Primitive. The name
Eps_i is a term of type
(set → prop) → set.
Axiom. (
Eps_i_ax) We take the following as an axiom:
∀P : set → prop, ∀x : set, P x → P (Eps_i P)
Definition. We define
True to be
∀p : prop, p → p of type
prop.
Definition. We define
False to be
∀p : prop, p of type
prop.
Definition. We define
not to be
λA : prop ⇒ A → False of type
prop → prop.
Notation. We use
¬ as a prefix operator with priority 700 corresponding to applying term
not.
Definition. We define
and to be
λA B : prop ⇒ ∀p : prop, (A → B → p) → p of type
prop → prop → prop.
Notation. We use
∧ as an infix operator with priority 780 and which associates to the left corresponding to applying term
and.
Definition. We define
or to be
λA B : prop ⇒ ∀p : prop, (A → p) → (B → p) → p of type
prop → prop → prop.
Notation. We use
∨ as an infix operator with priority 785 and which associates to the left corresponding to applying term
or.
Definition. We define
iff to be
λA B : prop ⇒ and (A → B) (B → A) of type
prop → prop → prop.
Notation. We use
↔ as an infix operator with priority 805 and no associativity corresponding to applying term
iff.
Beginning of Section Eq
Variable A : SType
Definition. We define
eq to be
λx y : A ⇒ ∀Q : A → A → prop, Q x y → Q y x of type
A → A → prop.
Definition. We define
neq to be
λx y : A ⇒ ¬ eq x y of type
A → A → prop.
End of Section Eq
Notation. We use
= as an infix operator with priority 502 and no associativity corresponding to applying term
eq.
Notation. We use
≠ as an infix operator with priority 502 and no associativity corresponding to applying term
neq.
Beginning of Section FE
Variable A B : SType
Axiom. (
func_ext) We take the following as an axiom:
∀f g : A → B, (∀x : A, f x = g x) → f = g
End of Section FE
Beginning of Section Ex
Variable A : SType
Definition. We define
ex to be
λQ : A → prop ⇒ ∀P : prop, (∀x : A, Q x → P) → P of type
(A → prop) → prop.
End of Section Ex
Notation. We use
∃ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using
ex.
Axiom. (
prop_ext) We take the following as an axiom:
∀p q : prop, iff p q → p = q
Primitive. The name
In is a term of type
set → set → prop.
Notation. We use
∈ as an infix operator with priority 500 and no associativity corresponding to applying term
In. Furthermore, we may write
∀ x ∈ A, B to mean
∀ x : set, x ∈ A → B.
Definition. We define
Subq to be
λA B ⇒ ∀x ∈ A, x ∈ B of type
set → set → prop.
Notation. We use
⊆ as an infix operator with priority 500 and no associativity corresponding to applying term
Subq. Furthermore, we may write
∀ x ⊆ A, B to mean
∀ x : set, x ⊆ A → B.
Axiom. (
set_ext) We take the following as an axiom:
∀X Y : set, X ⊆ Y → Y ⊆ X → X = Y
Axiom. (
In_ind) We take the following as an axiom:
∀P : set → prop, (∀X : set, (∀x ∈ X, P x) → P X) → ∀X : set, P X
Notation. We use
∃ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using
ex and handling ∈ or ⊆ ascriptions using
and.
Primitive. The name
Empty is a term of type
set.
Axiom. (
EmptyAx) We take the following as an axiom:
Primitive. The name
⋃ is a term of type
set → set.
Axiom. (
UnionEq) We take the following as an axiom:
∀X x, x ∈ ⋃ X ↔ ∃Y, x ∈ Y ∧ Y ∈ X
Primitive. The name
𝒫 is a term of type
set → set.
Axiom. (
PowerEq) We take the following as an axiom:
∀X Y : set, Y ∈ 𝒫 X ↔ Y ⊆ X
Primitive. The name
Repl is a term of type
set → (set → set) → set.
Notation.
{B| x ∈ A} is notation for
Repl A (λ x . B).
Axiom. (
ReplEq) We take the following as an axiom:
∀A : set, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ A} ↔ ∃x ∈ A, y = F x
Definition. We define
TransSet to be
λU : set ⇒ ∀x ∈ U, x ⊆ U of type
set → prop.
Definition. We define
Union_closed to be
λU : set ⇒ ∀X : set, X ∈ U → ⋃ X ∈ U of type
set → prop.
Definition. We define
Power_closed to be
λU : set ⇒ ∀X : set, X ∈ U → 𝒫 X ∈ U of type
set → prop.
Definition. We define
Repl_closed to be
λU : set ⇒ ∀X : set, X ∈ U → ∀F : set → set, (∀x : set, x ∈ X → F x ∈ U) → {F x|x ∈ X} ∈ U of type
set → prop.
Primitive. The name
UnivOf is a term of type
set → set.
Axiom. (
UnivOf_In) We take the following as an axiom:
Axiom. (
UnivOf_Min) We take the following as an axiom:
Axiom. (
FalseE) We take the following as an axiom:
Axiom. (
TrueI) We take the following as an axiom:
Axiom. (
andI) We take the following as an axiom:
∀A B : prop, A → B → A ∧ B
Axiom. (
andEL) We take the following as an axiom:
Axiom. (
andER) We take the following as an axiom:
Axiom. (
orIL) We take the following as an axiom:
Axiom. (
orIR) We take the following as an axiom:
Beginning of Section PropN
Variable P1 P2 P3 : prop
Axiom. (
and3I) We take the following as an axiom:
P1 → P2 → P3 → P1 ∧ P2 ∧ P3
Axiom. (
and3E) We take the following as an axiom:
P1 ∧ P2 ∧ P3 → (∀p : prop, (P1 → P2 → P3 → p) → p)
Axiom. (
or3I1) We take the following as an axiom:
Axiom. (
or3I2) We take the following as an axiom:
Axiom. (
or3I3) We take the following as an axiom:
Axiom. (
or3E) We take the following as an axiom:
P1 ∨ P2 ∨ P3 → (∀p : prop, (P1 → p) → (P2 → p) → (P3 → p) → p)
Variable P4 : prop
Axiom. (
and4I) We take the following as an axiom:
P1 → P2 → P3 → P4 → P1 ∧ P2 ∧ P3 ∧ P4
Variable P5 : prop
Axiom. (
and5I) We take the following as an axiom:
P1 → P2 → P3 → P4 → P5 → P1 ∧ P2 ∧ P3 ∧ P4 ∧ P5
End of Section PropN
Axiom. (
iffI) We take the following as an axiom:
∀A B : prop, (A → B) → (B → A) → (A ↔ B)
Axiom. (
iffEL) We take the following as an axiom:
∀A B : prop, (A ↔ B) → A → B
Axiom. (
iffER) We take the following as an axiom:
∀A B : prop, (A ↔ B) → B → A
Axiom. (
iff_refl) We take the following as an axiom:
Axiom. (
iff_sym) We take the following as an axiom:
∀A B : prop, (A ↔ B) → (B ↔ A)
Axiom. (
iff_trans) We take the following as an axiom:
∀A B C : prop, (A ↔ B) → (B ↔ C) → (A ↔ C)
Axiom. (
eq_i_tra) We take the following as an axiom:
∀x y z, x = y → y = z → x = z
Axiom. (
f_eq_i) We take the following as an axiom:
∀f : set → set, ∀x y, x = y → f x = f y
Axiom. (
neq_i_sym) We take the following as an axiom:
Definition. We define
nIn to be
λx X ⇒ ¬ In x X of type
set → set → prop.
Notation. We use
∉ as an infix operator with priority 502 and no associativity corresponding to applying term
nIn.
