Is there a reasonable way of defining the surreal number $\aleph_{-1}$ or $\omega_{-1}$?

Question

Is there a reasonable way of defining the surreal number $\aleph_{-1}$ or $\omega_{-1}$ ? Conway, in his book "On Numbers and Games", shows a way to define $\epsilon_{-1}$ on page 35 as the limit of $\epsilon_0 - 1$, $\omega^{\epsilon_{-1}}$ and so forth. I just wonder if there is a similar way of defining $\omega_{-1}$ or $\aleph_{-1}$ , or for that matter other such numbers such as $\theta_{-1}$, where ${\theta_\alpha}$ is the $\alpha$th inaccessible cardinal (or the ordinal that corresponds to it).

Answer 1

Here's an idea that might work; I haven't put too much thought into it, though, therefore there might be problems to it that I don't see. Also there are some points at the end where I'm basically guessing.

Preliminaries

I start out with the sign representation of the surreal number. As a reminder, the sign representation gives the surreal number as a function from its birthday (a ZFC ordinal) to the set $\{+,-\}$.

I think the sign representation is best suited for this task because the surreal ordinals are just the functions from the corresponding ZFC ordinals with constant value $+$. And for ZFC ordinals, we know exactly how to obtain the corresponding cardinals. I will proceed as follows:

1. Generalize the concept of equipotency (equal cardinality) from ordinals to surreal numbers and identify a specific surreal number as representing each equipotency class. Those will be the initial surreals (surreal cardinals). This is the part which I believe to be solid (but I still may be mistaken, of course).

2. Introduce a criterion when an initial surreal should be considered an omega number, that is, a number that should be named $\omega_x$ for some surreal number $x$, and figure out how to map surreal numbers to omega numbers. This step then finally gives meaning to terms like $\omega_{-1}$. This is the part where I'm guessing a lot (but would welcome proofs or refutations).

Equipotency of surreal numbers and initial surreals

Since in the sign representation, negation of a surreal number amounts just to reversing every individual sign, I restrict the following considerations to the positive surreal numbers. Generalization to negative numbers and zero should be obvious.

As already mentioned, the sign representation of a surreal number maps a ZFC ordinal to signs. For a positive number, this mapping starts with a $+$ (that is, $0$ is mapped to $+$). Now we can look at the number in a slightly different way: First there comes a batch of $+$, then possibly a batch of $-$, then again possibly a batch of $+$, and so on, until we reach the end of the number.

Obviously the sign representation is completely determined by the sequence of lengths of those individual batches (those lengths themselves are ordinal numbers). For example, take the number $3\omega/2 - 1$. This has the sign representation $$\underbrace{+++\ldots+++\ldots}_{\omega\cdot2}\underbrace{---\ldots-}_{\omega+1}$$ Now I define two surreal numbers as equipotent iff each corresponding batch of equal signs has an equipotent length. For example, $3\omega/2-1$ is equipotent to $\omega/2$ because both are a countably infinite number of $+$ followed by a countably infinite number of $-$.

Since surreal ordinals have only a single batch whose length is just the corresponding ZFC ordinal, this implies that two surreal ordinals are equipotent iff the corresponding ZFC ordinals are equipotent. That is, the definition of surreal number equipotency indeed is a proper generalization of ordinal equipotency.

Correspondingly I define a surreal number to be initial if each batch length is an initial ordinal. Again,it is immedaiately obvious that a surreal ordinal is an initial surreal iff its corresponding ZFC ordinal is an initial ordinal.

The definition of the cardinality of a given surreal number $x$ is then obvious: It is the unique initial surreal that is equipotent to $x$.

Let's denote the cardinality of $x$ with $c(x)$.

As an example, $c(3\omega/2-1) = \omega/2$, while $c(3\omega/2+1) = \omega/2+1$.

Omega numbers

The next question is which of the initial surreals should be assigned some $\omega_x$; I use the term “omega number” for that. Well, obviously the initial surreal ordinals should be omega numbers iff they are infinite.

Now the first possible criterion obviously is that the number shall be infinite. In the sign representation it means it starts with infinitely many $+$. However the simplest initial surreal with two batches would then be $\omega-1$, and that seems just too close, given that in the other direction, we have to go all the way until $\omega_1$ to get the next omega number with one batch.

Another possible criterion could be that all batches are of infinite length. Then the simplest two-batch initial surreal would be $\omega/2$. Which is better, but still seems unsatisfying to me for the same reason as before.

However there's some other thing all the infinite initial ordinals share: If $\alpha$ is an infinite initial ordinal, then $c(\alpha+\alpha)=\alpha$. Therefore I propose the following definition:

An initial surreal is an omega number if $c(x+x)=x$.

