LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A continuum-sized almost-disjoint family on omega

Statement

In ZFC there is an almost-disjoint family of infinite subsets of $ω$ of cardinality $2^{ℵ_{0}}$ .

Facts & Assumptions

Given: AC for the ambient cardinal comparison.

[F1]

Cardinal sum $κ \oplus λ$ , product $κ \otimes λ$ and exponentiation $κ^{λ}$ , and why they are written apart from the ordinal operations identifies $∣ 2^{ω} ∣ = 2^{ℵ_{0}}$ .

Proof

1.1

Fix an explicit bijection $e : 2^{< ω} \to ω$ . For each branch $x \in 2^{ω}$ , put $A_{x} = {e (x ↾ n) : n \in ω}$ . This set is infinite because distinct lengths give distinct nodes.

2.1

If $x \neq y$ , let $m$ be their first differing coordinate. Then $x ↾ n \neq y ↾ n$ for every $n > m$ , so $A_{x} \cap A_{y}$ is contained in the finite set of codes of their common initial segments. Also $A_{x} = A_{y}$ would force equality of every initial segment, so $x \mapsto A_{x}$ is injective. The family therefore has size $∣ 2^{ω} ∣ = 2^{ℵ_{0}}$ . The construction makes no arbitrary choice after $e$ is fixed.

F1step 1.1∎

Depends on

Used by

Cardinal exponentiation below the continuum under MA Theorem

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Karagila, Forcing & Symmetric Extensions, proof of Theorem 7.7 (standard reference, not scraped)