feat: add a typeclass for the continuum hypothesis

[eric-wieser]

This allows a proof from the Archive to be promoted to mathlib itself.

The proof strategy of iff_exists_sierpinski_pathological_pred was written almost entirely with public Gemini 3 Thinking, and then manually corrected.

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Migrated from #34075, which I no longer have push access to.

[github-actions]

PR summary 137deefa56

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.SetTheory.Cardinal.ContinuumHypothesis (new file)	1639

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Declarations diff (regex)

+ Cardinal.exists_sierpinski_pathological_partition_of_card_eq_aleph_one
+ Cardinal.exists_sierpinski_pathological_pred_of_card_eq_aleph_one
+ ContinuumHypothesis
+ aleph0_covby_continuum.{u}
+ continuum_eq_aleph_one.{u}
+ iff_aleph0_covby_continuum.{u}
+ iff_continuum_eq_aleph_one.{u}
+ iff_exists_sierpinski_pathological_partition
+ iff_exists_sierpinski_pathological_pred
+ of_aleph0_covby_continuum
+ of_continuum_eq_aleph_one

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean -- pending)

Computed after the build finishes.

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No changes to strong technical debt.

Increase in weak tech debt: (relative, absolute) = (1.00, 0.00)

Current number	Change	Type (weak)
exposed public sections	5060	1

Current commit 137deefa56
Reference commit 9939342d31

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.py pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).