universe is not empty:
And then define $0$ from axiom 4 applied to $\varphi(x) := x = x$. For a given $n$ we can also define $s(n)$ from axiom 4 with $\varphi(x) := n < x$ provided that we add the following axiom as well:
Now, with the above it should be possible to construct $\mathbb N$, but we need more for $\omega$. One obvious candidate is an axiom that postulates the existence of limit ordinals as ordinals without a predecessor.
This is for example presented as the infinity axiom for a theory of ordinals and sets of ordinals in [1]. I have changed it to postulate the existence of a limit ordinal larger than any other ordinal, in the hopes that it will help construct the other ordinals, but I'm not sure if it's safe from the Burali-Forti paradox.
In any case, with this axiom it should be possible to get $\omega$ using $$ \varphi(y) : = (\exists x: x < y) \land \forall z: z < y \to \exists t: z < t \land t < y $$ It seems to me also that we could get $\omega \cdot n$ too, but not $\omega^2$.
An alternative for axiom 7 could be an axiom scheme for the existence of limit ordinals, something along the lines of
7'. $ (\exists x: \varphi(x) \land (\forall y: \varphi(y) \to x < y) \land \forall z: \varphi(z) \to z = x \lor \exists t: \varphi(t) \land t < z \land \lnot \exists u: \varphi(u) \land t < u \land u < z) \\ \to \exists v: \forall w: \varphi(w) \to w < v $
which says that an ordinal larger than all elements $x$ satisfying $\varphi$ exists if the set of these elements is such that it has a first element and all other elements have a predecessor in $\varphi$. The problem with this is that it's not obvious how to write a formula $\varphi$ for $\mathbb N$ in order to get $\omega$ from it via axioms 7 and 4.
At least one more axiom would be needed here for transfinite induction, but its exact formulation depends on how limit ordinals are added.
What I would like about an axiom system like this (assuming it actually works, or course) is that each axiom serves a clear purpose related to the ordinals.
I hope this makes sense and contributes ideas for your question.
[1] https://arxiv.org/pdf/math/0502265.pdf