First fix the following notations:
$\mathcal{L}_{AR}:=$The first order language $\lbrace \overline{0},\overline{S},\overline{+},\overline{\times}, \sqsubset \rbrace$ which $\overline{0}$ is a constant symbol, $\overline{S}$ is a unary function symbol, $\overline{+}$ , $\overline{\times}$ are binary function symbols and $\sqsubset$ is a binary relation symbol.
In the usual set theoretic literature, ordinal numbers are generalization of natural numbers and the proper class of all ordinals ($Ord$) as an amorphous accumulation is very "similar" to the set of all natural numbers ($\omega$). But we need some "bones" to mold these bodies and complete their similarity by equipping them with a same structure. In this way we know one of the most important structures of $\omega$ is its well ordered Peano arithmetic $\langle \omega , 0, S, +, \times, < \rangle$ in the language of $\mathcal{L}_{AR}$ with the usual interpretations of symbols. In order to build some structural similarity between $\omega$ and $Ord$ we usually endow $Ord$ by some special generalizations of natural operations on $\omega$. This method gives us a (proper class) $\mathcal{L}_{AR}$ - structure $\langle Ord , 0_{ord}, S_{ord}, +_{ord}, \times_{ord}, <_{ord} \rangle$ by the following interpretations:
$<_{ord}\subseteq Ord\times Ord$
$<_{ord}:= \lbrace (\alpha , \beta )\in Ord \times Ord~|~ \alpha \in \beta \rbrace$
$0_{ord}\in Ord$
$0_{ord}:=\emptyset$
$S_{ord}:Ord\longrightarrow Ord$
$S_{ord}(\alpha):= \alpha \cup \lbrace \alpha \rbrace$
$+_{ord}:Ord\times Ord\longrightarrow Ord$
$\alpha +_{ord} \beta := ordertype\langle\alpha \times \lbrace 0 \rbrace \cup \beta \times \lbrace 1 \rbrace , <_{train~order}\rangle$ $\times_{ord}:Ord\times Ord\longrightarrow Ord$
$\alpha \times_{ord} \beta := ordertype \langle \beta \times \alpha , <_{lexographic~order} \rangle$
Now the following facts are clear by definitions:
Fact (1): On natural numbers,$0_{ord}$,$S_{ord}$,$+_{ord}$,$\times _{ord}$,$<_{ord}$ are equal to $0$,$S$,$+$,$\times$,$<$.
Fact (2):$\langle \omega , 0, S, +, \times , < \rangle \subseteq \langle Ord,0_{ord}, S_{ord},+_{ord},\times_{ord},<_{ord}\rangle$
By the fact (1) ordinary ordinal arithmetic "extends" the natural arithmetic of natural numbers. And the fact (2) says that this ordinal arithmetic has "primitive" (quantifier free) properties of natural number arithmetic. But something is uncomplete because we have the following facts:
Fact (3): $\langle \omega , 0\rangle \prec \langle Ord, 0_{ord}\rangle$
Proof: Easy induction on formulas.
Fact (4): $\langle \omega , S\rangle \nprec \langle Ord, S_{ord}\rangle$
Proof: Consider the sentence: $\exists x~\exists y~(\neg (x=y) \wedge \forall z~(\neg (\overline{S}(z)=x) \wedge \neg (\overline{S}(z)=y)))$ which is true in $\langle Ord, S_{ord}\rangle$ but false in $\langle \omega , S\rangle$.
Fact (5): $\langle \omega , +\rangle \nprec \langle Ord, +_{ord}\rangle$
Proof: Consider the sentence: $\forall x~\forall y~(x ~\overline{+}~y=y~\overline{+}~x)$ which is true in $\langle \omega , +\rangle$ but false in $\langle Ord, +_{ord}\rangle$.
Fact (6): $\langle \omega , \times \rangle \nprec \langle Ord, \times_{ord}\rangle$
Proof: Consider the sentence: $\forall x~\forall y~(x ~\overline{\times}~y=y~\overline{\times}~x)$ which is true in $\langle \omega , \times\rangle$ but false in $\langle Ord, \times_{ord}\rangle$.
Fact (7): $\langle \omega , <\rangle \nprec \langle Ord, <_{ord}\rangle$
Proof: Consider the sentence: $\exists x~\exists y~(\neg (x=y)\wedge \forall z~(z\sqsubset x \longrightarrow \exists t~(z\sqsubset t \wedge z\sqsubset x))\wedge \forall u~(u\sqsubset y \longrightarrow \exists v~(u\sqsubset v \wedge v\sqsubset y)))$ which is true in $\langle Ord, <_{ord}\rangle$ but false in $\langle \omega , <\rangle$.
Obviously by the facts we have: $\langle \omega ,0,S,+,\times,<\rangle \nprec \langle Ord, 0_{ord},S_{ord},+_{ord},\times_{ord},<_{ord} \rangle$.