If I'm not mistaken, the simplest two-batch initial surreal satisfying this equation is given by $\omega$-many $+$ followed by $\omega_1$-many $-$, which nicely mirrors the ordinal $\omega_1$ whose $\omega_1$-many $+$ could also be considered as $\omega$ many $+$ followed by $\omega_1$-many $+$. Turns out I was mistaken. See edit below.

Indeed, I suspect that such numbers will generally consist of batches of increasing length (but that's more of a guess than anything). If this is indeed the case, then the omega numbers should form a transfinite binary tree that can be matched to the transfinite binary tree of surreal numbers.

If all that works out the way I guess, then we would get: We can take the following as alternative definition to the one highlighted above, but without having the nice justification I thought it had.

• $\omega_{-1}$ is the surreal number whose sign representation is given by $\omega_0$-many $+$ followed by $\omega_1$-many $-$ (where the $-many$ ordinals are all ZFC ordinals).

• $\omega_{1/2}$ is the surreal number whose sign representation is given by $\omega_1$-many $+$ followed by $\omega_2$-many $-$.

• $\omega_{-1/2}$ is the surreal number whose sign representation is given by $\omega_0$-many $+$ followed by $\omega_1$-many $-$ followed by $\omega_2$-many $+$.

• Generally, for arbitrary $x$, you get $\omega_x$ by stating with $\omega$-many $+$, and then going through the sign representation of $x$ in order, and when the ordinal $\alpha$ is mapped to $+$, then add $\omega_{1+\alpha}$-many $+$, otherwise add $\omega_{1+\alpha}$-many $-$ (where here the $+$ in the index is ordinal addition, in particular, $1+\omega=\omega\ne\omega+1$).

Now if any of the previous considerations and/or guesses are wrong, then you could of course still define $\omega_x$ this way (after all, the last point is just a constructive rule), however there would be less justification for that construction.

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Edit: I now found out that I was wrong about the number $x$ I identified with $\omega_{-1}$ above solving the equation $c(x+x)=x$.

By construction, $\infty<x<\omega/2$. This implies $\infty<x+x<\omega$, which implies that its sign representation has exactly $\omega$-many $+$ preceding the first $-$. Now any surreal number of that form equipotent to $x$ would have to have $\aleph_1$-many $-$ signs following those $\omega$-many $+$ signs. But every additional $-$ sign gives a smaller number, and since $\omega_1$ is the initial ordinal of size $\aleph_1$, $x$ is therefore the largest number in its equipotency class. But since $x>0$, $x+x>x$, and therefore it cannot be equipotent to $x$.

This leaves several options:

• Keep the definition of omega numbers by $c(x+x)=x$. However I'm no longer convinced that any initial surreal other than the initial ordinals will fulfill them.

• Take the last bullet point above (the one conjecturing the form of $\omega_x$) as definition of $\omega_x$. This works, but has a certain ad-hoc vaalue to it.

• Find another cardinality-arithmetic definition of omega numbers that is better behaved. This may or may not result in $\omega_x$ as specified above.

Answer 2

No.

The easy answer is that the surreal numbers form a field, so the arithmetic is commutative and cancellative. Neither cardinal nor ordinal arithmetic is cancelleative, and only one of them is even commutative.

But there is a deeper question here. Why would you expect from $\aleph_{-1}$ to be? Okay, maybe it's just a formal object, kind of like how we can treat $-1$ as a formal additive inverse of $1$ if we want to only believe in the natural numbers. After further investigation we can conclude that $-1$ is somehow useful, and slowly we can start developing the feeling that it is as real as the rest of the natural numbers, and from there we can jump to $\Bbb{Z,Q,R,C}$ and so on.

So back to $\aleph_{-1}$. What would that be? Would that be a cardinal such that $\aleph_{-1}+\aleph_1=\aleph_0$ or is it a cardinal such that $\aleph_{-1}+\aleph_1=0$? In what sense is that a cardinal? Cardinals measure the size of sets, and since all sets already have cardinals assigned to them, in what sense is this new one a cardinal?

Suppose that it does end up useful, like $-1$ was, then we can extend our notion of sets to have a set $A$ such that $|A|=\aleph_{-1}$. Great, in that case what is the cardinality of $A\cup\omega_1$? Since the union is a conjunction, you don't have elements that cancel each other out.

There is no antimatter in the set theoretic universe.

But since unions, intersections, etc., are all based on Boolean algebras, this means that you need to throw out the rules of Boolean algebra in favour of something else. Can you do it? Sure, why not. Should you do it? It's your time and effort. But why would you do it? At some point we no longer have sets as collections of elements, now you have sets which are collections of negative, fractions, and other infinitesimal elements, whatever that means. These are no longer sets in any kind of traditional sense. And your cardinals are no longer cardinals in the traditional sense either.

