I've been searching online for a way of constructing lists from sets, but to no avail. However, I am aware of how to define ordered pairs and, more generally, ordered n-tuples from sets. My first preference being the Kuratowski definition:
$$ (a, b) := \big\lbrace\lbrace a\rbrace, \lbrace a,b\rbrace \big\rbrace $$
As I'm sure you are all aware, ordered pairs can then be used to define Cartesian products. Using Cartesian products, I believe I've find out how to construct/define lists using sets. It's essentially just repeated Cartesian products with sets containing only one constant member. E.g.
$$ [10, 3, \emptyset] := \lbrace 10 \rbrace \times \lbrace 3 \rbrace \times \lbrace \emptyset \rbrace $$
What I'm stuck on is how to represent the list as a single set for any given number of members. For three members I'm guessing
$$ [a,b,c] := \Big\lbrace \big\lbrace\big\lbrace \lbrace a \rbrace, \lbrace a,b \rbrace \big\rbrace \big\rbrace, \big\lbrace \big\lbrace\lbrace a\rbrace, \lbrace a,b\rbrace \big\rbrace, c \big\rbrace \Big\rbrace $$
based on the idea that $(a,b) \times \lbrace c \rbrace = ((a,b),c)$. If you're having difficulty understanding the repeated nested braces, I recommend replacing the ordered pair $(a,b)$ with some unused, arbitrary symbol, e.g. $x$, figure out the ordered triplet as an ordered pair of $x$ and $c$, then use substitution to replace the arbitrary letter with the ordered pair.
Just to be clear, I'm using the following definition of list: an ordered collection of well-defined objects.
- Am I correct?
- How does one define a list in terms of sets for any arbitrary number of members of that list?