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#deck:Math::Sets::The definition on trial
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In set theory, which two words are taken as undefined and described only by axioms?	'Set' and 'is a member of' (∈). Every definition of set in words (a collection, a box) is a picture, not a definition; the axioms say what sets do.	definition
State the axiom of extensionality, and say what question it settles.	Two sets with the same members are the same set. It settles when two sets are equal: membership is the only thing a set records.	axiom
Why is {2, 5} = {5, 2}?	Same members, so equal by extensionality. Order is not membership: a set does not record which member was written first.	axiom
Is {1, 1, 2} a legal way to write a set? Which set is it?	Yes: it is {1, 2}. Being listed twice is the same as being a member once, because a set records only whether each object is in it.	axiom
The textbook says a set is a collection of 'distinct objects'. What is that word really doing?	Not forbidding repeats in the list, since {1, 1, 2} is legal. It is reaching for extensionality, the rule for when two sets are equal, which the textbook never states.	trap
Why is it correct to say THE empty set rather than AN empty set?	Any two empty sets have the same members (none), so by extensionality they are equal. There is exactly one, written ∅.	theorem
What does 'well-defined' mean when applied to a set?	For every object it is settled whether it is a member or not. It does not mean anyone knows the answer.	definition
Give a set that is well-defined even though nobody knows whether it is empty.	The odd perfect numbers (odd n equal to the sum of its proper divisors). Any given n can be checked, so membership is settled; whether any exists is an open problem.	example
State Russell's paradox.	Let R = {x : x ∉ x}, the set of all sets that are not members of themselves. Then R ∈ R holds exactly when R ∉ R, a contradiction, so no such set exists.	theorem
What does Russell's paradox show about the definition 'a set is a well-defined collection'?	That a perfectly sharp rule (x ∉ x) can still fail to determine a set. Being well-defined is not enough; the rule must select from a set you already have.	trap
Whose foundational system did Russell's paradox break, and how did he hear of it?	Frege's. Russell found the paradox in 1901 and wrote to Frege in 1902; Frege's system assumed every property defines a set.	history
State the axiom of separation (Zermelo, 1908).	For any set A and property P, {x ∈ A : P(x)} is a set. A rule may select members from an existing set, never from 'everything'.	axiom
With separation in place, what does Russell's argument prove instead of a contradiction?	For any set U, the set R = {x ∈ U : x ∉ x} is not a member of U. So no set contains every set: there is no universal set.	theorem
Which older argument has the same shape as Russell's paradox?	Cantor's diagonal argument: build an object that differs from the y-th item at position y, so it is on no list. Russell's R differs from each set y on the question 'is y in y?'.	connection
Is {1} ∈ {1, 2}? Is {1} ⊆ {1, 2}?	No and yes. ∈ asks whether the object {1} is one of the members (they are 1 and 2); ⊆ asks whether every member of {1} is in {1, 2}.	trap
Which sets is ∅ a subset of, and why?	Every set, including ∅ itself. A ⊆ B means every member of A is in B, and ∅ has no member that could fail the test.	theorem
What is the difference between A ⊆ B and A ⊂ B?	A ⊆ B allows A = B. A ⊂ B (proper subset) means A ⊆ B and A ≠ B: B has at least one member A lacks.	definition
Is every set a subset of itself? A proper subset of itself?	Yes: every member of A is in A. No: a proper subset must be missing something.	theorem
Define A ∪ B, A ∩ B and A ∖ B in one line each.	A ∪ B: members of A or of B (or both). A ∩ B: members of both. A ∖ B: members of A that are not in B.	definition
When are two sets disjoint?	When they have no member in common: A ∩ B = ∅. ∅ is disjoint from every set, including itself.	definition
Does 'null set' mean the same thing in every book?	No. In algebra and logic it means ∅. In measure theory and probability it means a set of measure zero, which can be infinite: the Cantor set is a null set.	trap
Verdict: is 'a set is a well-defined collection of distinct objects' a good definition?	A useful first description, not a definition. 'Collection' is a synonym; when two sets are equal is left out; and Russell shows a well-defined rule can fail to give a set.	principle
What does the 'box' definition (a set is a box with things inside) do better than 'a well-defined collection'?	It is built around ∈ (a ∈ A, c ∉ A), it treats the picture as a picture, and it says a set is a single object, so a set can be a member of another set.	principle
What does the 'box' definition of a set leave out?	Any rule for when two sets are equal (extensionality) and any notion of well-defined. It cannot explain why {1, 1} = {1} or why there is only one empty set.	trap
Is ∈ transitive? Give the standard counterexample.	No. 1 ∈ {1} and {1} ∈ {{1}}, yet 1 ∉ {{1}}: the only member of {{1}} is {1}. A member of a member is not a member. (⊆ is transitive.)	trap
Is ∅ = {∅}?	No. ∅ has no members; {∅} has exactly one member, the empty set. An empty box inside a box is a box with something in it.	theorem
Name three ways the 'box' picture of a set misleads.	A box in a box is inside the outer box, but ∈ is not transitive; a thing sits in one box, but one element belongs to many sets; a box can hold two identical marbles, but a set has no copies.	trap
What does a, b ∈ A abbreviate?	a ∈ A and b ∈ A.	definition
Does Russell's paradox depend on the step 'C is either ordinary or extraordinary'?	No. With P meaning C ∈ C, the definition gives P ↔ ¬P. Assuming P gives ¬P, a contradiction, so ¬P; then ¬P gives P. No case split is used, and the argument is valid even in intuitionistic logic, which rejects excluded middle.	theorem
A contradiction refutes its premises together. Which premise does ZF set theory give up to escape Russell?	That C, the set of all ordinary sets, exists. Separation only lets you collect the ordinary members of a set you already have.	axiom
In ZF, which sets are 'ordinary' (not members of themselves)?	All of them, by the axiom of foundation. So the set of all ordinary sets would be the set of everything, which does not exist.	theorem
Which many-valued logic gives 'C ∈ C' a consistent truth value, and which do not?	Łukasiewicz logic: the value ½, since ½ ↔ ¬½ comes out fully true. Kleene's K3 and Gödel's logics also reject excluded middle, but no value there satisfies P ↔ ¬P.	logic
Why does no finite-valued Łukasiewicz logic rescue naive set theory?	Curry's sets C_k = {x : x ∈ x → (x ∈ x → … → false)} with k arrows need the truth value k/(k+1). A logic with finitely many values misses some of these, so some C_k gets no value at all.	logic
Name two non-classical escapes from Russell's paradox, and what each gives up.	Paraconsistent logic keeps the contradiction but stops it from proving everything. Contraction-free logic blocks the step 'P gives ¬P, therefore ¬P'. Both are research programmes; mainstream mathematics instead denies that R exists.	logic