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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? ‎‎‎

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Isn't question (1) answered by the Lowenheim-Skolem theorem? (One would want a proper class version of LS, which is provable using global choice.) –  Joel David Hamkins Jul 22 '13 at 15:00
    
And how could it be unique? Any permutation of the infinite ordinals would provide another way of imposing the structure. –  Joel David Hamkins Jul 22 '13 at 15:02

2 Answers 2

I think you'll be interested in the surreal numbers -- see Philip Ehrlich's paper, "The absolute arithmetic continuum and the unification of all numbers great and small" (http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bsl/1327328438). If my understanding is right, the surreals form the "maximal" elementary extension of the ordered field $\mathbb{R}$. In particular, in the surreal numbers many odd expressions like those in your Question 6 are meaningful.

Of course, since things are class-sized now, how the surreals behave -- or in general, how anything you're looking for behaves -- will depend on the ambient set theory, to some extent; I don't know of any particular point of interest, but it's worth keeping in mind.

One final remark, towards your Question 7: many logics stronger than first-order -- including higher-order logic (with the full, as opposed to Henkin, semantics) and infinitary logic -- can characterize $\mathbb{N}$ up to isomorphism, so the answer is trivially "no." You'd probably want to restrict your attention to compact logics extending first-order logic; there are some interesting ones here, but they tend to be a little odd. Saharon Shelah has a number of (difficult) papers on the subject; as examples, the logic gotten from first-order logic by allowing quantification over automorphisms of definable fields is compact, as are some logics with messier generalized quantifiers: see shelah.logic.at/files/375.ps and http://arxiv.org/pdf/math/0009080.pdf, page 3, respectively. It's worth noting that there is no logic stronger than first-order logic which is compact and has the Downward Lowenheim-Skolem property (this is Lindstrom's Theorem).

If you're interested in the area of logics beyond first-order, I heartily recommend the book "Model-Theoretic Logics" ed. Barwise and Feferman, which is chapter-by-chapter available at ProjectEuclid: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.pl/1235417263.

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I just realized that my "final remark" is actually longer than the rest of my answer. Oh well. :P –  Noah S Jul 22 '13 at 15:55

As Joel commented, you're asking for a class-sized elementary extension of the structure $\langle\omega,0,S,+,\times,<\rangle$. Such things exist (under reasonable assumptions about proper classes), but they have essentially nothing to do with ordinal numbers. In particular, the order relation $<$ cannot be a well-ordering.

There are reasonable operations on ordinals that look somewhat more like the familiar operations on natural numbers. For example, the "natural sum" of two ordinals, $\alpha\oplus\beta$, is defined as the largest ordinal that can be partitioned into a subset of order-type $\alpha$ and a subset of order-type $\beta$. Unlike genuine ordinal addition, $\oplus$ is commutative. If you're interested in this sort of thing, look up "natural sum" and "natural product" or the equivalent names "Hessenberg sum" and "Hessenberg product". But don't expect to find elementary extensions of the natural numbers in this way. These Hessenberg operations are still compatible with the well-ordering of the ordinals.

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