User module:Kingdon/Sandbox

Interface:Classical propositional calculus, Interface:First-order logic with quantifiability, Interface:Peano axioms, User interface:Kingdon/Sandbox, Interface:Basic arithmetic. Interface:Tarski's geometry axioms, User interface:Kingdon/Logic, Interface:Set theory.

 import (CLASSICAL Interface:Classical_propositional_calculus ) import (QUANTIFY_OVER_SETS Interface:First-order_logic_with_quantifiability (CLASSICAL) set.) import (SETS Interface:Set_theory (CLASSICAL QUANTIFY_OVER_SETS) )
 * 1) import (QUANTIFY_OVER_NUMBERS Interface:First-order_logic_with_quantifiability (CLASSICAL) number.)

import (QUANTIFY_OVER_NUMBERS User_interface:Kingdon/Logic (CLASSICAL QUANTIFY_OVER_SETS SETS) )
 * 1) import (TARSKI Interface:Tarski's_geometry_axioms (CLASSICAL FIRSTORDER) )
 * 2) import (PEANO Interface:Peano_axioms (CLASSICAL FIRSTORDER) )
 * 3) import (ARITHMETIC Interface:Basic_arithmetic (CLASSICAL FIRSTORDER) )
 * 4) import (BASIC Interface:Basic_operations_of_general_set_theory (CLASSICAL FIRSTORDER) )
 * 5) import (ZF Interface:Zermelo–Fraenkel_set_theory (CLASSICAL FIRSTORDER) )

var (formula p q φ ψ)
 * 1) var (variable x y x0 y0 x1 y1 z k n)
 * 2) var (object s t u)
 * 3) var (variable result x′)
 * 4) var (object A B C D)

var (variable x a) var (object X Y Z A)

def ((0) (∅)) def ((1) (singleton (∅))) def ((< X Y) (X ∈ Y))

def ((subst X a φ) (set.subst X a φ))

thm (SetEmptyMembership  ((∅) ∈ (singleton (∅))) ( (∅) SingletonMembership ))

thm (OneInUniverse  ((1) ∈ (universe)) ( (1) (0) UnorderedPairRightMembership ))

thm (EmptyMembershipWithSubst   (subst (1) a (((value a) ∈ (universe)) ∧ ((0) ∈ (value a)))) ( OneInUniverse SetEmptyMembership introduceConjunction  The substitution is.  (value a) (1) (universe) MembershipBuilderRR (value a) (1) (∅) MembershipBuilderLL buildConjunctionInConsequent

set.makeSubstExplicit eliminateBiconditionalForward applyModusPonens ))

 Here we prove.  thm (ThereExistsUniverse  ((∃ x φ) ↔ (set.∃ x (((value x) ∈ (universe)) ∧ φ))) (  is, by definition, equivalent to  and in turn to. We move the opening negation inside the quantifier to get.  x (((value x) ∈ (universe)) → (¬ φ)) set.NotForAll

((value x) ∈ (universe)) φ ConjunctionImplication x set.buildThereExists swapBiconditional applyBiconditionalTransitivity ))

thm (NotOne  (∃ x ((0) < (value x))) ( x EmptyMembershipWithSubst set.introduceThereExistsFromObject

x ((0) < (value x)) ThereExistsUniverse eliminateBiconditionalForward applyModusPonens ))

export (RESULT User_interface:Kingdon/Sandbox (CLASSICAL QUANTIFY_OVER_NUMBERS) ) 