3 \* Feferman's system T0 *\
5 \open elements \* [1] 2.1. 2.2. 2.4. *\
7 \decl "rule application" App: *Obj => *Obj => *Obj -> *Prop
9 \decl "classification predicate" Cl: *Obj -> *Prop
11 \decl "classification membership" Eta: *Obj => *Obj -> *Prop
13 \* we must make an explicit coercion from *Obj to *Term *\
14 \decl "object-to-term-coercion" T: *Obj -> *Term
16 \decl "term application" At: *Term => *Term -> *Term
18 \decl "term-object equivalence" E: *Term => *Obj -> *Prop
22 \open logical_abbreviations \* [1] 2.3. 2.5. *\
24 \def "logical comprehension restricted to classifications"
25 CAll = [q:*Obj->*Prop] [x:*Obj] Cl(x) -> q(x)
26 : (*Obj -> *Prop) -> *Prop
28 \def "logical existence restricted to classifications"
29 CEx = [q:*Obj->*Prop] Ex([x:*Obj] And(Cl(x), q(x)))
30 : (*Obj -> *Prop) -> *Prop
32 \def "logical comprehension restricted to a classification"
33 EAll = [a:*Obj, q:*Obj->*Prop] [x:*Obj] Eta(x, a) -> q(x)
34 : *Obj => (*Obj -> *Prop) -> *Prop
36 \def "logical existence restricted to a classification"
37 EEx = [a:*Obj, q:*Obj->*Prop] Ex([x:*Obj] And(Eta(x, a), q(x)))
38 : *Obj => (*Obj -> *Prop) -> *Prop
42 \open non_logical_abbreviations \* [1] 2.4. 2.7 *\
44 \def "object application"
45 OAt = [f:*Obj, x:*Obj] At(T(f), T(x)) : *Obj => *Obj -> *Term
47 \def "convergence of a term to an object"
48 Conv = [t:*Term] EX([y:*Obj] E(t, y)) : *Term -> *Prop
50 \def "term-term equivalence"
51 Eq = [t1:*Term, t2:*Term] [y:*Obj] Iff(E(t1, y), E(t2, y))
52 : *Term => *Term -> *Prop
54 \def "classification membership of a term"
55 TEta = [t:*Term, a:*Obj] EEx(a, [y:*Obj] E(t, y))
56 : *Term => *Obj -> *Prop
58 \def "operation (rule with inhabited domain)"
59 Op = [f:*Obj] Ex([x:*Obj] Conv(OAt(f, x))) : *Obj -> *Prop
61 \def "classification inclusion"
62 ESub = [a1:*Obj, a2:*Obj] EAll(a1, [x:*Obj] Eta(x, a2))
63 : *Obj => *Obj -> *Prop
65 \def "classification morphism"
66 ETo = [f:*Obj, a:*Obj, b:*Obj] EAll(a, [x:*Obj] TEta(OAt(f, x), b))
67 : *Obj => *Obj => *Obj -> *Prop
71 \open non_logical_axioms \* [1] 2.4. 3.2 *\
73 \* we axiomatize E because *Term is not inductively generated *\
74 \ax e_refl: [y:*Obj] E(T(y), y)
76 \ax e_at_in: [t1:*Term][t2:*Term][f:*Obj][x:*Obj][y:*Obj]
77 E(t1, f) -> E(t2, x) -> App(f, x, y) -> E(At(t1, t2), y)
79 \ax e_at_out: [f:*Obj][x:*Obj][y:*Obj] E(At(T(f), T(x)), y) -> App(f,x,y)
81 \ax "I (i)" id_dec: [x:*Obj][y:*Obj] Or(Id(x, y), NId(x, y))
83 \ax "I (ii)" at_mono: [f:*Obj][x:*Obj][y1:*Obj][y2:*Obj]
84 E(OAt(f, x), y1) -> E(OAt(f, x), y2) -> Id(y1, y2)
86 \ax "I (iii)" eta_cl: [x:*Obj][a:*Obj] Eta(x, a) -> Cl(a)