Applications of cardinal numbers
Solution 1:
The answer is an obvious no. For two main reasons:
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.
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?
Solution 2:
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.