Axiom. (
Eps_i_ex) We take the following as an axiom:
∀P : set → prop, (∃x, P x) → P (Eps_i P)
Axiom. (
pred_ext) We take the following as an axiom:
∀P Q : set → prop, (∀x, P x ↔ Q x) → P = Q
Axiom. (
prop_ext_2) We take the following as an axiom:
∀p q : prop, (p → q) → (q → p) → p = q
Axiom. (
Subq_ref) We take the following as an axiom:
Axiom. (
Subq_tra) We take the following as an axiom:
∀X Y Z : set, X ⊆ Y → Y ⊆ Z → X ⊆ Z
Axiom. (
Subq_contra) We take the following as an axiom:
∀X Y z : set, X ⊆ Y → z ∉ Y → z ∉ X
Axiom. (
EmptyE) We take the following as an axiom:
Axiom. (
Subq_Empty) We take the following as an axiom:
Axiom. (
Empty_eq) We take the following as an axiom:
Axiom. (
UnionI) We take the following as an axiom:
∀X x Y : set, x ∈ Y → Y ∈ X → x ∈ ⋃ X
Axiom. (
UnionE) We take the following as an axiom:
∀X x : set, x ∈ ⋃ X → ∃Y : set, x ∈ Y ∧ Y ∈ X
Axiom. (
UnionE_impred) We take the following as an axiom:
∀X x : set, x ∈ ⋃ X → ∀p : prop, (∀Y : set, x ∈ Y → Y ∈ X → p) → p
Axiom. (
PowerI) We take the following as an axiom:
∀X Y : set, Y ⊆ X → Y ∈ 𝒫 X
Axiom. (
PowerE) We take the following as an axiom:
∀X Y : set, Y ∈ 𝒫 X → Y ⊆ X
Axiom. (
xm) We take the following as an axiom:
Axiom. (
dneg) We take the following as an axiom:
Axiom. (
eq_or_nand) We take the following as an axiom:
Primitive. The name
exactly1of2 is a term of type
prop → prop → prop.
Axiom. (
exactly1of2_E) We take the following as an axiom:
∀A B : prop, exactly1of2 A B → ∀p : prop, (A → ¬ B → p) → (¬ A → B → p) → p
Axiom. (
ReplI) We take the following as an axiom:
∀A : set, ∀F : set → set, ∀x : set, x ∈ A → F x ∈ {F x|x ∈ A}
Axiom. (
ReplE) We take the following as an axiom:
∀A : set, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ A} → ∃x ∈ A, y = F x
Axiom. (
ReplE_impred) We take the following as an axiom:
∀A : set, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ A} → ∀p : prop, (∀x : set, x ∈ A → y = F x → p) → p
Axiom. (
ReplE') We take the following as an axiom:
∀X, ∀f : set → set, ∀p : set → prop, (∀x ∈ X, p (f x)) → ∀y ∈ {f x|x ∈ X}, p y
Axiom. (
Repl_Empty) We take the following as an axiom:
Axiom. (
ReplEq_ext_sub) We take the following as an axiom:
∀X, ∀F G : set → set, (∀x ∈ X, F x = G x) → {F x|x ∈ X} ⊆ {G x|x ∈ X}
Axiom. (
ReplEq_ext) We take the following as an axiom:
∀X, ∀F G : set → set, (∀x ∈ X, F x = G x) → {F x|x ∈ X} = {G x|x ∈ X}
Axiom. (
Repl_inv_eq) We take the following as an axiom:
∀P : set → prop, ∀f g : set → set, (∀x, P x → g (f x) = x) → ∀X, (∀x ∈ X, P x) → {g y|y ∈ {f x|x ∈ X}} = X
Axiom. (
Repl_invol_eq) We take the following as an axiom:
∀P : set → prop, ∀f : set → set, (∀x, P x → f (f x) = x) → ∀X, (∀x ∈ X, P x) → {f y|y ∈ {f x|x ∈ X}} = X
Primitive. The name
If_i is a term of type
prop → set → set → set.
Notation.
if cond then T else E is notation corresponding to
If_i type cond T E where
type is the inferred type of
T.
Axiom. (
If_i_0) We take the following as an axiom:
Axiom. (
If_i_1) We take the following as an axiom:
Axiom. (
If_i_or) We take the following as an axiom:
Primitive. The name
UPair is a term of type
set → set → set.
Notation.
{x,y} is notation for
UPair x y.
Axiom. (
UPairE) We take the following as an axiom:
∀x y z : set, x ∈ {y,z} → x = y ∨ x = z
Axiom. (
UPairI1) We take the following as an axiom:
Axiom. (
UPairI2) We take the following as an axiom:
Primitive. The name
Sing is a term of type
set → set.
Notation.
{x} is notation for
Sing x.
Axiom. (
SingI) We take the following as an axiom:
Axiom. (
SingE) We take the following as an axiom:
∀x y : set, y ∈ {x} → y = x
Primitive. The name
binunion is a term of type
set → set → set.
Notation. We use
∪ as an infix operator with priority 345 and which associates to the left corresponding to applying term
binunion.
Axiom. (
binunionI1) We take the following as an axiom:
∀X Y z : set, z ∈ X → z ∈ X ∪ Y
Axiom. (
binunionI2) We take the following as an axiom:
∀X Y z : set, z ∈ Y → z ∈ X ∪ Y
Axiom. (
binunionE) We take the following as an axiom:
∀X Y z : set, z ∈ X ∪ Y → z ∈ X ∨ z ∈ Y
Axiom. (
binunionE') We take the following as an axiom:
∀X Y z, ∀p : prop, (z ∈ X → p) → (z ∈ Y → p) → (z ∈ X ∪ Y → p)
Axiom. (
binunion_asso) We take the following as an axiom:
∀X Y Z : set, X ∪ (Y ∪ Z) = (X ∪ Y) ∪ Z
Axiom. (
binunion_com) We take the following as an axiom:
∀X Y : set, X ∪ Y = Y ∪ X
Definition. We define
SetAdjoin to be
λX y ⇒ X ∪ {y} of type
set → set → set.
Notation. We now use the set enumeration notation
{...,...,...} in general. If 0 elements are given, then
Empty is used to form the corresponding term. If 1 element is given, then
Sing is used to form the corresponding term. If 2 elements are given, then
UPair is used to form the corresponding term. If more than elements are given, then
SetAdjoin is used to reduce to the case with one fewer elements.
Primitive. The name
famunion is a term of type
set → (set → set) → set.
Notation. We use
⋃ x [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using
famunion.
Axiom. (
famunionI) We take the following as an axiom:
∀X : set, ∀F : (set → set), ∀x y : set, x ∈ X → y ∈ F x → y ∈ ⋃x ∈ XF x
Axiom. (
famunionE) We take the following as an axiom:
∀X : set, ∀F : (set → set), ∀y : set, y ∈ (⋃x ∈ XF x) → ∃x ∈ X, y ∈ F x
Axiom. (
famunionE_impred) We take the following as an axiom:
∀X : set, ∀F : (set → set), ∀y : set, y ∈ (⋃x ∈ XF x) → ∀p : prop, (∀x, x ∈ X → y ∈ F x → p) → p
Beginning of Section SepSec
Variable X : set
Variable P : set → prop
Let z : set ≝ Eps_i (λz ⇒ z ∈ X ∧ P z)
Primitive. The name
Sep is a term of type
set.
End of Section SepSec
Notation.
{x ∈ A | B} is notation for
Sep A (λ x . B).
Axiom. (
SepI) We take the following as an axiom:
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ X → P x → x ∈ {x ∈ X|P x}
Axiom. (
SepE) We take the following as an axiom:
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → x ∈ X ∧ P x
Axiom. (
SepE1) We take the following as an axiom:
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → x ∈ X
Axiom. (
SepE2) We take the following as an axiom:
∀X : set, ∀P : (set → prop), ∀x : set, x ∈ {x ∈ X|P x} → P x
Axiom. (
Sep_Subq) We take the following as an axiom:
∀X : set, ∀P : set → prop, {x ∈ X|P x} ⊆ X
Axiom. (
Sep_In_Power) We take the following as an axiom:
∀X : set, ∀P : set → prop, {x ∈ X|P x} ∈ 𝒫 X
Primitive. The name
ReplSep is a term of type
set → (set → prop) → (set → set) → set.
Notation.
{B| x ∈ A, C} is notation for
ReplSep A (λ x . C) (λ x . B).
Axiom. (
ReplSepI) We take the following as an axiom:
∀X : set, ∀P : set → prop, ∀F : set → set, ∀x : set, x ∈ X → P x → F x ∈ {F x|x ∈ X, P x}
Axiom. (
ReplSepE) We take the following as an axiom:
∀X : set, ∀P : set → prop, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ X, P x} → ∃x : set, x ∈ X ∧ P x ∧ y = F x
Axiom. (
ReplSepE_impred) We take the following as an axiom:
∀X : set, ∀P : set → prop, ∀F : set → set, ∀y : set, y ∈ {F x|x ∈ X, P x} → ∀p : prop, (∀x ∈ X, P x → y = F x → p) → p
Primitive. The name
binintersect is a term of type
set → set → set.
Notation. We use
∩ as an infix operator with priority 340 and which associates to the left corresponding to applying term
binintersect.
Axiom. (
binintersectI) We take the following as an axiom:
∀X Y z, z ∈ X → z ∈ Y → z ∈ X ∩ Y
Axiom. (
binintersectE) We take the following as an axiom:
∀X Y z, z ∈ X ∩ Y → z ∈ X ∧ z ∈ Y
Primitive. The name
setminus is a term of type
set → set → set.
