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FALSE: ZFC fixes the value of $2^{κ}$ for every infinite regular $κ$

Statement

False claim: ZFC fixes the value of $2^{κ}$ for every infinite regular cardinal $κ$ ; that is, the function $κ \mapsto 2^{κ}$ on the infinite regular cardinals is determined by the axioms of ZFC (Cardinal (initial ordinal) and cardinality).

The claim is refuted at the single regular cardinal $κ = ℵ_{1}$ : the two consistency pictures below, read externally in the finite-fragment sense, give $2^{ℵ_{1}} = ℵ_{2}$ and $2^{ℵ_{1}} = ℵ_{3}$ respectively, and $ℵ_{2}$ and $ℵ_{3}$ are distinct cardinals (The successor cardinal $κ^{+}$ , the alephs $ℵ_{α}$ , the beths $ℶ_{α}$ , successor and limit cardinals, and the identifications $ℵ_{0} = ω$ and $ℵ_{1} = ω_{1}$ ). What ZFC does prove is the necessary constraints on the function — $κ < 2^{κ}$ and $cf (2^{κ}) > κ$ — and the refutation here concerns the value, not those constraints.

Facts & Assumptions

Given: The metatheoretic hypothesis that ZF is consistent, in the finite-fragment sense of [F1] and [F3], and the Axiom of Choice inside the forcing constructions of [F2].

[F1]

A verified proof transformation establishes $Con (Z F) \Rightarrow Con (Z F C + G C H)$ and $Con (Z F) \Rightarrow Con (Z F C + C H)$ , with no assumption of a transitive set model of ZF. (Formal consistency of ZFC plus GCH relative to ZF, Positive relative consistency of CH and GCH)

[F2]

In ZFC, if $κ$ is infinite regular and $λ > κ$ satisfies $2^{< κ} = κ$ and $λ^{κ} = λ$ , then the forcing $Add (κ, λ)$ preserves all cardinals and forces $2^{κ} = λ$ ; over a ground model of GCH, $κ = ℵ_{1}$ and $λ = ℵ_{3}$ satisfy these hypotheses and give a cardinal-preserving extension with CH and $2^{ℵ_{1}} = ℵ_{3}$ . In particular a value $λ \geq κ^{+ +}$ is compatible with ZFC, while GCH asserts $2^{κ} = κ^{+}$ . (Higher Cohen forcing violates GCH at a regular cardinal, The Axiom of Choice)

[F3]

Externally, $Con (Z F C)$ implies $Con (Z F C + \neg C H)$ and $Con (Z F C + \neg G C H)$ ; the implication is a metatheorem obtained by applying a finite-fragment construction to any purported contradiction proof, and no PA proof of a uniform refutation transformer and no external transitive model is claimed. (Externally fixed-fragment relative consistency of not CH and not GCH)

[F4]

The aleph operation is strictly increasing, $ℵ_{α + 1} = ℵ_{α}^{+}$ is the least cardinal strictly above $ℵ_{α}$ and is the successor cardinal of $ℵ_{α}$ ; every $ℵ_{α}$ is an infinite cardinal, so $ℵ_{2} \neq ℵ_{3}$ and both exceed $ℵ_{1}$ ; in particular $c = 2^{ℵ_{0}}$ is a cardinal. (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, Cardinal (initial ordinal) and cardinality)

Refutation

technique · direct

1.1

First picture. [F1] gives $Con (Z F) \Rightarrow Con (Z F C + G C H)$ by a proof transformation, so the finite fragments of $Z F C + G C H$ are consistent whenever those of ZF are; in such a picture GCH holds, that is $2^{κ} = κ^{+}$ at every infinite cardinal $κ$ , so at $κ = ℵ_{1}$ the value is $2^{ℵ_{1}} = ℵ_{1}^{+} = ℵ_{2}$ .

F1F2F4

1.2

Second picture. Over a ground model of ZFC+GCH the cardinal parameters $κ = ℵ_{1}$ , $λ = ℵ_{3}$ satisfy the hypotheses of [F2], since $2^{< ℵ_{1}} = ℵ_{1}$ and $ℵ_{3}^{ℵ_{1}} = ℵ_{3}$ under GCH; the extension by $Add (ℵ_{1}, ℵ_{3})$ preserves all cardinals, keeps CH, and forces $2^{ℵ_{1}} = ℵ_{3}$ .

F2F4

1.3

Choice is used, and only where stated. The forcing of the second picture is a ZFC construction: $Add (ℵ_{1}, ℵ_{3})$ and its cardinal-preservation proof use the Axiom of Choice, and the identification of cardinals with alephs uses it as well; the first picture's relative-consistency theorem is a syntactic transformation that needs no choice in the metatheory.

F1F2F4

2.1

The two pictures disagree. One picture has $2^{ℵ_{1}} = ℵ_{2}$ and the other has $2^{ℵ_{1}} = ℵ_{3}$ , and $ℵ_{2} \neq ℵ_{3}$ because the aleph operation is strictly increasing; so the value of $2^{ℵ_{1}}$ is not the same in all pictures of ZFC, and no single value of $2^{ℵ_{1}}$ is a theorem of ZFC.

step 1.1step 1.2F4

3.1

By step 2.1 the claim is false at the regular cardinal $ℵ_{1}$ , and by [F3] the negative consistency statements are exactly the kind of metatheorem that records such failures; nothing here infers the existence of an external transitive model from $Con (Z F C)$ , and no independence or completeness claim beyond the two pictures is made. ∎

step 2.1step 1.3F3

Depends on

Used by

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Dependency tree · two levels

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Sources

T. Jech, Set Theory, Chapter 15 (Easton's theorem and the independence of the continuum function), printed pp.232-237 (standard reference, not scraped)

FALSE: ZFC fixes the value of 2κ2κ for every infinite regular κκ