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#deck:Math::Sets::What is a set
#tags:sets math
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Define a set in four rules.	'Set' and 'member of' are undefined. (1) Extensionality: two sets with the same members are the same set. (2) A set is one object, so it can be a member of another set. (3) Separation: a rule picks the members of a set you already have, {x ∈ A : P(x)}, never {x : P(x)}. (4) There is an empty set, and by (1) only one.	definition
Everything about sets is defined through one relation. Which, and how does the program check that nothing is lost?	Membership, ∈. Equality is 'each member of either is in the other' and subset is 'every member of A is in B'; written in 'in' alone, they agree with Python's == and <= on all 64 pairs of subsets of {0, 1, 2}.	definition
What does a set record, and what does it forget?	It records which objects are its members. It forgets the order they were named in, how many times, and the rule that picked them: {2, 5} = {5, 2}, {1, 1, 2} = {1, 2}, and two different rules can give one set.	axiom
Why is 'a set is a collection of things' not a definition, even though it is a fair description?	'Collection' is a synonym: it names the idea without reducing it to anything simpler. Something has to be the first undefined word, and for sets it is 'set' and 'member of'; the axioms say what they do, not what they are.	trap
Which rule of the definition makes 'THE empty set' a correct phrase, and why?	Extensionality. Two rules that select nothing have the same members, none, so they select the same set. The program finds one distinct empty set among {n in 0..9 : n² < 0} and {w in [set] : len(w) > 5}.	axiom
How many members does {∅} have, and which rule of the definition is at work?	One. A set is a single object, so ∅ can be a member of another set; {∅} is a box holding an empty box, and ∅ ≠ {∅}.	trap
How many levels down does ∈ look?	Exactly one. 1 ∈ {1} and {1} ∈ {{1}}, but 1 ∉ {{1}}: a member of a member is not a member. ⊆ is the relation that chains.	trap
Run Russell's rule under separation on A = {∅, {∅}, {{∅}}}. What is R = {x ∈ A : x ∉ x}, and where is it not?	R is a harmless set (here R = A, since no set is a member of itself), and R ∉ A: if R were a member y of A, 'y ∈ y' would get opposite answers from A's membership and from R's rule. So no set has every set as a member.	theorem
Why does 'a set is one object' matter beyond ∅ versus {∅}?	Because a set can then be a member of a set: functions become sets of pairs, numbers become sets, and Russell's question 'is this set a member of itself?' can be asked. It is what makes set theory a foundation and not a notation.	connection