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Russell.v
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Russell.v
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(* This program is free software; you can redistribute it and/or *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1 *)
(* of the License, or (at your option) any later version. *)
(* *)
(* This program is distributed in the hope that it will be useful, *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *)
(* GNU General Public License for more details. *)
(* *)
(* You should have received a copy of the GNU Lesser General Public *)
(* License along with this program; if not, write to the Free *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA *)
(* Contribution to the Coq Library V6.3 (July 1999) *)
(* Just for fun : a proof that there is no set of all sets, using *)
(* Russell's paradox construction *)
(* There, of course, are other proofs (foundation, etc) *)
Require Import Sets.
Require Import Axioms.
Theorem Russell : forall E : Ens, (forall E' : Ens, IN E' E) -> False.
intros U HU.
cut
((fun x : Ens => IN x x -> False) (Comp U (fun x : Ens => IN x x -> False))).
intros HR.
apply HR.
apply IN_P_Comp; auto with zfc.
intros w1 w2 HF e i; apply HF; apply IN_sound_left with w2; auto with zfc;
apply IN_sound_right with w2; auto with zfc.
intros H.
cut
(IN (Comp U (fun x : Ens => IN x x -> False))
(Comp U (fun x : Ens => IN x x -> False))).
change
((fun x : Ens => IN x x -> False) (Comp U (fun x : Ens => IN x x -> False)))
in |- *.
cut
(forall w1 w2 : Ens, (IN w1 w1 -> False) -> EQ w1 w2 -> IN w2 w2 -> False).
intros ww.
exact
(IN_Comp_P U (Comp U (fun x : Ens => IN x x -> False))
(fun x : Ens => IN x x -> False) ww H).
intros w1 w2 HF e i; apply HF; apply IN_sound_left with w2; auto with zfc;
apply IN_sound_right with w2; auto with zfc.
assumption.
Qed.