Notation. We use
∖ as an infix operator with priority 350 and no associativity corresponding to applying term
setminus.
Axiom. (
setminusI) We take the following as an axiom:
∀X Y z, (z ∈ X) → (z ∉ Y) → z ∈ X ∖ Y
Axiom. (
setminusE) We take the following as an axiom:
∀X Y z, (z ∈ X ∖ Y) → z ∈ X ∧ z ∉ Y
Axiom. (
setminusE1) We take the following as an axiom:
∀X Y z, (z ∈ X ∖ Y) → z ∈ X
Axiom. (
In_irref) We take the following as an axiom:
Axiom. (
In_no2cycle) We take the following as an axiom:
Primitive. The name
ordsucc is a term of type
set → set.
Axiom. (
ordsuccI1) We take the following as an axiom:
Axiom. (
ordsuccI2) We take the following as an axiom:
Axiom. (
ordsuccE) We take the following as an axiom:
Notation. Natural numbers 0,1,2,... are notation for the terms formed using
Empty as 0 and forming successors with
ordsucc.
Axiom. (
ordsucc_inj) We take the following as an axiom:
Axiom. (
In_0_1) We take the following as an axiom:
Axiom. (
In_0_2) We take the following as an axiom:
Axiom. (
In_1_2) We take the following as an axiom:
Definition. We define
nat_p to be
λn : set ⇒ ∀p : set → prop, p 0 → (∀x : set, p x → p (ordsucc x)) → p n of type
set → prop.
Axiom. (
nat_0) We take the following as an axiom:
Axiom. (
nat_ordsucc) We take the following as an axiom:
Axiom. (
nat_1) We take the following as an axiom:
Axiom. (
nat_2) We take the following as an axiom:
Axiom. (
nat_ind) We take the following as an axiom:
Axiom. (
nat_inv) We take the following as an axiom:
Axiom. (
nat_p_trans) We take the following as an axiom:
Axiom. (
nat_trans) We take the following as an axiom:
Axiom. (
cases_1) We take the following as an axiom:
∀i ∈ 1, ∀p : set → prop, p 0 → p i
Axiom. (
cases_2) We take the following as an axiom:
∀i ∈ 2, ∀p : set → prop, p 0 → p 1 → p i
Axiom. (
cases_3) We take the following as an axiom:
∀i ∈ 3, ∀p : set → prop, p 0 → p 1 → p 2 → p i
Axiom. (
neq_0_1) We take the following as an axiom:
Axiom. (
neq_1_0) We take the following as an axiom:
Axiom. (
neq_0_2) We take the following as an axiom:
Axiom. (
neq_2_0) We take the following as an axiom:
Axiom. (
neq_1_2) We take the following as an axiom:
Axiom. (
ZF_closed_E) We take the following as an axiom:
Primitive. The name
ω is a term of type
set.
Axiom. (
omega_nat_p) We take the following as an axiom:
Axiom. (
nat_p_omega) We take the following as an axiom:
Axiom. (
ordinal_1) We take the following as an axiom:
Axiom. (
ordinal_2) We take the following as an axiom:
Axiom. (
ordinal_ind) We take the following as an axiom:
∀p : set → prop, (∀alpha, ordinal alpha → (∀beta ∈ alpha, p beta) → p alpha) → ∀alpha, ordinal alpha → p alpha
Definition. We define
inj to be
λX Y f ⇒ (∀u ∈ X, f u ∈ Y) ∧ (∀u v ∈ X, f u = f v → u = v) of type
set → set → (set → set) → prop.
Definition. We define
bij to be
λX Y f ⇒ (∀u ∈ X, f u ∈ Y) ∧ (∀u v ∈ X, f u = f v → u = v) ∧ (∀w ∈ Y, ∃u ∈ X, f u = w) of type
set → set → (set → set) → prop.
Axiom. (
bijI) We take the following as an axiom:
∀X Y, ∀f : set → set, (∀u ∈ X, f u ∈ Y) → (∀u v ∈ X, f u = f v → u = v) → (∀w ∈ Y, ∃u ∈ X, f u = w) → bij X Y f
Axiom. (
bijE) We take the following as an axiom:
∀X Y, ∀f : set → set, bij X Y f → ∀p : prop, ((∀u ∈ X, f u ∈ Y) → (∀u v ∈ X, f u = f v → u = v) → (∀w ∈ Y, ∃u ∈ X, f u = w) → p) → p
Primitive. The name
inv is a term of type
set → (set → set) → set → set.
Axiom. (
surj_rinv) We take the following as an axiom:
∀X Y, ∀f : set → set, (∀w ∈ Y, ∃u ∈ X, f u = w) → ∀y ∈ Y, inv X f y ∈ X ∧ f (inv X f y) = y
Axiom. (
inj_linv) We take the following as an axiom:
∀X, ∀f : set → set, (∀u v ∈ X, f u = f v → u = v) → ∀x ∈ X, inv X f (f x) = x
Axiom. (
bij_inv) We take the following as an axiom:
∀X Y, ∀f : set → set, bij X Y f → bij Y X (inv X f)
Axiom. (
bij_id) We take the following as an axiom:
Axiom. (
bij_comp) We take the following as an axiom:
∀X Y Z : set, ∀f g : set → set, bij X Y f → bij Y Z g → bij X Z (λx ⇒ g (f x))
Definition. We define
equip to be
λX Y : set ⇒ ∃f : set → set, bij X Y f of type
set → set → prop.
Axiom. (
equip_ref) We take the following as an axiom:
Axiom. (
equip_sym) We take the following as an axiom:
Axiom. (
equip_tra) We take the following as an axiom:
Beginning of Section SchroederBernstein
Axiom. (
KnasterTarski_set) We take the following as an axiom:
∀A, ∀F : set → set, (∀U ∈ 𝒫 A, F U ∈ 𝒫 A) → (∀U V ∈ 𝒫 A, U ⊆ V → F U ⊆ F V) → ∃Y ∈ 𝒫 A, F Y = Y
Axiom. (
image_In_Power) We take the following as an axiom:
∀A B, ∀f : set → set, (∀x ∈ A, f x ∈ B) → ∀U ∈ 𝒫 A, {f x|x ∈ U} ∈ 𝒫 B
End of Section SchroederBernstein
Beginning of Section PigeonHole
End of Section PigeonHole
Definition. We define
finite to be
λX ⇒ ∃n ∈ ω, equip X n of type
set → prop.
Axiom. (
finite_ind) We take the following as an axiom:
Axiom. (
Subq_finite) We take the following as an axiom:
Axiom. (
exandE_i) We take the following as an axiom:
∀P Q : set → prop, (∃x, P x ∧ Q x) → ∀r : prop, (∀x, P x → Q x → r) → r
Axiom. (
exandE_ii) We take the following as an axiom:
∀P Q : (set → set) → prop, (∃x : set → set, P x ∧ Q x) → ∀p : prop, (∀x : set → set, P x → Q x → p) → p
Axiom. (
exandE_iii) We take the following as an axiom:
∀P Q : (set → set → set) → prop, (∃x : set → set → set, P x ∧ Q x) → ∀p : prop, (∀x : set → set → set, P x → Q x → p) → p
Axiom. (
exandE_iiii) We take the following as an axiom:
∀P Q : (set → set → set → set) → prop, (∃x : set → set → set → set, P x ∧ Q x) → ∀p : prop, (∀x : set → set → set → set, P x → Q x → p) → p
Beginning of Section Descr_ii
Variable P : (set → set) → prop
Primitive. The name
Descr_ii is a term of type
set → set.
Hypothesis Pex : ∃f : set → set, P f
Hypothesis Puniq : ∀f g : set → set, P f → P g → f = g
End of Section Descr_ii
Beginning of Section Descr_iii
Variable P : (set → set → set) → prop
Primitive. The name
Descr_iii is a term of type
set → set → set.
Hypothesis Pex : ∃f : set → set → set, P f
Hypothesis Puniq : ∀f g : set → set → set, P f → P g → f = g
End of Section Descr_iii
Beginning of Section Descr_Vo1
Primitive. The name
Descr_Vo1 is a term of type
Vo 1.
Hypothesis Pex : ∃f : Vo 1, P f
Hypothesis Puniq : ∀f g : Vo 1, P f → P g → f = g
End of Section Descr_Vo1
Beginning of Section If_ii
Variable p : prop
Variable f g : set → set
Primitive. The name
If_ii is a term of type
set → set.
