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. *\
44 \def "convergence of a term to an object"
45 Conv = [t:*Term] EX([y:*Obj] E(t, y)) : *Term -> *Prop
47 \def "term-term equivalence"
48 Eq = [t1:*Term, t2:*Term] [y:*Obj] Iff(E(t1, y), E(t2, y))
49 : *Term => *Term -> *Prop
53 \open non_logical_axioms \* [1] 2.4. *\
55 \* we axiomatize E because *Term is not inductively generated *\
56 \ax e_refl: [y:*Obj] E(T(y), y)
58 \ax e_at_in: [f:*Obj][x:*Obj][y:*Obj] App(f,x,y) -> E(At(T(f), T(x)), y)
60 \ax e_at_out: [f:*Obj][x:*Obj][y:*Obj] E(At(T(f), T(x)), y) -> App(f,x,y)