# Applications of Countable Infinite Sets and Power Sets

What are the possible applications of Countable Infinite Sets and Power Sets in areas that are not strictly mathematical?

Also I want to know the significance they carry. What was not possible before the concepts of countability and power sets were introduced and what became possible afterwards? How did the introduction of these concepts change our thinking and outlook and the areas they affect?

Please explain in plain language, possibly with examples, as I am from a non-mathematical background. Thank you in advance.

PS: The applications need not be in strictly practical fields either (as Asaf Karagila keeps pointing there aren't any). Please help!

• My comments from the previous thread still apply. (For 10k users: here's a link.) – Asaf Karagila Jun 13 '14 at 11:45
• I can imagine a (mathematical) world in which all sets are countable. I can't imagine one in which all sets are uncountable. You want sets of 5 elements to be uncountable? or maybe you want there to be no such thing as sets of 5 elements? – Gerry Myerson Jun 13 '14 at 12:28
• @Gerry Myerson, I do not WANT them to be uncountable. I just wondered what if, just trying to get a better appreciation of countable infinite sets. And maybe I did not make it clear in my query, but I am talking in terms of infinite sets. Finite sets are definitely countable. – user51013 Jun 13 '14 at 16:31
• I'm glad we agree that finite sets are countable! But what about the set of natural numbers? Unless you have some weird definition of countable (or of infinite), that's going to be a countable infinite set. So are you asking for a system where there is no such thing as the set of natural numbers? – Gerry Myerson Jun 14 '14 at 4:40
• If your question is, "what changes were brought in our thinking after this distinction was made," then that should be the body of your question, and not "What would happen if all sets were uncountable?" Please rewrite your question so it makes sense, then go to the meta site to propose reopening the revised question. – Gerry Myerson Jun 14 '14 at 13:34

The notion of cardinality as formalized and explored by Cantor introduced a whole new set of questions about the reals that wouldn't have occurred to mathematicians before Cantor. Before Cantor, it was well-known that the reals and the rationals were very different infinite sets, but the difference was generally described by the fact that the reals have a completeness property (loosely, convergent sequences of reals converge to another real -- not so in the rationals, it isn't too difficult to construct a sequence of rationals that converges to $\sqrt{2}$). Once you can distinguish countable and uncountable sets (so after Cantor), the whole picture changes.

For example, in terms of the first-order theory of linear order types (in particular, ignoring algebraic properties), the reals can be completely characterized by the following properties:

1) uncountable

2) no endpoints

3) dense in itself (between any two distinct reals there is another real)

4) no uncountable strictly monotonic sequences

5) contains a countable dense subset (the rationals -- i.e. between any two distinct reals there is a rational)

(If you're wondering why completeness isn't on this list, completeness is a second-order property, quantifying over sets of reals -- "all sets which have an upper bound have a least upper bound.")

Note, though, how many of these properties rely on cardinality. The next interesting question is whether all of these properties are necessary. In particular, consider dropping the last property. Is there a linear order type which satisfies the first 4 properties but contains no subset order-isomorphic to the reals? The answer turns out to be yes -- first construct an uncountable tree with no uncountable branches or levels (actually a counterexample to what would be a natural generalization of Konig's lemma), then look at the lexicographic order on nodes of that tree. The result is a linear order as described above, known as an Aronszajn line (or Specker type). This whole construction can be done in ZFC.

There are other interesting questions here. Consider whether there are alternatives to the last property above. Perhaps it is enough to consider that the linear order has a countable chain condition -- i.e. there are no uncountable collections of pairwise disjoint open intervals. Does this give order types which contain a suborder isomorphic to the reals? This question turns out to be independent of ZFC -- the counterexample is called a Souslin line, and whether such a thing exists depends on the model of ZFC you are using (some have them, some don't).

Again, the point is that these questions about the reals, and possible alternatives to them, would not have occurred to anyone before the development of a formal definition of cardinality for infinite sets, in terms of bijections. Cantor's ideas brought on something of a crisis in our understanding of the reals, and the results (mostly independence results lately) continue to flood in.

• The five axioms you wrote do not "completely characterize the real numbers". They also characterize $\Bbb R\setminus\{0\}$ for example. – Asaf Karagila Jun 15 '14 at 6:59
• Also, I don't see how these are applications of uncountable sets to non-mathematical contexts. Which is what the question is all about. – Asaf Karagila Jun 15 '14 at 7:00
• $\Bbb R\setminus\{0\}$ contains a suborder isomorphic to the reals -- in fact, lots of them, like $\Bbb R^+$, $(0,1)$, etc.. That is the sense in which the axioms determine the reals as a first-order characterization of a linear order type. And any context where the reals are relevant, axiomatically reasonable alternatives like an Aronszajn line would also be relevant -- what is it about the reals that makes them the "right" place to talk about physics, and what about physics could rule out using an Aronszajn line instead? – user128390 Jun 15 '14 at 10:23
• I think that you don't quite understand what order isomorphic means. If an ordered set is isomorphic to the real numbers it must be complete. The irrational numbers are not a complete order. Therefore they are nor isomorphic, nor they contain a subset isomorphic to the real numbers. Moreover, when you say that an theory completely determines a structure it doesn't mean that everything other structure has an isomorphic substructure. It means there is exactly one model up to isomorphism. – Asaf Karagila Jun 15 '14 at 10:49
• If you're talking about first-order then you can't say "uncountable". Your first and fourth axioms are not first-order. Also "countable dense subset" is not something that you can express in first-order logic. So three of your five axioms are not first-order. And again, order types means order isomorphism the order types of $\Bbb{R,R\setminus\{0\},R\setminus Q}$ are pairwise different. All satisfy your axioms, and the last one doesn't have any subset isomorphic to $\Bbb R$ itself. – Asaf Karagila Jun 15 '14 at 13:38