Axiom. (
If_ii_1) We take the following as an axiom:
Axiom. (
If_ii_0) We take the following as an axiom:
End of Section If_ii
Beginning of Section If_iii
Variable p : prop
Variable f g : set → set → set
Primitive. The name
If_iii is a term of type
set → set → set.
Axiom. (
If_iii_1) We take the following as an axiom:
Axiom. (
If_iii_0) We take the following as an axiom:
End of Section If_iii
Beginning of Section EpsilonRec_i
Variable F : set → (set → set) → set
Primitive. The name
In_rec_i is a term of type
set → set.
Hypothesis Fr : ∀X : set, ∀g h : set → set, (∀x ∈ X, g x = h x) → F X g = F X h
Axiom. (
In_rec_i_eq) We take the following as an axiom:
End of Section EpsilonRec_i
Beginning of Section EpsilonRec_ii
Variable F : set → (set → (set → set)) → (set → set)
Primitive. The name
In_rec_ii is a term of type
set → (set → set).
Hypothesis Fr : ∀X : set, ∀g h : set → (set → set), (∀x ∈ X, g x = h x) → F X g = F X h
End of Section EpsilonRec_ii
Beginning of Section EpsilonRec_iii
Variable F : set → (set → (set → set → set)) → (set → set → set)
Primitive. The name
In_rec_iii is a term of type
set → (set → set → set).
Hypothesis Fr : ∀X : set, ∀g h : set → (set → set → set), (∀x ∈ X, g x = h x) → F X g = F X h
End of Section EpsilonRec_iii
Beginning of Section NatRec
Variable z : set
Variable f : set → set → set
Let F : set → (set → set) → set ≝ λn g ⇒ if ⋃ n ∈ n then f (⋃ n) (g (⋃ n)) else z
Axiom. (
nat_primrec_r) We take the following as an axiom:
∀X : set, ∀g h : set → set, (∀x ∈ X, g x = h x) → F X g = F X h
End of Section NatRec
Beginning of Section NatArith
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_nat.
Axiom. (
add_nat_0R) We take the following as an axiom:
Axiom. (
add_nat_SR) We take the following as an axiom:
Axiom. (
add_nat_p) We take the following as an axiom:
Definition. We define
mul_nat to be
λn m : set ⇒ nat_primrec 0 (λ_ r ⇒ n + r) m of type
set → set → set.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_nat.
Axiom. (
mul_nat_0R) We take the following as an axiom:
Axiom. (
mul_nat_SR) We take the following as an axiom:
Axiom. (
mul_nat_p) We take the following as an axiom:
End of Section NatArith
Axiom. (
Inj1_eq) We take the following as an axiom:
Axiom. (
Inj1I1) We take the following as an axiom:
Axiom. (
Inj1I2) We take the following as an axiom:
Axiom. (
Inj1E) We take the following as an axiom:
Axiom. (
Inj1NE1) We take the following as an axiom:
Axiom. (
Inj1NE2) We take the following as an axiom:
Definition. We define
Inj0 to be
λX ⇒ {Inj1 x|x ∈ X} of type
set → set.
Axiom. (
Inj0I) We take the following as an axiom:
Axiom. (
Inj0E) We take the following as an axiom:
Axiom. (
Unj_eq) We take the following as an axiom:
Axiom. (
Unj_Inj1_eq) We take the following as an axiom:
Axiom. (
Inj1_inj) We take the following as an axiom:
Axiom. (
Unj_Inj0_eq) We take the following as an axiom:
Axiom. (
Inj0_inj) We take the following as an axiom:
Axiom. (
Inj0_0) We take the following as an axiom:
Notation. We use
+ as an infix operator with priority 450 and which associates to the left corresponding to applying term
setsum.
Axiom. (
Inj0_setsum) We take the following as an axiom:
∀X Y x : set, x ∈ X → Inj0 x ∈ X + Y
Axiom. (
Inj1_setsum) We take the following as an axiom:
∀X Y y : set, y ∈ Y → Inj1 y ∈ X + Y
Axiom. (
setsum_0_0) We take the following as an axiom:
Beginning of Section pair_setsum
Axiom. (
pairI0) We take the following as an axiom:
∀X Y x, x ∈ X → pair 0 x ∈ pair X Y
Axiom. (
pairI1) We take the following as an axiom:
∀X Y y, y ∈ Y → pair 1 y ∈ pair X Y
Axiom. (
pairE) We take the following as an axiom:
∀X Y z, z ∈ pair X Y → (∃x ∈ X, z = pair 0 x) ∨ (∃y ∈ Y, z = pair 1 y)
Axiom. (
pairE0) We take the following as an axiom:
∀X Y x, pair 0 x ∈ pair X Y → x ∈ X
Axiom. (
pairE1) We take the following as an axiom:
∀X Y y, pair 1 y ∈ pair X Y → y ∈ Y
Axiom. (
proj0I) We take the following as an axiom:
Axiom. (
proj0E) We take the following as an axiom:
Axiom. (
proj1I) We take the following as an axiom:
Axiom. (
proj1E) We take the following as an axiom:
Definition. We define
Sigma to be
λX Y ⇒ ⋃x ∈ X{pair x y|y ∈ Y x} of type
set → (set → set) → set.
Notation. We use
∑ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using
Sigma.
Axiom. (
pair_Sigma) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀x ∈ X, ∀y ∈ Y x, pair x y ∈ ∑x ∈ X, Y x
Axiom. (
proj0_Sigma) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → proj0 z ∈ X
Axiom. (
proj1_Sigma) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → proj1 z ∈ Y (proj0 z)
Axiom. (
pair_Sigma_E1) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀x y : set, pair x y ∈ (∑x ∈ X, Y x) → y ∈ Y x
Axiom. (
Sigma_E) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → ∃x ∈ X, ∃y ∈ Y x, z = pair x y
Definition. We define
setprod to be
λX Y : set ⇒ ∑x ∈ X, Y of type
set → set → set.
Notation. We use
⨯ as an infix operator with priority 440 and which associates to the left corresponding to applying term
setprod.
Let lam : set → (set → set) → set ≝ Sigma
Definition. We define
ap to be
λf x ⇒ {proj1 z|z ∈ f, ∃y : set, z = pair x y} of type
set → set → set.
Notation. When
x is a set, a term
x y is notation for
ap x y.
Notation.
λ x ∈ A ⇒ B is notation for the set
Sigma A (λ x : set ⇒ B).
Notation. We now use n-tuple notation (
a0,...,
an-1) for n ≥ 2 for λ i ∈
n .
if i = 0
then a0 else ... if i =
n-2 then an-2 else an-1.
Axiom. (
lamI) We take the following as an axiom:
∀X : set, ∀F : set → set, ∀x ∈ X, ∀y ∈ F x, pair x y ∈ λx ∈ X ⇒ F x
Axiom. (
lamE) We take the following as an axiom:
∀X : set, ∀F : set → set, ∀z : set, z ∈ (λx ∈ X ⇒ F x) → ∃x ∈ X, ∃y ∈ F x, z = pair x y
Axiom. (
apI) We take the following as an axiom:
∀f x y, pair x y ∈ f → y ∈ f x
Axiom. (
apE) We take the following as an axiom:
∀f x y, y ∈ f x → pair x y ∈ f
Axiom. (
beta) We take the following as an axiom:
∀X : set, ∀F : set → set, ∀x : set, x ∈ X → (λx ∈ X ⇒ F x) x = F x
Axiom. (
proj0_ap_0) We take the following as an axiom:
Axiom. (
proj1_ap_1) We take the following as an axiom:
Axiom. (
pair_ap_0) We take the following as an axiom:
∀x y : set, (pair x y) 0 = x
Axiom. (
pair_ap_1) We take the following as an axiom:
∀x y : set, (pair x y) 1 = y
Axiom. (
ap0_Sigma) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → (z 0) ∈ X
Axiom. (
ap1_Sigma) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀z : set, z ∈ (∑x ∈ X, Y x) → (z 1) ∈ (Y (z 0))
Definition. We define
pair_p to be
λu : set ⇒ pair (u 0) (u 1) = u of type
set → prop.
Axiom. (
pair_p_I) We take the following as an axiom:
Axiom. (
tuple_pair) We take the following as an axiom:
∀x y : set, pair x y = (x,y)
Definition. We define
Pi to be
λX Y ⇒ {f ∈ 𝒫 (∑x ∈ X, ⋃ (Y x))|∀x ∈ X, f x ∈ Y x} of type
set → (set → set) → set.
Notation. We use
∏ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using
Pi.
