Union of Uncountably Many Uncountable Sets

Solution 1:

It generalizes very naturally.

If $\kappa$ and $\lambda$ are cardinal numbers such that $\kappa$ is infinite and $0<\lambda\le\kappa$, the union of $\lambda$ sets of cardinality $\kappa$ has cardinality $\kappa$. In other words, the union of at least one and at most $\kappa$ sets of cardinality $\kappa$ has cardinality $\kappa$.

An alternative generalization is that if $\kappa$ is an infinite cardinal number, the union of at most $\kappa$ sets, each of cardinality at most $\kappa$, also has cardinality at most $\kappa$. In symbols, if $\lambda<\kappa$, and $\{A_\xi:\xi<\lambda\}$ is a family of sets such that $|A_\xi|\le\kappa$ for each $\xi<\lambda$, then $$\left|\bigcup_{\xi<\lambda}A_\xi\right|\le\kappa\;.$$

Solution 2:

To understand this property correctly we need to realize that "The union of finitely many finite sets is finite" first. If we replace "finite" by the word "small" then it would say $$\text{The cardinality of a union of a small amount of small sets is small.}$$

When jumping from finite numbers to infinite numbers this is also true, for example countable sets are very small in this fashion, and the countable union of countable sets is indeed countable.

Uncountable set is just a set which is not countable, it is like saying $x\neq 0$ for a real number. We have no way of knowing what $x$ is, just that it is nonzero. So talking about "an uncountable union of uncountable sets" makes very little sense.

However, we can "extend" the notion of being small. We can say that sets are small if they have cardinality $\aleph_1$ or less, e.g. if we work in a context of much larger cardinalities. In this case the union of $\aleph_1$ many sets of cardinality $\aleph_1$ is still with cardinality $\aleph_1$.

The same can be said about $\aleph_2$ as small, and so on. However it does not always hold. Having gone through $\aleph_1,\aleph_2,\ldots,\aleph_n,\ldots$ we reach $\aleph_\omega$ which is the least cardinal not being an $\aleph_n$.

If we have $|A_n|=\aleph_n$ then $\bigcup_{n} A_n$ has a cardinality larger than $A_n$ for all $n$, in particular it is larger than $\aleph_n$ for all $n$. In fact it is a countable union of uncountable sets, but the result is not with bijection with any $A_n$. This shows that if we say that a set is small if it has cardinality of $\aleph_n$ for some $n<\omega$ then a small family can be unified into a large set.

Definition: Let $\kappa$ be a cardinal, we say that $\kappa$ is regular if for every $\lambda<\kappa$ and $\lambda_i<\kappa$ for $i<\lambda$ we have: $$|\bigcup_{i<\lambda}\lambda_i|<\kappa$$

The first paragraph is to say that $\aleph_0$ is regular, the second is to say that $\aleph_1$ is indeed regular as well. The later part shows that $\aleph_\omega$ is not regular.

Theorem: If $\kappa$ is an infinite cardinal number then $\kappa^+$ (the successor cardinal of $\kappa$) is regular.

The result of this theorem is that if $\kappa$ is a regular cardinal we can extend the notion of "small" to "being of cardinality less than $\kappa$", while if $\kappa$ is singular we need to change some things in the definition of "small".

Of course that all of the above requires and relies heavily on the axiom of choice, without it we have that a countable union of finite sets can be uncountable; the countable union of countable sets can be of size $\aleph_1$; and even that every uncountable $\aleph$ cardinal is singular, which says that the notion of "small" behaves very strangely in the universe.