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Under CH the classical cardinal invariants all equal aleph one

Statement

In ZFC (with the Axiom of Choice The Axiom of Choice), assume the Continuum Hypothesis, in the form $c = 2^{ℵ_{0}} = ℵ_{1}$ (Cardinal sum $κ \oplus λ$ , product $κ \otimes λ$ and exponentiation $κ^{λ}$ , and why they are written apart from the ordinal operations, The successor cardinal $κ^{+}$ , the alephs $ℵ_{α}$ , the beths $ℶ_{α}$ , successor and limit cardinals, and the identifications $ℵ_{0} = ω$ and $ℵ_{1} = ω_{1}$ ). Then the six combinatorial invariants and the eight ideal invariants of this pair of pages all collapse to $ℵ_{1}$ :

$p = t = b = d = s = r = ℵ_{1}, add (I) = cov (I) = non (I) = cof (I) = ℵ_{1} (I = N, M) .$

Facts & Assumptions

Given: ZFC and the Continuum Hypothesis, stated as $c = 2^{ℵ_{0}} = ℵ_{1}$ .

[F1]

$p$ and $t$ are cardinals with $ℵ_{1} \leq p \leq t \leq c = 2^{ℵ_{0}}$ . (Basic bounds for p and t, The pseudointersection and tower numbers)

[F2]

$b$ and $d$ are cardinals with $ℵ_{1} \leq b = cf (b) \leq cf (d) \leq d \leq c$ . (Basic bounding and dominating relations, Eventual domination and the numbers b and d)

[F3]

$s$ and $r$ are cardinals with $ℵ_{1} \leq s \leq d \leq c$ and $ℵ_{1} \leq b \leq r \leq c$ . (Splitting and reaping comparisons with b and d, The splitting and reaping numbers)

[F4]

For $I = N$ and $I = M$ the eight numbers satisfy $ℵ_{1} \leq add (I) \leq \min (cov (I), non (I)) \leq \max (cov (I), non (I)) \leq cof (I) \leq c$ . (Elementary bounds on ideal cardinal invariants, Add, cov, non and cof for null and meagre ideals)

[F5]

In ZFC, hence under the Axiom of Choice, all the numbers above are cardinals, hence ordinals, and for ordinals $α, β, γ$ with $α \leq β \leq α$ one has $α = β$ , because exactly one of $α \in β$ , $α = β$ , $β \in α$ holds. (The Axiom of Choice, Cardinal (initial ordinal) and cardinality, Trichotomy and well-ordering of the ordinals)

Proof

technique · direct

1.1

By [F1], $ℵ_{1} \leq p \leq t \leq c = ℵ_{1}$ , so $p$ and $t$ are pinched between $ℵ_{1}$ and $ℵ_{1}$ .

givenF1

1.2

By [F2], $ℵ_{1} \leq b \leq d \leq c = ℵ_{1}$ , so $b$ and $d$ are pinched between $ℵ_{1}$ and $ℵ_{1}$ .

givenF2

1.3

By [F4], for $I = N$ and $I = M$ each of the four numbers lies between $ℵ_{1}$ and $c = ℵ_{1}$ .

givenF4

2.1

By [F3] together with step 1.2, $ℵ_{1} \leq s \leq d \leq ℵ_{1}$ and $ℵ_{1} \leq b \leq r \leq c = ℵ_{1}$ , so $s$ and $r$ are pinched between $ℵ_{1}$ and $ℵ_{1}$ .

step 1.2givenF3

3.1

Every one of the fourteen numbers is an ordinal $x$ with $ℵ_{1} \leq x \leq ℵ_{1}$ , so $x = ℵ_{1}$ by the antisymmetry clause of [F5]; this gives $p = t = b = d = s = r = ℵ_{1}$ and $add (I) = cov (I) = non (I) = cof (I) = ℵ_{1}$ for both ideals $I = N, M$ , as claimed. ∎

step 1.1step 1.2step 1.3step 2.1F5

Depends on

Used by

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Dependency tree · two levels

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Sources

J. D. Monk, Continuum cardinals, Theorem 1 and the surrounding elementary bounds, printed pp.1, 8, 15, 19 (standard reference, not scraped)
Tomek Bartoszynski, Invariants of Measure and Category, Sections 1-2, printed pp.1-3 (standard reference, not scraped)