Axiom. (
PiI) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀f : set, (∀u ∈ f, pair_p u ∧ u 0 ∈ X) → (∀x ∈ X, f x ∈ Y x) → f ∈ ∏x ∈ X, Y x
Axiom. (
lam_Pi) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀F : set → set, (∀x ∈ X, F x ∈ Y x) → (λx ∈ X ⇒ F x) ∈ (∏x ∈ X, Y x)
Axiom. (
ap_Pi) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀f : set, ∀x : set, f ∈ (∏x ∈ X, Y x) → x ∈ X → f x ∈ Y x
Definition. We define
setexp to be
λX Y : set ⇒ ∏y ∈ Y, X of type
set → set → set.
Notation. We use
:^: as an infix operator with priority 430 and which associates to the left corresponding to applying term
setexp.
Axiom. (
lamI2) We take the following as an axiom:
∀X, ∀F : set → set, ∀x ∈ X, ∀y ∈ F x, (x,y) ∈ λx ∈ X ⇒ F x
Beginning of Section Tuples
Variable x0 x1 : set
End of Section Tuples
Axiom. (
ReplEq_setprod_ext) We take the following as an axiom:
∀X Y, ∀F G : set → set → set, (∀x ∈ X, ∀y ∈ Y, F x y = G x y) → {F (w 0) (w 1)|w ∈ X ⨯ Y} = {G (w 0) (w 1)|w ∈ X ⨯ Y}
Axiom. (
tuple_2_Sigma) We take the following as an axiom:
∀X : set, ∀Y : set → set, ∀x ∈ X, ∀y ∈ Y x, (x,y) ∈ ∑x ∈ X, Y x
Axiom. (
tuple_2_setprod) We take the following as an axiom:
∀X : set, ∀Y : set, ∀x ∈ X, ∀y ∈ Y, (x,y) ∈ X ⨯ Y
End of Section pair_setsum
Notation. We use
∑ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using
Sigma.
Notation. We use
⨯ as an infix operator with priority 440 and which associates to the left corresponding to applying term
setprod.
Notation. We use
∏ x...y [possibly with ascriptions] , B as a binder notation corresponding to a term constructed using
Pi.
Notation. We use
:^: as an infix operator with priority 430 and which associates to the left corresponding to applying term
setexp.
Primitive. The name
DescrR_i_io_1 is a term of type
(set → (set → prop) → prop) → set.
Primitive. The name
DescrR_i_io_2 is a term of type
(set → (set → prop) → prop) → set → prop.
Definition. We define
PNoEq_ to be
λalpha p q ⇒ ∀beta ∈ alpha, p beta ↔ q beta of type
set → (set → prop) → (set → prop) → prop.
Axiom. (
PNoEq_ref_) We take the following as an axiom:
∀alpha, ∀p : set → prop, PNoEq_ alpha p p
Axiom. (
PNoEq_sym_) We take the following as an axiom:
∀alpha, ∀p q : set → prop, PNoEq_ alpha p q → PNoEq_ alpha q p
Axiom. (
PNoEq_tra_) We take the following as an axiom:
Definition. We define
PNoLt_ to be
λalpha p q ⇒ ∃beta ∈ alpha, PNoEq_ beta p q ∧ ¬ p beta ∧ q beta of type
set → (set → prop) → (set → prop) → prop.
Axiom. (
PNoLt_E_) We take the following as an axiom:
Axiom. (
PNoLt_irref_) We take the following as an axiom:
∀alpha, ∀p : set → prop, ¬ PNoLt_ alpha p p
Axiom. (
PNoLt_mon_) We take the following as an axiom:
Axiom. (
PNoLt_tra_) We take the following as an axiom:
Primitive. The name
PNoLt is a term of type
set → (set → prop) → set → (set → prop) → prop.
Axiom. (
PNoLtI1) We take the following as an axiom:
Axiom. (
PNoLtI2) We take the following as an axiom:
Axiom. (
PNoLtI3) We take the following as an axiom:
Axiom. (
PNoLtE) We take the following as an axiom:
Axiom. (
PNoLt_irref) We take the following as an axiom:
∀alpha, ∀p : set → prop, ¬ PNoLt alpha p alpha p
Axiom. (
PNoLtEq_tra) We take the following as an axiom:
Axiom. (
PNoEqLt_tra) We take the following as an axiom:
Axiom. (
PNoLt_tra) We take the following as an axiom:
Definition. We define
PNoLe to be
λalpha p beta q ⇒ PNoLt alpha p beta q ∨ alpha = beta ∧ PNoEq_ alpha p q of type
set → (set → prop) → set → (set → prop) → prop.
Axiom. (
PNoLeI1) We take the following as an axiom:
Axiom. (
PNoLeI2) We take the following as an axiom:
∀alpha, ∀p q : set → prop, PNoEq_ alpha p q → PNoLe alpha p alpha q
Axiom. (
PNoLe_ref) We take the following as an axiom:
∀alpha, ∀p : set → prop, PNoLe alpha p alpha p
Axiom. (
PNoLtLe_tra) We take the following as an axiom:
Axiom. (
PNoLeLt_tra) We take the following as an axiom:
Axiom. (
PNoEqLe_tra) We take the following as an axiom:
Axiom. (
PNoLe_tra) We take the following as an axiom:
Definition. We define
PNo_downc to be
λL alpha p ⇒ ∃beta, ordinal beta ∧ ∃q : set → prop, L beta q ∧ PNoLe alpha p beta q of type
(set → (set → prop) → prop) → set → (set → prop) → prop.
Definition. We define
PNo_upc to be
λR alpha p ⇒ ∃beta, ordinal beta ∧ ∃q : set → prop, R beta q ∧ PNoLe beta q alpha p of type
(set → (set → prop) → prop) → set → (set → prop) → prop.
Axiom. (
PNoLe_downc) We take the following as an axiom:
Axiom. (
PNo_downc_ref) We take the following as an axiom:
∀L : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, L alpha p → PNo_downc L alpha p
Axiom. (
PNo_upc_ref) We take the following as an axiom:
∀R : set → (set → prop) → prop, ∀alpha, ordinal alpha → ∀p : set → prop, R alpha p → PNo_upc R alpha p
Axiom. (
PNoLe_upc) We take the following as an axiom:
Definition. We define
PNoLt_pwise to be
λL R ⇒ ∀gamma, ordinal gamma → ∀p : set → prop, L gamma p → ∀delta, ordinal delta → ∀q : set → prop, R delta q → PNoLt gamma p delta q of type
(set → (set → prop) → prop) → (set → (set → prop) → prop) → prop.
Axiom. (
PNo_extend0_eq) We take the following as an axiom:
∀alpha, ∀p : set → prop, PNoEq_ alpha p (λdelta ⇒ p delta ∧ delta ≠ alpha)
Axiom. (
PNo_extend1_eq) We take the following as an axiom:
∀alpha, ∀p : set → prop, PNoEq_ alpha p (λdelta ⇒ p delta ∨ delta = alpha)
Definition. We define
PNo_lenbdd to be
λalpha L ⇒ ∀beta, ∀p : set → prop, L beta p → beta ∈ alpha of type
set → (set → (set → prop) → prop) → prop.
Definition. We define
PNo_least_rep2 to be
λL R beta p ⇒ PNo_least_rep L R beta p ∧ ∀x, x ∉ beta → ¬ p x of type
(set → (set → prop) → prop) → (set → (set → prop) → prop) → set → (set → prop) → prop.
Primitive. The name
PNo_bd is a term of type
(set → (set → prop) → prop) → (set → (set → prop) → prop) → set.
Primitive. The name
PNo_pred is a term of type
(set → (set → prop) → prop) → (set → (set → prop) → prop) → set → prop.
Axiom. (
PNo_bd_pred) We take the following as an axiom:
Axiom. (
PNo_bd_In) We take the following as an axiom:
Beginning of Section TaggedSets
Notation. We use
' as a postfix operator with priority 100 corresponding to applying term
tag.
Definition. We define
SNoElts_ to be
λalpha ⇒ alpha ∪ {beta '|beta ∈ alpha} of type
set → set.
Axiom. (
SNoElts_mon) We take the following as an axiom:
Axiom. (
PNoEq_PSNo) We take the following as an axiom:
Axiom. (
SNo_PSNo) We take the following as an axiom:
Primitive. The name
SNo is a term of type
set → prop.
Axiom. (
SNo_SNo) We take the following as an axiom:
Primitive. The name
SNoLev is a term of type
set → set.
Axiom. (
SNoLev_uniq) We take the following as an axiom:
Axiom. (
SNoLev_prop) We take the following as an axiom:
Axiom. (
SNoLev_) We take the following as an axiom:
Axiom. (
SNoLev_PSNo) We take the following as an axiom:
Axiom. (
SNo_Subq) We take the following as an axiom:
Definition. We define
SNoEq_ to be
λalpha x y ⇒ PNoEq_ alpha (λbeta ⇒ beta ∈ x) (λbeta ⇒ beta ∈ y) of type
set → set → set → prop.
