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In the MA model the additivity of null and meagre equals the continuum

Statement

Let $P_{ω_{2}}$ be the $ω_{2}$ -length finite-support bookkeeping iteration over a ground model of ZFC+GCH and let $G$ be generic for it (The omega_2 iteration forces MA and continuum aleph_2). In the resulting model Martin's Axiom holds and $c = 2^{ℵ_{0}} = ℵ_{2}$ , and there the additivities of the two ideals of Add, cov, non and cof for null and meagre ideals are both equal to the continuum:

$add (N) = ℵ_{2} = add (M) .$

Facts & Assumptions

Given: The $ω_{2}$ -length bookkeeping iteration over a ZFC+GCH ground model, with generic $G$ , and the resulting model of ZFC, in which the Axiom of Choice holds.

[F1]

Over a ZFC+GCH ground model the $ω_{2}$ bookkeeping iteration is ccc and forces MA together with $2^{ℵ_{0}} = ℵ_{2}$ , hence not CH. (The omega_2 iteration forces MA and continuum aleph_2)

[F2]

In ZFC+MA the union of fewer than $2^{ℵ_{0}}$ Lebesgue-null subsets of the real line is null; in particular every set of reals of cardinality below the continuum is null. (MA makes unions of fewer than continuum many null sets null)

[F3]

In ZFC+MA the union of fewer than $2^{ℵ_{0}}$ meagre subsets of the real line is meagre; in particular every set of reals of cardinality below the continuum is meagre. (MA makes unions of fewer than continuum many meagre sets meagre)

[F4]

$add (I)$ for $I = N, M$ is the least cardinality of a subfamily of $I$ whose union is not in $I$ , the minimum being attained, and $add (I) \leq cof (I) \leq c = 2^{ℵ_{0}}$ . (Add, cov, non and cof for null and meagre ideals, Elementary bounds on ideal cardinal invariants)

[F5]

$ℵ_{2}$ is a cardinal and $ℵ_{2} = 2^{ℵ_{0}}$ in the model of [F1]; cardinalities are cardinals under the Axiom of Choice. (The successor cardinal $κ^{+}$ , the alephs $ℵ_{α}$ , the beths $ℶ_{α}$ , successor and limit cardinals, and the identifications $ℵ_{0} = ω$ and $ℵ_{1} = ω_{1}$ , Cardinal sum $κ \oplus λ$ , product $κ \otimes λ$ and exponentiation $κ^{λ}$ , and why they are written apart from the ordinal operations, The Axiom of Choice)

Proof

technique · direct

1.1

In the model of the given iteration, MA holds and $c = 2^{ℵ_{0}} = ℵ_{2}$ by [F1]; in particular the continuum is the cardinal $ℵ_{2} > 0$ , so the phrase "fewer than $2^{ℵ_{0}}$ " in [F2] and [F3] means "of cardinality below $ℵ_{2}$ ".

givenF1F5

2.1

$add (N) \leq ℵ_{2}$ and $add (M) \leq ℵ_{2}$ : by [F4] the additivity is at most the continuum, which is $ℵ_{2}$ by step 1.1.

step 1.1F4

2.2

$add (N) \geq ℵ_{2}$ : let $A \subseteq N$ with $∣ A ∣ < ℵ_{2}$ ; by step 1.1 the family has fewer than $2^{ℵ_{0}}$ members, so [F2] makes $⋃ A$ null, that is, $⋃ A \in N$ . Hence no subfamily of $N$ of size below $ℵ_{2}$ witnesses the additivity, and the minimum clause of [F4] gives $add (N) \geq ℵ_{2}$ .

step 1.1F2F4

2.3

$add (M) \geq ℵ_{2}$ : the same argument with [F3] in place of [F2] gives that every subfamily of $M$ of size below $ℵ_{2}$ has its union in $M$ , hence $add (M) \geq ℵ_{2}$ by the minimum clause of [F4].

step 1.1F3F4

3.1

Combining step 2.1 with steps 2.2 and 2.3 gives $add (N) = ℵ_{2} = add (M)$ , and $ℵ_{2} = c$ by step 1.1; the Axiom of Choice is used in the iteration and in reading the cardinalities of [F4] as cardinals. ∎

step 1.1step 2.1step 2.2step 2.3F5

Depends on

Used by

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Sources

Tomek Bartoszynski, Invariants of Measure and Category, Section 4 (MA and the additivity of the ideals), printed pp.8-9 (standard reference, not scraped)