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I know basic things about cardinality (I'm only in High School) like that since $\mathbb{Q}$ is countable, its cardinality is $\aleph_0$. Also that the cardinality of $\mathbb{R}$ is $2^{\aleph_0}$.

Are there any direct applications of these numbers outside of theoretical math?

I know this can be convenient for certain proofs and help understanding sets of numbers, but are there any applications of this?

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Yes. Plenty of set theorists got a job because infinite cardinals. –  Asaf Karagila Mar 7 '13 at 20:24
    
Well, an important application is that if you have $\,\aleph_0\,$ dollars then you can form $\,2^{\aleph_0}\,$ different sets with them. This has proven to be very handy to Scrooge McDuck, for example... –  DonAntonio Mar 7 '13 at 20:25
    
Asaf's application is way more important than mine... –  DonAntonio Mar 7 '13 at 20:26
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For all intents and purposes, there is no such thing as infinity outside of mathematical (or philosophical) thought. All measurements and interactions we encounter every day are finite. –  rschwieb Mar 7 '13 at 20:29
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3 Answers

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The answer is an obvious no. For two main reasons:

  1. With the exception of occasional naive approach to sets, there is little to no use of set theory outside theoretical mathematics. So any application would be indirect and purely coincidental.

  2. Applied mathematics is not concerned with infinite objects. Let alone "vastly huge beyond any reasonable visualization and imagination of a human being" sizes of infinity.

It is important to understand that mathematics is not "merely a tool for engineers" (or physicists). It is a world filled with magic and mind boggling ideas which have absolutely nothing to do with this physical reality. Infinite sets is one of them. These ideas trickle slowly and some of them eventually get to the point where they have some use, but these uses are far from being "direct" in any sense of the word.

For example, by plain cardinality arguments it is easy to see that almost any function from $\Bbb R$ to itself is not continuous, or even Borel measurable. Almost any continuous function is nowhere differentiable, and almost all the differentiable functions are not continuously differentiable, and so on and so forth (although some of these arguments require more than sheer cardinality).

But have you ever seen someone "applying" everywhere-discontinuous functions to a real world situation? I can't recall anything like that (although it might be in some quantum theory sort of application I am unaware of).

As long as mankind is limited by a finite powers of perception we cannot even distinguish between $100^{100^{100^{100^{100}}}}$ and $\aleph_0$.

Might also be relevant: Can we distinguish $\aleph_0$ from $\aleph_1$ in Nature?

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I would love to hear the complaints against this answer. Also, if someone does have direct applications of infinite cardinals outside theoretical mathematics, I'd love to know about them! –  Asaf Karagila Mar 7 '13 at 20:47
    
Maybe cardinality have something to do with theory of possible worlds. I'm not sure "direct application" in your mind is applicable to such things.tau.ac.il/~samet/papers/knowing-whether.pdf –  Metta World Peace Mar 8 '13 at 0:12
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Quoted from Christian Marks blog(blog seems to be gone now):

In an unexpected development for the depressed market for mathematical logicians, Wall Street has begun quietly and aggressively recruiting proof theorists and recursion theorists for their expertise in applying ordinal notations and ordinal collapsing functions to high-frequency algorithmic trading. Ordinal notations, which specify sequences of ordinal numbers of ever increasing complexity, are being used by elite trading operations to parameterize families of trading strategies of breathtaking sophistication.

The monetary advantage of the current strategy is rapidly exhausted after a lifetime of approximately four seconds — an eternity for a machine, but barely enough time for a human to begin to comprehend what happened. The algorithm then switches to another trading strategy of higher ordinal rank, and uses this for a few seconds on one or more electronic exchanges, and so on, while opponent algorithms attempt the same maneuvers, risking billions of dollars in the process.

The elusive and highly coveted positions for proof theorists on Wall Street, where they are known as trans-quantitative analysts, have not been advertised, to the chagrin of executive recruiters who work on commission. Elite hedge funds and bank holding companies have been discreetly approaching mathematical logicians who have programming experience and who are familiar with arcane software such as the ordinal calculator. A few logicians were offered seven figure salaries, according to a source who was not authorized to speak on the matter.

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Ordinal$\neq$cardinals. –  Asaf Karagila Mar 7 '13 at 21:03
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Yes and no.

The distinction between different infinite cardinals is crucial in probability theory. If you have a sequence of sets $A_1,A_2,\cdots$ (necessarily countable) that are each assigned a probability, then according to the Kolmogorov axioms $A_1 \cup A_2 \cup \cdots$ is necessarily assigned a probability. On the other hand, if you have a $2^{\aleph_0}$ many sets, their union needn't be assigned a probability. This is it to prevent certain subtle paradoxes, whereby its impossible to assign a "length" (or "mass") to every subset of $[a,b]$.

One possible rebuttal is that the cardinal numbers are only used to prevent very subtle problems in probability theory, and so they don't really have direct applications to the real world. However, I think this is wrong-headed. (Classical) logic is brittle. One contradiction and the whole system falls to pieces. So if probability theory had even one contradiction, this would be a SERIOUS problem, in my opinion.

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Have you ever seen anyone outside theoretical mathematics refer to non-measurable sets? Or even non-Borel sets? –  Asaf Karagila Mar 7 '13 at 21:06
    
No: the Kolmogorov axioms protect us from these things. –  user18921 Mar 7 '13 at 21:45
    
I really don't see how that comment addressed mine; and in a re-read of your answer, I also don't see how the fact that we only require probability to be countably additive is related to cardinals, or protects us from paradoxes. –  Asaf Karagila Mar 7 '13 at 21:49
    
Yes my answer isn't very clear. Anyway, consider the following. If arbitrary unions of events were assumed to be events, and if the singleton subsets of $[a,b]$ were assumed to be events, then every subset of $[a,b]$ would be an event. Including the non-measurable ones. –  user18921 Mar 7 '13 at 21:55
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@Metta: I am happy to see that the axiom of choice has real world applications. –  Asaf Karagila Mar 8 '13 at 0:34
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