Could there be an extension of the ordinals?

surreal number- oh the first comment mentioned them
Sure. Things like this essentially get made up all the time in set theory and its applications.

As a simple example, take the simple ordinal &omega; and look at its one-point compactification under the order topology. We just add a single point at “infinity” and get the ordinal &omega;+1. I.e. the natural numbers, but with something at the end.

As a simple-*ish* example, the Tikhonov corkscrew, an example of regular, but not completely regular space, is constructed using the ordinals &omega; and &omega;&#x2081; by taking

- two copies of &omega; as the y-axis of an xy-plane and having them point/converge towards the origin, and

- two copies of &omega;&#x2081; as the x-axis also pointing towards the origin.

The product of these axes gives an ordinal-valued xy-plane with “long” horizontal lines and  “short” vertical lines. (The rest of the construction is more complicated.)

Another, much, much more complicated example is Shelah’s construction of a linear order L used to get the cardinal inequality &dfr;<&afr;. It works by choosing a pair of starting cardinals (which is an ordinal) &mu; and &lambda; and letting &mu; be the “base” order. Then we take two copies of &lambda; and flip one around and call it &lambda;\* and stick them together at 0 in a horizontal line H. (We also remove 0 from H for technical reasons.) Then stick a copy of H in at every ordinal &alpha;<&mu; as a sort of “infinitesimal neighborhood” of &alpha;, obtaining a new order L^(1). Then we do this again for every point p of L^(1) creating a linear order L^(2). We do this countably many times and consider the final order L^(&infin;) as the order consisting of all points which “dive” finitely many neighborhoods into L^(&infin;). (So literally L^(&infin;) should be something like a union of all the L^(n).) This final structure can be thought of as a “<&omega;^(th)-order” hyperreal type structure on the cardinal &mu;.

There are tons more, but this kind of a gauge of the extremes of this type of thing.

Edit: Oh and since you mentioned rationals, I assume you are potentially interested in doing arithmetic and algebra using ordinals. This can be done, but not in the same way as we are used to with, say, real numbers. Standard ordinal algebra is not actually particularly nice. Addition on any ordinal gives a non-abelian semigroup. Multiplication is non-commutative. Exponentials are always defined, but become quite complicated. Division and logarithms are only partial operations on **Ord**, meaning some ordinals can’t be consistently divided or have logarithms taken.

Cardinal arithmetic is a bit nicer. Addition and multiplication become commutative, but a bit boring at the surface level since for infinite cardinals these just reduce to the max operation. Cardinal exponentiation is a fiercely bad algebraic operation as lots of interesting cardinal exponential values are independent of ZFC. The most famous of course being the Continuum Hypothesis that 2^(&omega;)=&omega;&#x2081;.

However, none of this stops us from constructing entirely new types of operations using ordinals that may have nice algebraic structure. So perhaps there are ordinal structures that can be thought of as analogous to the reals, or the complexes, or some field of fractions of an interesting polynomial ring, etc. In fact, we can simply take an appropriately-sized cardinal and endow it with a structure that makes it isomorphic to whatever structure of that size you like, so long as it consistently exists within a model of the theory you want to work in. So we could take &omega; with a carefully chosen order and turn it into an isomorphic copy of the the rationals &Qopf;. The simplest way to do this is by taking any bijection from &Qopf; to &omega; and simply “pushing forward” the order on the rationals to &omega;.
by
One thing to notice is that addition is different with the ordinals and the surreals. With the ordinals, addition is noncommutative, and 1+ω=ω≠ω+1, but in the surreals, addition is commutative and 1+ω=ω+1.

(You can still think of one as a ~~subset~~ subclass of the other; there's just two different addition operators on the ordinals.)

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