Axiom. (
SNoEq_I) We take the following as an axiom:
Axiom. (
SNo_eq) We take the following as an axiom:
End of Section TaggedSets
Notation. We use
< as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLt.
Notation. We use
≤ as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLe.
Axiom. (
SNoLtLe) We take the following as an axiom:
Axiom. (
SNoLeE) We take the following as an axiom:
Axiom. (
SNoEq_sym_) We take the following as an axiom:
Axiom. (
SNoEq_tra_) We take the following as an axiom:
Axiom. (
SNoLtE) We take the following as an axiom:
Axiom. (
SNoLtI2) We take the following as an axiom:
Axiom. (
SNoLtI3) We take the following as an axiom:
Axiom. (
SNoLt_irref) We take the following as an axiom:
Axiom. (
SNoLt_tra) We take the following as an axiom:
Axiom. (
SNoLe_ref) We take the following as an axiom:
Axiom. (
SNoLtLe_tra) We take the following as an axiom:
Axiom. (
SNoLeLt_tra) We take the following as an axiom:
Axiom. (
SNoLe_tra) We take the following as an axiom:
Axiom. (
SNoLtLe_or) We take the following as an axiom:
Definition. We define
SNoCutP to be
λL R ⇒ (∀x ∈ L, SNo x) ∧ (∀y ∈ R, SNo y) ∧ (∀x ∈ L, ∀y ∈ R, x < y) of type
set → set → prop.
Axiom. (
SNoCutP_L_0) We take the following as an axiom:
Axiom. (
SNoCutP_0_R) We take the following as an axiom:
Axiom. (
SNoCutP_0_0) We take the following as an axiom:
Axiom. (
SNoS_E) We take the following as an axiom:
Beginning of Section TaggedSets2
Notation. We use
' as a postfix operator with priority 100 corresponding to applying term
tag.
Axiom. (
SNoS_I) We take the following as an axiom:
Axiom. (
SNoS_I2) We take the following as an axiom:
Axiom. (
SNoS_Subq) We take the following as an axiom:
Axiom. (
SNoS_E2) We take the following as an axiom:
Axiom. (
SNoS_In_neq) We take the following as an axiom:
Axiom. (
SNoS_SNoLev) We take the following as an axiom:
Axiom. (
SNoL_E) We take the following as an axiom:
Axiom. (
SNoR_E) We take the following as an axiom:
Axiom. (
SNoL_SNoS_) We take the following as an axiom:
Axiom. (
SNoR_SNoS_) We take the following as an axiom:
Axiom. (
SNoL_SNoS) We take the following as an axiom:
Axiom. (
SNoR_SNoS) We take the following as an axiom:
Axiom. (
SNoL_I) We take the following as an axiom:
Axiom. (
SNoR_I) We take the following as an axiom:
Axiom. (
SNo_eta) We take the following as an axiom:
Axiom. (
SNoCut_Le) We take the following as an axiom:
Axiom. (
SNoCut_ext) We take the following as an axiom:
Axiom. (
ordinal_SNo) We take the following as an axiom:
Axiom. (
nat_p_SNo) We take the following as an axiom:
Axiom. (
omega_SNo) We take the following as an axiom:
Axiom. (
SNo_0) We take the following as an axiom:
Axiom. (
SNo_1) We take the following as an axiom:
Axiom. (
SNo_2) We take the following as an axiom:
Axiom. (
SNoLev_0) We take the following as an axiom:
Axiom. (
SNoCut_0_0) We take the following as an axiom:
Axiom. (
SNoL_0) We take the following as an axiom:
Axiom. (
SNoR_0) We take the following as an axiom:
Axiom. (
SNoL_1) We take the following as an axiom:
Axiom. (
SNoR_1) We take the following as an axiom:
Axiom. (
eps_0_1) We take the following as an axiom:
Axiom. (
SNo__eps_) We take the following as an axiom:
Axiom. (
SNo_eps_) We take the following as an axiom:
Axiom. (
SNo_eps_1) We take the following as an axiom:
Axiom. (
SNoLev_eps_) We take the following as an axiom:
Axiom. (
SNo_eps_pos) We take the following as an axiom:
Axiom. (
eps_SNo_eq) We take the following as an axiom:
Axiom. (
eps_SNoCutP) We take the following as an axiom:
Axiom. (
eps_SNoCut) We take the following as an axiom:
End of Section TaggedSets2
Axiom. (
SNo_etaE) We take the following as an axiom:
Axiom. (
SNo_ind) We take the following as an axiom:
∀P : set → prop, (∀L R, SNoCutP L R → (∀x ∈ L, P x) → (∀y ∈ R, P y) → P (SNoCut L R)) → ∀z, SNo z → P z
Beginning of Section SurrealRecI
Variable F : set → (set → set) → set
Primitive. The name
SNo_rec_i is a term of type
set → set.
Hypothesis Fr : ∀z, SNo z → ∀g h : set → set, (∀w ∈ SNoS_ (SNoLev z), g w = h w) → F z g = F z h
End of Section SurrealRecI
Beginning of Section SurrealRecII
Variable F : set → (set → (set → set)) → (set → set)
Let G : set → (set → set → (set → set)) → set → (set → set) ≝ λalpha g ⇒ If_iii (ordinal alpha) (λz : set ⇒ If_ii (z ∈ SNoS_ (ordsucc alpha)) (F z (λw ⇒ g (SNoLev w) w)) default) (λz : set ⇒ default)
Primitive. The name
SNo_rec_ii is a term of type
set → (set → set).
Hypothesis Fr : ∀z, SNo z → ∀g h : set → (set → set), (∀w ∈ SNoS_ (SNoLev z), g w = h w) → F z g = F z h
End of Section SurrealRecII
Beginning of Section SurrealRec2
Variable F : set → set → (set → set → set) → set
Let G : set → (set → set → set) → set → (set → set) → set ≝ λw f z g ⇒ F w z (λx y ⇒ if x = w then g y else f x y)
Primitive. The name
SNo_rec2 is a term of type
set → set → set.
Axiom. (
SNo_rec2_eq) We take the following as an axiom:
End of Section SurrealRec2
Axiom. (
SNoLev_ind) We take the following as an axiom:
Axiom. (
SNoLev_ind2) We take the following as an axiom:
Axiom. (
SNoLev_ind3) We take the following as an axiom:
∀P : set → set → set → prop, (∀x y z, SNo x → SNo y → SNo z → (∀u ∈ SNoS_ (SNoLev x), P u y z) → (∀v ∈ SNoS_ (SNoLev y), P x v z) → (∀w ∈ SNoS_ (SNoLev z), P x y w) → (∀u ∈ SNoS_ (SNoLev x), ∀v ∈ SNoS_ (SNoLev y), P u v z) → (∀u ∈ SNoS_ (SNoLev x), ∀w ∈ SNoS_ (SNoLev z), P u y w) → (∀v ∈ SNoS_ (SNoLev y), ∀w ∈ SNoS_ (SNoLev z), P x v w) → (∀u ∈ SNoS_ (SNoLev x), ∀v ∈ SNoS_ (SNoLev y), ∀w ∈ SNoS_ (SNoLev z), P u v w) → P x y z) → ∀x y z, SNo x → SNo y → SNo z → P x y z
Axiom. (
SNo_omega) We take the following as an axiom:
Axiom. (
SNoLt_0_1) We take the following as an axiom:
Axiom. (
SNoLt_0_2) We take the following as an axiom:
Axiom. (
SNoLt_1_2) We take the following as an axiom:
Axiom. (
restr_SNo_) We take the following as an axiom:
Axiom. (
restr_SNo) We take the following as an axiom:
Axiom. (
restr_SNoEq) We take the following as an axiom:
Beginning of Section SurrealMinus
Primitive. The name
minus_SNo is a term of type
set → set.
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_SNo.
Notation. We use
≤ as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLe.
End of Section SurrealMinus
Beginning of Section SurrealAdd
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_SNo.
Primitive. The name
add_SNo is a term of type
set → set → set.
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_SNo.