Moreover, you haven't used the surreal numbers to define these entities. Instead you relied on the idea that instead of a Boolean algebra, your truth values form a field, redeveloped the basics of mathematical logic, and then redeveloped set theory, to find yourself arriving at this place. In this strange new world, you may be able to develop a new theory of surreal numbers, I don't know. I'm not familiar with the rules of logic you've yet-to-develop. But then you can ask, why not add those so-called cardinals as well?

And in either case, you'd be missing a criminal amount of understanding of what set theory is used for, and how set theory is actually done.

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Addendum

Why could Conway define $\varepsilon_{-1}$? After all, he is somehow extending the definition of ordinals.

There is a natural order embedding of the ordinals into the surreal numbers. But it is only the order that embeds, not the arithmetic. After embedding the order, we can look at the equation defining $\varepsilon$ numbers, $x=\omega^x$, and we can solve it "in the surreal numbers".

Arguably, this is a reasonable definition of $\varepsilon$ numbers. But it does not extend the idea of ordinals or cardinals, it extends the idea of $\varepsilon$-numbers to a domain beyond the ordinals.

Answer 3

Disclaimer I'm not a professional mathematician. I am however actively learning about Surreals, ordinals & cardinals. At the risk of presenting a possibly unpopular opinion, for this answer I will take the role of Conway's advocate. Note: I'm sure someone could do a much better job of this than me.

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In the surreal number system, we create new numbers by filling in the gaps between previously created numbers. While I tend to think about fundamental sequences for ordinals (eg. the sequence of ordinals approaching the limit ordinal from below), it has recently been suggested to instead work with the concept of closure. I mention this because I think it would be useful for someone trying to more rigoursly define uncountables & inaccessibles in the surreal system.

Wrt $\varepsilon_{-1}$, from ONAG:

What is $\varepsilon_{-1}$? This must be to the left of $\varepsilon$, and (being a leader) therefore to the left of $\varepsilon-1$, thence of $\omega^{\varepsilon-1}$, $\omega^{\omega^{\varepsilon-1}}$, and so on.

In a slightly modified format from what Conway presents in ONAG we could say: $$\varepsilon_{-1}=\{\omega, \omega^\omega, \omega^{\omega^\omega},...|\varepsilon-1, \omega^{\varepsilon-1}, \omega^{\omega^{\varepsilon-1}},...\}$$

This is reminiscent of $\omega-1$, $\frac{\omega}{2}$ & $\sqrt{\omega}$: $$\omega-1=\{1,2,3,...|\omega\}$$ $$\frac{\omega}{2}=\{1,2,3,...|\omega-1,\omega-2,\omega-3,...\}$$ $$\sqrt{\omega} = \{1,2,3,... |\frac{\omega}{2},\frac{\omega}{4},\frac{\omega}{8},...\}$$

So what about $\omega_{-1}$? If I had to propose a form it would be: $$\omega_{-1}=\{0|\omega^{-\alpha} : \forall \alpha < \omega_1\}$$

I feel confident a better form could be proposed. This is just to illustrate an example. This raises the quesiton, what would this thing be? Well, we know $\omega_1$ is uncountably larger than $\omega$, so it stands to reason $\omega_{-1}$ would be uncountably smaller than $\omega$ (& anything in $\mathbb{N^+}$) - an uncountable infinitesimal, if you will.

We could also consider things like: $$\omega_{\varepsilon}=\{\omega_0|\omega_1, \omega_{1/2}, \omega_{1/4},...\}$$ $$\omega_{-\varepsilon}=\{\omega_{-1}, \omega_{-1/2}, \omega_{-1/4},... |\omega_0\}$$

Note: $\varepsilon$ here is an infinitesimal which is equal to $\frac{1}{\omega}$, not $\varepsilon_0$ which is written as $\varepsilon$ in the quote from ONAG.

As far as how we can do all this, I advise checking out the appendix to part zero of ONAG which begins with:

This is Liberty-hall gentlemen!

Conway proceeds to introduce the Mathematician's Liberation Movement * (pg. 66):

• Objects may be created from earlier objects in any reasonably constructive fashion.
• Equality among the created objects can be any desired equivalence relation.

Also, it's probably useful to state that we are working with surordinals & surcardinals so that it isn't confused with the normal ordinals & cardinals (just like when working with the surreals where we can do things like $1-\varepsilon$, which isn't in $\mathbb{R}$). Working in the sur(ordinal/cardinal) context, I think definitions for inaccessibles (& beyond) can be reasonably constructed.

Note: I'm not sure about the implications of surcardinals wrt the continuum hypothesis since it seems feasible to have $\aleph_{1/2}$, $\aleph_\varepsilon$ & (infinitely) more.