June 25, 2004
Slightly Modified: February 20, 2005
Perfect Subset Property for co-analytic sets in ZF\P
Abstract: We prove that second order arithmetic plus every uncountable co-analytic set has a perfect subset is equiconsistent with ZFC.
Introduction: The question of whether every uncountable co-analytic set has a perfect subset (in other words, whether every co-analytic set has perfect subset property) has been an important problem in classical descriptive set theory. Eventually, the question has been shown undecidable in ZFC and a positive answer equiconsistent with the existence of an inaccessible cardinal. Nevertheless, based on strong set existence axioms (Σ11 determinacy would suffice), a consensus has emerged that every uncountable co-analytic set has a perfect subset.
In ZFC, perfect subset property for co-analytic sets is equivalent to the claim that ω1 is inaccessible to the reals, that is ω1 in V is inaccessible in L(r) for every real number r (which implies that ω1 is inaccessible in the constructible universe L). Conversely, given a model of ZFC with an inaccessible cardinal κ, by using forcing, specifically Levy collapse, we can collapse all cardinals below κ to ω to produce a model of ZFC + ω1 is inaccessible to the reals.
Second order arithmetic is the standard theory about integers and sets of integers. It consists of basic facts about integers, induction axiom (0∈X and ∀n (n ∈ X → n+1 ∈ X) implies every natural number I ∈ X), and comprehension axiom schema, which is the universal closure of ∃X ∀n (x∈X ↔ φ(n)) for every formula φ with parameters. Simpson's book  explains how to formalize and prove much of mathematics, including descriptive set theory and theory of constructible sets in (weak subsystems of) second order arithmetic. The membership relation for co-analytic sets is formalized by a Π11 formula with parameters. A perfect set is the set of all paths through a perfect tree, and can be coded by such perfect tree. A formula with one free real number variable defines an uncountable set of reals if there is no real number that enumerates all of the real numbers that satisfy the formula.
Second order arithmetic is inter-interpretable ZF\P + "every set is
countable" and hence equiconsistent with ZF minus power-set.
For convenience, we will work in ZF\P. A reference to an uncountable
set in ZF\P should be interpreted as a reference to a corresponding
Foundational Significance: The proof illustrates that as far as ZFC "knows", the class of all ordinals Ord is inaccessible: The proof uses Ord in place of an inaccessible cardinal in the Levy collapse. More importantly, the results contribute to the program of proving consistency and existence of well-behaved models of higher set theory from natural and intuitively true statements about integers and real numbers. A number of philosophers believe that the notion of an arbitrary set of real numbers is inherently vague or that uncountable sets exist only as syntactical constructs. At any rate, there is a greater doubt about higher set theory than about integers and real numbers, and it is important to place the study of higher set theory on a firm foundation.
Further Areas Study:
A model obtained by Col(ω, <Ord)
should have every projective set measurable, have Baire property, and
perfect subset property, but verifying this requires checking that
the usual proof can be modified to work for class forcing.
Restrictions of Separation and Replacement to ΣnZF
formulas should correspond to corresponding restrictions in ZFC\P +
(proper formalization of) every uncountable co-analytic set has a
perfect subset. A more ambitious project toward establishing
consistency of higher infinities would be to show inside second order
Σ1n+1 determinacy is equivalent to "Mn#(r) exists for every real r" (which implies that ZFC + there are n Woodin cardinals has a well-behaved model).
Theorem: ZF\P + "every co-analytic set/class has perfect subset property" is equiconsistent with ZFC.
Proof: Perfect subset property for co-analytic sets is equivalent to "for every real number r, the set of real numbers constructible from r is countable". The only uses of uncountable sets in the standard proof are routine manipulations of projective sets, so the proof goes through (with syntactic modifications) inside ZF\P.
Consider a model M of ZF\P. The notions in this paragraph are assumed to be relativized to M. The constructible sets under the membership relation form a model L of ZFC\P. Suppose that every set is countable and perfect subset property holds for co-analytic sets. We show that L satisfies ZFC by showing that if α is a (countable) ordinal, then L satisfies the existence of the power-set of α. Let r be a real number coding a bijection between α and ω. The set of real numbers in L(r) is countable, so the set of subsets of α in L(r)M and hence in LM is countable in M. Therefore, there is an ordinal β such that Lβ contains all constructible subsets of α, so Lβ +1 and hence L contains the constructible power set of α.
For the other direction, let M be a (countable) model of ZFC. Let G=Col(ω,
<Ord), which is the product of generic collapses Col(ω,
κ) for every infinite regular cardinal
κ. Since the forcing is class
forcing, the generic extension M[G] does not have to satisfy ZFC;
however, G is sufficiently tame for M[G] to satisfy ZFC\P. Because
all uncountable cardinals are collapsed, in M[G], every set is
countable. Let r be a real number in M[G]. Let g be a sufficiently
long initial segment of G--sufficiently long for M[g] to contain r.
Since M[g] satisfies ZFC, ω1L(r)
exists in M[g]. By absoluteness, the same ordinal is ω1L(r)
in M[G], so the set of all reals constructible from r is countable in
M[G]. Thus, in M[G] co-analytic sets satisfy perfect subset
 Simpson, Stephen. Subsystems of Second Order Arithmetic. Springer-Verlag: New York, 1999.