TITLE: Understanding what is a set.
QUESTION [6 upvotes]: I thought I understand what is a set long ago, that is, a collection $S$ of some stuff satisfying some "well defined" rule so that either $a\in S$ or $a\notin S$. But recently I became less sure.
For example, consider subsets of integers. I thought it is something like
$S=\{n\in{\mathbb Z}\,|\, property\}$, where "property" is some "rule" to decide if $a\in S$ or not, like " $n$ is an even number that cannot be written as the sum of two primes". However, each "rule" can be formalized into an English paragraph so that it uniquely decides the rule, but the set of English paragraphs is countable, this means there are at most countably many "rules". Now there are uncountably many subsets of ${\mathbb Z}$. So there are subsets in ${\mathbb Z}$ without "rules", I guess meaning only God knows whether $n\in S$ or not, or even He does not know? It is disturbing to think like this, especially when we often use languages like "let $S$ be an arbitrary subset of ${\mathbb Z}$".
I am aware that there is no formal definition of set. I would like to hear any comments on how to think about what is a set, hopefully regain some peace in mind.

REPLY [4 votes]: I find the reason you lost your piece of mind is rather problematic... Applying your reasoning to the real numbers we can reach the same "conclusion", namely there are real numbers that "only God knows" what they are. The example of the reals can even become equivalent with the subsets of $ \mathbb{Z} $, if we order $ \mathbb{Z} $ then every subset of $ \mathbb{Z} $ can be described by a sequence of zeros and ones (its characteristic function) - which can be thought of as the binary representation of a real number in $ [0,1] $. So every subset of $ \mathbb{Z} $ is now merely a point on an interval (there are some minor technical problems with this mapping but they don't affect the essence of the argument)...
Sets can be really mind-bending (axiom of choice, non-measurable sets...) and it is a fact that Cantor wrote to Dedekind "I see it, but don't believe it!" on a set theory question but these difficulties don't have to do with how and if we can "express" every single set in writing.
PS the set of English paragraphs may be countable, but the set of English paragraphs that can be read by a human being during a period of 20.000 years is actually finite - so until we figure out a way to live forever we are bounded as a species by finiteness