http://openproblemgarden.org/op/distributivity_of_a_lattice_of_funcoids_is_not_provable_without_axiom_of_choice Comments for "Distributivity of a lattice of funcoids is not provable without axiom of choice" en http://openproblemgarden.org/op/distributivity_of_a_lattice_of_funcoids_is_not_provable_without_axiom_of_choice <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/porton_victor">Porton</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/topology">Topology</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; Distributivity of the lattice <img class="teximage" src="/files/tex/d051d7da40c4a6b1d4234c0f74689d1bc7c994f1.png" alt="$ \mathsf{FCD}(A;B) $" /> of funcoids (for arbitrary sets <img class="teximage" src="/files/tex/7a8d9782350e8eb5a84c149576d83160492cbdd3.png" alt="$ A $" /> and <img class="teximage" src="/files/tex/4369e4eb2b0938fb27436a8c4f4a062f83d4d49e.png" alt="$ B $" />) is not provable in ZF (without axiom of choice). </div> <p>A similar conjecture:</p> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; <img class="teximage" src="/files/tex/c6fc3b6da0655ddeaeffe670703a33edcb4650f6.png" alt="$ a\setminus^{\ast} b = a\#b $" /> for arbitrary filters <img class="teximage" src="/files/tex/b1d91efbd5571a84788303f1137fb33fe82c43e2.png" alt="$ a $" /> and <img class="teximage" src="/files/tex/b94226d9717591da8122ae1467eda72a0f35d810.png" alt="$ b $" /> on a powerset cannot be proved in ZF (without axiom of choice). </div> </tr></td> </table> </td> </tr> </table> Porton, Victor axiom of choice distributive lattice distributivity funcoid reverse math reverse mathematics ZF ZFC Topology http://openproblemgarden.org/op/distributivity_of_a_lattice_of_funcoids_is_not_provable_without_axiom_of_choice#comment Sat, 24 Aug 2013 17:58:56 +0200 porton 56163 at http://openproblemgarden.org