Set notation
When defining a set, something is normally put between curly braces. There are a few different conventions for what can go inside:
- All the elements of the set, separated by commas: \( \{a,b,c, \ldots\} \)
- Set-builder notation: a name for a general element of the set, then a condition that all elements must satisfy, with either a vertical bar or a colon in between: \( \{ x \mid P(x) \} \) or \( \{ x : P(x) \} \).
Piper H asks[1], in the notation \( \{ a_1, a_2, \ldots \} \), where the elements inside are a sequence, are you making an implicit claim that the elements are countable?
The answer is of course no, this is not a claim of countability. Only well-orderability. This is where the ambiguity lies in the notation above. The length of the sequence is not given, and if the sequence is meant to be interpreted in a ZFC structure, then we need to know explicitly how long this sequence is, i.e. which ordinal indexes its elements. If we don’t, it can represent any nonempty set. (We call this a sequence despite the use of curly braces for an unordered set of elements because the enumeration of elements implies some function mapping ordinals to the \a_i.
Colons are used in set notation to indicate a condition the elements must satisfy. Colons are also used when defining functions to show which sets a function maps between (eg \(f:\mathbb{R}\to\mathbb{R}\). This can cause ambiguities when defining a set containing functions, for example:
$$\{f:\mathbb{R}\to\mathbb{R}:f>0\}$$