Axiom. (
add_SNo_eq) We take the following as an axiom:
Axiom. (
SNo_add_SNo) We take the following as an axiom:
Axiom. (
add_SNo_Lt1) We take the following as an axiom:
Axiom. (
add_SNo_Le1) We take the following as an axiom:
Axiom. (
add_SNo_Lt2) We take the following as an axiom:
Axiom. (
add_SNo_Le2) We take the following as an axiom:
Axiom. (
add_SNo_Lt3) We take the following as an axiom:
Axiom. (
add_SNo_Le3) We take the following as an axiom:
Axiom. (
add_SNo_com) We take the following as an axiom:
Axiom. (
add_SNo_0L) We take the following as an axiom:
Axiom. (
add_SNo_0R) We take the following as an axiom:
Axiom. (
minus_SNo_0) We take the following as an axiom:
Axiom. (
add_SNo_Lt4) We take the following as an axiom:
End of Section SurrealAdd
Beginning of Section SurrealMul
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_SNo.
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_SNo.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_SNo.
Axiom. (
mul_SNo_eq) We take the following as an axiom:
Axiom. (
mul_SNo_eq_2) We take the following as an axiom:
∀x y, SNo x → SNo y → ∀p : prop, (∀L R, (∀u, u ∈ L → (∀q : prop, (∀w0 ∈ SNoL x, ∀w1 ∈ SNoL y, u = w0 * y + x * w1 + - w0 * w1 → q) → (∀z0 ∈ SNoR x, ∀z1 ∈ SNoR y, u = z0 * y + x * z1 + - z0 * z1 → q) → q)) → (∀w0 ∈ SNoL x, ∀w1 ∈ SNoL y, w0 * y + x * w1 + - w0 * w1 ∈ L) → (∀z0 ∈ SNoR x, ∀z1 ∈ SNoR y, z0 * y + x * z1 + - z0 * z1 ∈ L) → (∀u, u ∈ R → (∀q : prop, (∀w0 ∈ SNoL x, ∀z1 ∈ SNoR y, u = w0 * y + x * z1 + - w0 * z1 → q) → (∀z0 ∈ SNoR x, ∀w1 ∈ SNoL y, u = z0 * y + x * w1 + - z0 * w1 → q) → q)) → (∀w0 ∈ SNoL x, ∀z1 ∈ SNoR y, w0 * y + x * z1 + - w0 * z1 ∈ R) → (∀z0 ∈ SNoR x, ∀w1 ∈ SNoL y, z0 * y + x * w1 + - z0 * w1 ∈ R) → x * y = SNoCut L R → p) → p
Axiom. (
mul_SNo_prop_1) We take the following as an axiom:
∀x, SNo x → ∀y, SNo y → ∀p : prop, (SNo (x * y) → (∀u ∈ SNoL x, ∀v ∈ SNoL y, u * y + x * v < x * y + u * v) → (∀u ∈ SNoR x, ∀v ∈ SNoR y, u * y + x * v < x * y + u * v) → (∀u ∈ SNoL x, ∀v ∈ SNoR y, x * y + u * v < u * y + x * v) → (∀u ∈ SNoR x, ∀v ∈ SNoL y, x * y + u * v < u * y + x * v) → p) → p
Axiom. (
SNo_mul_SNo) We take the following as an axiom:
Axiom. (
mul_SNo_eq_3) We take the following as an axiom:
∀x y, SNo x → SNo y → ∀p : prop, (∀L R, SNoCutP L R → (∀u, u ∈ L → (∀q : prop, (∀w0 ∈ SNoL x, ∀w1 ∈ SNoL y, u = w0 * y + x * w1 + - w0 * w1 → q) → (∀z0 ∈ SNoR x, ∀z1 ∈ SNoR y, u = z0 * y + x * z1 + - z0 * z1 → q) → q)) → (∀w0 ∈ SNoL x, ∀w1 ∈ SNoL y, w0 * y + x * w1 + - w0 * w1 ∈ L) → (∀z0 ∈ SNoR x, ∀z1 ∈ SNoR y, z0 * y + x * z1 + - z0 * z1 ∈ L) → (∀u, u ∈ R → (∀q : prop, (∀w0 ∈ SNoL x, ∀z1 ∈ SNoR y, u = w0 * y + x * z1 + - w0 * z1 → q) → (∀z0 ∈ SNoR x, ∀w1 ∈ SNoL y, u = z0 * y + x * w1 + - z0 * w1 → q) → q)) → (∀w0 ∈ SNoL x, ∀z1 ∈ SNoR y, w0 * y + x * z1 + - w0 * z1 ∈ R) → (∀z0 ∈ SNoR x, ∀w1 ∈ SNoL y, z0 * y + x * w1 + - z0 * w1 ∈ R) → x * y = SNoCut L R → p) → p
Axiom. (
mul_SNo_Lt) We take the following as an axiom:
Axiom. (
mul_SNo_Le) We take the following as an axiom:
Axiom. (
mul_SNo_Subq_lem) We take the following as an axiom:
∀x y X Y Z W, ∀U U', (∀u, u ∈ U → (∀q : prop, (∀w0 ∈ X, ∀w1 ∈ Y, u = w0 * y + x * w1 + - w0 * w1 → q) → (∀z0 ∈ Z, ∀z1 ∈ W, u = z0 * y + x * z1 + - z0 * z1 → q) → q)) → (∀w0 ∈ X, ∀w1 ∈ Y, w0 * y + x * w1 + - w0 * w1 ∈ U') → (∀w0 ∈ Z, ∀w1 ∈ W, w0 * y + x * w1 + - w0 * w1 ∈ U') → U ⊆ U'
Axiom. (
mul_SNo_com) We take the following as an axiom:
Beginning of Section mul_SNo_assoc_lems
Variable M : set → set → set
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
M.
Hypothesis SNo_M : ∀x y, SNo x → SNo y → SNo (x * y)
Hypothesis DL : ∀x y z, SNo x → SNo y → SNo z → x * (y + z) = x * y + x * z
Hypothesis DR : ∀x y z, SNo x → SNo y → SNo z → (x + y) * z = x * z + y * z
Hypothesis M_Lt : ∀x y u v, SNo x → SNo y → SNo u → SNo v → u < x → v < y → u * y + x * v < x * y + u * v
Hypothesis M_Le : ∀x y u v, SNo x → SNo y → SNo u → SNo v → u ≤ x → v ≤ y → u * y + x * v ≤ x * y + u * v
Axiom. (
mul_SNo_assoc_lem1) We take the following as an axiom:
∀x y z, SNo x → SNo y → SNo z → (∀u ∈ SNoS_ (SNoLev x), u * (y * z) = (u * y) * z) → (∀v ∈ SNoS_ (SNoLev y), x * (v * z) = (x * v) * z) → (∀w ∈ SNoS_ (SNoLev z), x * (y * w) = (x * y) * w) → (∀u ∈ SNoS_ (SNoLev x), ∀v ∈ SNoS_ (SNoLev y), u * (v * z) = (u * v) * z) → (∀u ∈ SNoS_ (SNoLev x), ∀w ∈ SNoS_ (SNoLev z), u * (y * w) = (u * y) * w) → (∀v ∈ SNoS_ (SNoLev y), ∀w ∈ SNoS_ (SNoLev z), x * (v * w) = (x * v) * w) → (∀u ∈ SNoS_ (SNoLev x), ∀v ∈ SNoS_ (SNoLev y), ∀w ∈ SNoS_ (SNoLev z), u * (v * w) = (u * v) * w) → ∀L, (∀u ∈ L, ∀q : prop, (∀v ∈ SNoL x, ∀w ∈ SNoL (y * z), u = v * (y * z) + x * w + - v * w → q) → (∀v ∈ SNoR x, ∀w ∈ SNoR (y * z), u = v * (y * z) + x * w + - v * w → q) → q) → ∀u ∈ L, u < (x * y) * z
Axiom. (
mul_SNo_assoc_lem2) We take the following as an axiom:
∀x y z, SNo x → SNo y → SNo z → (∀u ∈ SNoS_ (SNoLev x), u * (y * z) = (u * y) * z) → (∀v ∈ SNoS_ (SNoLev y), x * (v * z) = (x * v) * z) → (∀w ∈ SNoS_ (SNoLev z), x * (y * w) = (x * y) * w) → (∀u ∈ SNoS_ (SNoLev x), ∀v ∈ SNoS_ (SNoLev y), u * (v * z) = (u * v) * z) → (∀u ∈ SNoS_ (SNoLev x), ∀w ∈ SNoS_ (SNoLev z), u * (y * w) = (u * y) * w) → (∀v ∈ SNoS_ (SNoLev y), ∀w ∈ SNoS_ (SNoLev z), x * (v * w) = (x * v) * w) → (∀u ∈ SNoS_ (SNoLev x), ∀v ∈ SNoS_ (SNoLev y), ∀w ∈ SNoS_ (SNoLev z), u * (v * w) = (u * v) * w) → ∀R, (∀u ∈ R, ∀q : prop, (∀v ∈ SNoL x, ∀w ∈ SNoR (y * z), u = v * (y * z) + x * w + - v * w → q) → (∀v ∈ SNoR x, ∀w ∈ SNoL (y * z), u = v * (y * z) + x * w + - v * w → q) → q) → ∀u ∈ R, (x * y) * z < u
End of Section mul_SNo_assoc_lems
Axiom. (
SNo_foil) We take the following as an axiom:
Axiom. (
SNo_foil_mm) We take the following as an axiom:
End of Section SurrealMul
Beginning of Section SurrealExp
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_SNo.
