The set of irrational numbers is G_δ (Π⁰₂) but not F_σ (Σ⁰₂). What does this demonstrate about the Borel hierarchy?
AThe Borel hierarchy collapses at level 2 — all Borel sets are either open or closed
BThe Borel hierarchy is strict: there are sets at each level that are not in any lower level, so the classification genuinely captures increasing complexity
CThe irrationals are not a Borel set, since they cannot be expressed as a countable union of closed sets
DG_δ and F_σ are interchangeable names for the same class of sets
The irrationals are G_δ (a countable intersection of open sets) but provably not F_σ (not a countable union of closed sets). This shows the hierarchy is genuine and does not collapse: each level contains sets not captured by lower levels. If the hierarchy collapsed, the classification program would be trivially uninteresting. The non-collapse result requires a Baire category argument and is fundamental to the subject.
Question 2 Multiple Choice
A set A ⊆ ℝ is analytic (Σ¹₁) and its complement is also analytic. What does Suslin's theorem conclude about A?
AA must be either open or closed
BA is Borel — it lies in the Borel σ-algebra, below the analytic level in the hierarchy
CA is Lebesgue measurable but not necessarily Borel
DA must be countable, since analytic sets that are also coanalytic are small
Suslin's theorem states: a set is Borel if and only if it is both analytic (Σ¹₁) and coanalytic (Π¹₁). Coanalytic means its complement is analytic. So if A is analytic and its complement is analytic (hence A is coanalytic), then A is Borel. This is one of the key boundary results in descriptive set theory — it characterizes the Borel sets from one level above via a clean intersection condition.
Question 3 True / False
Most subset of ℝ that can be explicitly described in a few sentences of mathematical English is a Borel set.
TTrue
FFalse
Answer: False
Analytic sets (Σ¹₁) can be explicitly described as projections of Borel sets, yet they need not be Borel — there exist analytic non-Borel sets, constructible by diagonalization via universal sets. More generally, 'describable' is a vague notion; the hierarchy formalizes exactly which descriptions (open, Gδ, Fσ, analytic, etc.) correspond to which level of definitional complexity. Not all explicitly described sets land in the Borel σ-algebra.
Question 4 True / False
The Axiom of Determinacy (AD) implies that all projective sets of reals are Lebesgue measurable, but AD contradicts the Axiom of Choice (AC).
TTrue
FFalse
Answer: True
This is correct. AD states that for every subset A of Baire space, one of the two players in the associated infinite game has a winning strategy. AD implies remarkable regularity: every set of reals is measurable, has the Baire property, and has the perfect set property. However, AC allows construction of non-measurable sets (like Vitali sets), which AD prohibits — so they genuinely contradict each other. AD is consistent with ZF (just not ZF + AC). Large cardinal axioms can imply determinacy for restricted projective classes while preserving AC.
Question 5 Short Answer
Explain the central theme connecting the Borel hierarchy, the projective hierarchy, and the regularity properties of sets. Why does definability matter for measurability?
Think about your answer, then reveal below.
Model answer: The central theme is that a set's position in the definability hierarchy — how it is built from open sets by countable union, complementation, and projection — determines its regularity properties (measurability, Baire property, perfect set property). Borel sets are well-behaved in all three senses. Analytic sets retain these properties. At higher projective levels (Σ¹₂ and beyond), whether sets are measurable depends on axioms beyond ZFC. Sets constructed using the Axiom of Choice non-constructively can be non-measurable — they escape the definability hierarchy entirely.
Definability matters for measurability because measurability is a regularity condition — a constraint on how sets interact with the σ-algebra. Sets with explicit combinatorial descriptions inherit structure that forces them to be measurable. Non-measurable sets like Vitali sets require a non-constructive choice function, which is precisely the source of their irregularity. Descriptive set theory's insight is: the more explicit the definition, the more controlled the behavior.