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Distributivity of a lattice of funcoids is not provable without axiom of choice (Solved)

Importance: Low ✭
Author(s): Porton, Victor
Subject: Topology
Keywords: axiom of choice
distributive lattice
distributivity
funcoid
reverse math
reverse mathematics
ZF
ZFC
Recomm. for undergrads: no
Posted by: porton
on: August 24th, 2013
Solved by: Todd Trimble
Conjecture   Distributivity of the lattice $ \mathsf{FCD}(A;B) $ of funcoids (for arbitrary sets $ A $ and $ B $) is not provable in ZF (without axiom of choice).

A similar conjecture:

Conjecture   $ a\setminus^{\ast} b = a\#b $ for arbitrary filters $ a $ and $ b $ on a powerset cannot be proved in ZF (without axiom of choice).

See this blog post for a rationale of this conjecture.

See here for used notation.

The first conjecture is shown false (that is a proof without AC exists) by Todd Trimble.

Bibliography

The blog post where the conjecture have been introduced


* indicates original appearance(s) of problem.
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