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_SNo.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_SNo.
Notation. We use
^ as an infix operator with priority 342 and which associates to the right corresponding to applying term
exp_SNo_nat.
Axiom. (
eps_bd_1) We take the following as an axiom:
Axiom. (
SNoS_finite) We take the following as an axiom:
End of Section SurrealExp
Beginning of Section Int
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_SNo.
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_SNo.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_SNo.
Primitive. The name
int is a term of type
set.
Axiom. (
int_SNo_cases) We take the following as an axiom:
∀p : set → prop, (∀n ∈ ω, p n) → (∀n ∈ ω, p (- n)) → ∀x ∈ int, p x
Axiom. (
int_SNo) We take the following as an axiom:
Axiom. (
int_add_SNo) We take the following as an axiom:
Axiom. (
int_mul_SNo) We take the following as an axiom:
End of Section Int
Beginning of Section SurrealAbs
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_SNo.
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_SNo.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_SNo.
Axiom. (
abs_SNo_0) We take the following as an axiom:
Axiom. (
pos_abs_SNo) We take the following as an axiom:
Axiom. (
neg_abs_SNo) We take the following as an axiom:
Axiom. (
SNo_abs_SNo) We take the following as an axiom:
Axiom. (
abs_SNo_Lev) We take the following as an axiom:
End of Section SurrealAbs
Beginning of Section SNoMaxMin
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_SNo.
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_SNo.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_SNo.
Notation. We use
^ as an infix operator with priority 342 and which associates to the right corresponding to applying term
exp_SNo_nat.
Notation. We use
< as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLt.
Notation. We use
≤ as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLe.
Definition. We define
SNo_max_of to be
λX x ⇒ x ∈ X ∧ SNo x ∧ ∀y ∈ X, SNo y → y ≤ x of type
set → set → prop.
Definition. We define
SNo_min_of to be
λX x ⇒ x ∈ X ∧ SNo x ∧ ∀y ∈ X, SNo y → x ≤ y of type
set → set → prop.
End of Section SNoMaxMin
Beginning of Section DiadicRationals
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_SNo.
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_SNo.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_SNo.
Notation. We use
< as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLt.
Notation. We use
≤ as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLe.
Notation. We use
^ as an infix operator with priority 342 and which associates to the right corresponding to applying term
exp_SNo_nat.
End of Section DiadicRationals
Beginning of Section Reals
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_SNo.
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_SNo.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_SNo.
Notation. We use
< as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLt.
Notation. We use
≤ as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLe.
Notation. We use
^ as an infix operator with priority 342 and which associates to the right corresponding to applying term
exp_SNo_nat.
Primitive. The name
real is a term of type
set.
Axiom. (
real_I) We take the following as an axiom:
Axiom. (
real_E) We take the following as an axiom:
Axiom. (
real_SNo) We take the following as an axiom:
Axiom. (
real_0) We take the following as an axiom:
Axiom. (
real_1) We take the following as an axiom:
Axiom. (
SNo_approx_real_rep) We take the following as an axiom:
∀x ∈ real, ∀p : prop, (∀f g ∈ SNoS_ ωω, (∀n ∈ ω, f n < x) → (∀n ∈ ω, x < f n + eps_ n) → (∀n ∈ ω, ∀i ∈ n, f i < f n) → (∀n ∈ ω, g n + - eps_ n < x) → (∀n ∈ ω, x < g n) → (∀n ∈ ω, ∀i ∈ n, g n < g i) → SNoCutP {f n|n ∈ ω} {g n|n ∈ ω} → x = SNoCut {f n|n ∈ ω} {g n|n ∈ ω} → p) → p
End of Section Reals
Beginning of Section ComplexSNo
Beginning of Section CTaggedSets
Notation. We use
' as a postfix operator with priority 100 corresponding to applying term
tag.
Notation. We use
'' as a postfix operator with priority 100 corresponding to applying term
ctag.
Definition. We define
SNo_pair to be
λx y ⇒ x ∪ {u ''|u ∈ y} of type
set → set → set.
Axiom. (
SNo_pair_0) We take the following as an axiom:
End of Section CTaggedSets
Primitive. The name
CSNo is a term of type
set → prop.
Axiom. (
CSNo_I) We take the following as an axiom:
Axiom. (
CSNo_E) We take the following as an axiom:
Axiom. (
SNo_CSNo) We take the following as an axiom:
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_SNo.
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_SNo.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_SNo.
Primitive. The name
Complex_i is a term of type
set.
Primitive. The name
CSNo_Re is a term of type
set → set.
Primitive. The name
CSNo_Im is a term of type
set → set.
Axiom. (
CSNo_Re1) We take the following as an axiom:
Axiom. (
CSNo_Re2) We take the following as an axiom:
∀x y, SNo x → SNo y → Re (pa x y) = x
Axiom. (
CSNo_Im1) We take the following as an axiom:
∀z, CSNo z → SNo (Im z) ∧ z = pa (Re z) (Im z)
Axiom. (
CSNo_Im2) We take the following as an axiom:
∀x y, SNo x → SNo y → Im (pa x y) = y
Axiom. (
CSNo_ReR) We take the following as an axiom:
Axiom. (
CSNo_ImR) We take the following as an axiom:
Axiom. (
CSNo_ReIm) We take the following as an axiom:
∀z, CSNo z → z = pa (Re z) (Im z)
Primitive. The name
minus_CSNo is a term of type
set → set.
Primitive. The name
add_CSNo is a term of type
set → set → set.
Primitive. The name
mul_CSNo is a term of type
set → set → set.
Notation. We use
:-: as a prefix operator with priority 358 corresponding to applying term
minus_CSNo.
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_CSNo.
Notation. We use
⨯ as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_CSNo.
Axiom. (
SNo_Re) We take the following as an axiom:
Axiom. (
SNo_Im) We take the following as an axiom:
Axiom. (
Re_0) We take the following as an axiom:
Axiom. (
Im_0) We take the following as an axiom:
Axiom. (
Re_1) We take the following as an axiom:
Axiom. (
Im_1) We take the following as an axiom:
Axiom. (
Re_i) We take the following as an axiom:
Axiom. (
Im_i) We take the following as an axiom:
Axiom. (
add_CSNo_0L) We take the following as an axiom:
Axiom. (
add_CSNo_0R) We take the following as an axiom:
Axiom. (
mul_CSNo_CRe) We take the following as an axiom:
∀z w, CSNo z → CSNo w → Re (mult' z w) = Re z * Re w + - (Im z * Im w)
Axiom. (
mul_CSNo_CIm) We take the following as an axiom:
∀z w, CSNo z → CSNo w → Im (mult' z w) = Re z * Im w + Im z * Re w
Axiom. (
CSNo_0) We take the following as an axiom:
Axiom. (
CSNo_1) We take the following as an axiom:
End of Section ComplexSNo
Beginning of Section Complex
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_CSNo.
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_CSNo.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_CSNo.
Notation. We use
< as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLt.
Notation. We use
≤ as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLe.
Primitive. The name
complex is a term of type
set.
Axiom. (
complex_I) We take the following as an axiom:
Axiom. (
complex_E) We take the following as an axiom:
Axiom. (
complex_0) We take the following as an axiom:
Axiom. (
complex_1) We take the following as an axiom:
Axiom. (
complex_i) We take the following as an axiom:
Axiom. (
real_Re_eq) We take the following as an axiom:
Axiom. (
real_Im_eq) We take the following as an axiom:
Axiom. (
complex_eta) We take the following as an axiom:
End of Section Complex
Notation. We use
< as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLt.
Notation. We use
≤ as an infix operator with priority 490 and no associativity corresponding to applying term
SNoLe.
Notation. We use
- as a prefix operator with priority 358 corresponding to applying term
minus_SNo.
Notation. We use
+ as an infix operator with priority 360 and which associates to the right corresponding to applying term
add_SNo.
Notation. We use
* as an infix operator with priority 355 and which associates to the right corresponding to applying term
mul_SNo.