So we can observe that ordinary ordinal arithmetic hasn't "all" (first order) properties of standard arithmetic on natural numbers and some properties are "missed". Now there are some natural questions here:
Question (1): Is this "incompleteness" of ordinal arithmetic fundamental? In other words, are there some interpretations $S^{*}$, $+^{*}$, $\times^{*}$, $<^{*}$ for $\mathcal{L}_{AR}$ symbols such that $\langle \omega , 0, S, +, \times , < \rangle \prec \langle Ord , 0, S^{*}, +^{*}, \times^{*} , <^{*} \rangle$? (We call this interpretation, a "good" arithmetic on ordinals.)
Remak (1): Note that possitive answer of above question means that we can find a "good" ordinal arithmetic which "extends" natural number arithmetic and satisfies "all" of its properties too. So it seems that we must "desert" the "classic" ordinal arithmetic and build a "modern" set theory based on this (these) "well behavior" ordinal arithmetic(s) which will be a "renaissance" in set theory!
Now consider the following questions in two cases dependent on the answer of question (1):
If the answer of question (1) be negative:
Question (2): Is there any non trivial sub language $\lbrace \overline{0} \rbrace \varsubsetneq \mathcal{L} \varsubsetneq \mathcal{L}_{AR}$ such that the answer of question (1) be positive up to $\mathcal{L}$? If yes, what are the maximal languages between $\lbrace \overline{0} \rbrace$ and $\mathcal{L}_{AR}$?
If the answer of question (1) be possitive:
Question (3): Is the "good" ordinal arithmetic in question (1) unique?
Question (4): How much is the degree of (first order) "unifiability" between the collections $\omega$ and $Ord$? Precisely is the following sentence true?
"For all first order language $\mathcal{L}$ and for all $\mathcal{L}$ - structure $\mathcal{M}$ with $Dom(\mathcal{M})=\omega$, there is an $\mathcal{L}$ - structure $\mathcal{N}$ with $Dom(\mathcal{N})=Ord$ which $\mathcal{M} \prec \mathcal{N}$ ".
Question (5): Let $\langle Ord , 0 , S^{*} ,+^{*}, \times^{*}, <^{*} \rangle$ be a "good" arithmetic on $Ord$. Define $\mathbb{N}^{ord}:=\langle Ord , 0 , S^{*} ,+^{*}, \times^{*}, <^{*} \rangle$ and build $\mathbb{Z}^{ord}$, $\mathbb{Q}^{ord}$, $\mathbb{R}^{ord}$ and $\mathbb{C}^{ord}$ from $\mathbb{N}^{ord}$ in the usual methods which we produce $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{C}$ from $\mathbb{N}=\langle \omega , 0 , S ,+, \times, < \rangle$. Is there any well behavior real and complex "analysis" on $\mathbb{R}^{ord}$ and $\mathbb{C}^{ord}$? In other words, is the behavior of "continous line of infinities" ($\mathbb{R}^{ord}$) and "complex plane of infinities" ($\mathbb{C}^{ord}$) as same as $\mathbb{R}$ and $\mathbb{C}$? More precisely, do we have $\mathbb{R}\prec \mathbb{R}^{ord}$ and $\mathbb{C}\prec \mathbb{C}^{ord}$ in the language of (ordered) fields?
Remak (2): Note that we can produce $\mathbb{Z}^{ord}$, $\mathbb{Q}^{ord}$, $\mathbb{R}^{ord}$ and $\mathbb{C}^{ord}$ by the usual structure $\mathbb{N}^{ord}:=\langle Ord , 0 , S_{ord} ,+_{ord}, \times_{ord}, <_{ord} \rangle$ but the arithmetics on $\mathbb{Z}^{ord}$, $\mathbb{Q}^{ord}$, $\mathbb{R}^{ord}$ and $\mathbb{C}^{ord}$ will be very bad and complicated. So at first we need to fix our arithmetic on $Ord$ by choosing the best one. Anyway, if the answer of above question be possitive, we can go beyond current infinitary "combinatorics" (number theory) and build a "good behavior" infinitary real and complex "analysis" which could be an extra revolutionary development in our set theory and mathematics.
Question (6): Assume the answer of question (5) be possitive, what is the interpretation of "transcendental" infinitary real or "imaginary" infinitary complex numbers? For example what is the meaninig of $\sqrt[\beth_{1}]{\aleph_{\omega}}$ or $i^{\aleph_{1}}$? Is there any fundamental transcendental infinitary real numbers such as $\pi^{ord}$ and $e^{ord}$, "hidden" between "integer" infinitary numbers such as $\aleph_{2}$ and $\aleph_{4}$ with a fundamental relation similar to $e^{i\pi}+1=0$?
Question (7): What is the answer of above questions in logics with more "expression power" than first order, such as higher order or infinitary logics?
