Is there any infinite quantity small enough to be affected by finite changes?

Question

Hilbert's paradox of the Grand Hotel shows us, among other useful things, that the cardinality of any infinite set is a quantity equal to n more than itself for any finite n. I am interested in finding an infinite quantity that is too small to be the cardinality of any infinite set; that is, one for which this property does not hold.

Is there a quantity x for which the following two qualities are both true? - For all finite numbers y, x > y. - For all finite numbers z, x + z > x.

To visualize what I'm describing, imagine the following scenario. There is a tower of carefully stacked tin cans. It towers so high that no matter how many tin cans you stacked up beside it, you would never exceed its height. And yet, if you picked it up from the bottom, and stacked just one more tin can underneath it, it would still get taller. Disregarding issues of physics - that's another conversation entirely, and a far more trivial one - could such a tower exist?

For another example, consider the following. I'm playing Mario Party with my friend. I spent all my coins to get my first star. He has no stars, but enough coins to get a star and have some left over. No matter how many coins he gets, it'll never measure up to my star. I'm in first place. But if he lands on a star space and buys a star, he'll have as many stars as I have, and more coins, and then, rather than us being tied, he'll be in first place. Given this strange behavior, how can we conceive of my score in the game as a single quantity?

For yet another example, consider a robot from Isaac Asimov's "I, Robot." For those unfamiliar with the book, a robot within this fictional world must follow three laws, prioritized in order of first to third, so that, in the event of a conflict among the laws, it may never disobey a higher priority law in favor of a lower priority law, no matter how badly it has to disobey the lower priority law in order to maintain obedience of the higher priority law. Suppose a robot encounters such a conflict. Let its deviation from its laws be a quantity n. Its mission is to behave in such a way as to minimize n. How can it treat a deviation from the second law as infinitely larger than a deviation from the third law, so that under no circumstance will it deviate from the second law to conform to the third law; and the same with the first and second laws; and yet, at the same time, be able to distinguish between different severities of deviations from the same law?

Answer 1

We say that a set is Dedekind-finite, if it has one (equivalently, all) of the following properties:

1. There is no injection from $\Bbb N$ into the set.
2. There is no injection from the set into any of its proper subsets.
3. There is no injection from the set into the set obtained by removing a single element.

We say that a set is Dedekind-infinite if it doesn't have these properties. In terms of cardinal numbers this means that a cardinal $a$ is a cardinal number for a Dedekind-infinite set if and only if $a+1=a$, and by induction this means $a+n=a$ for any finite $n$.

We can show that if a set is finite, then it is Dedekind-finite, and so if it is Dedekind-infinite then it is not finite.

Assuming the axiom of choice (whatever it means) every infinite set is Dedekind-infinite. So if $A$ is an infinite set, adding or removing a finite amount to it would not change its cardinality.

But it is consistent with the failure of the axiom of choice, as Andres points out, that there are infinite Dedekind-finite sets. This means that it is possible to have infinite hotels where you can't quite move the people around like Hilbert is doing in his hotel (Hilbert is a shrewd businessman, but seems to be treating his guests in a rude manner).

With these sort of sets you get a real difference between "Stacking any finite height of cans won't reach to the height of this stack" and "Stacking an infinite number of cans at once" (where stacking seems to be implied as indexed by $\Bbb N$).

The problem is that the axiom of choice is very useful to us, and in fact it is very natural to assume (many of the people who opposed it were implicitly using a fragment of the axiom of choice, one sufficient for proving the equivalence between finite and Dedekind-finite). So in most modern mathematics you won't find this notion of infinite Dedekind-finite sets.

However in some parts of the mathematics, where the axiom of choice is not assumed, Dedekind-finite sets are a source for many counterexamples and are objects of interest. The problem is that by removing the axiom of choice we invariably add more restrictions on what we can and cannot do, so things become much more difficult (which is why the axiom of choice is often assumed: utilitarianism). So getting into the details here might end up being excessive.

Answer 2

It depends what you mean by "quantity".

In the Hilbert hotel paradox, we conceptualize infinity as a number measuring the quantity of an infinite set, and allow certain manipulations of infinite sets in order to show that they have the same quantity.

So, for example, if $\aleph_0$ is the cardinality of $\Bbb N$ (the capacity of the infinite hotel), then we would say that $\aleph_0 + \aleph_0 = \aleph_0$. That is, we can fit a whole hotel of people into a full hotel by moving people around appropriately. In particular, we can "just" have the previous occupants move to the even numbered rooms, and have the new occupants take the odd numbered rooms.

You might say, however, that moving infinitely many things infinitely far away isn't really "possible", per se, so that $\aleph_0 + \aleph_0 ">" \aleph_0$. Making such distinctions between infinities leads to ideas such as Dedekind finite sets (as described above by A. Caicedo and A. Karagila) and the ordinal numbers.

If you are willing to give up the notion that all infinite numbers should describe the sizes of sets, you could end up with something like the surreal numbers. This idea is probably the closest to what you have in mind.

As for the way a good-ol' axiom-of-choice-fearing mathematician would conceptualize situations such as those you describe, I would look into lexicographical order.