RCF2: Evaluation and Consistency

0.51 MB·

by  Michael Pfender

RCF 2
Evaluation and Consistency∗
ε&C ∗ π R ∗ π• R
O O
Michael Pfender†
9
0
0 July 2008‡
2

n
a
J
Abstract: Weconstructhereaniterative evaluation ofall(coded)PRmaps:

0
progress of this iteration can be measured by descending complexity, within
3
Ordinal O := N[ω], of polynomials in one indeterminate, called “ω”. As (well)

]
T order on this Ordinal we choose the lexicographical one. Non-infinit descent
C of such iterations is added as a mild additional axiom schema (π ) to Theory
O
. PR = PR + (abstr) of Primitive Recursion with predicate abstraction, out
h A
t of foregoing part RFC 1. This then gives (correct) on-termination of iterative
a
m evaluation of argumented deduction trees as well: for theories PR and π R =
A O
[ PR + (π ). By means of this constructive evaluation the Main Theorem
A O
is proved, on Termination-conditioned (Inner) Soundness for Theories π R, O
2 O
v extending N[ω]. As a consequence we get in fact Self-Consistency for theories
1
π R, namely π R-derivability of π R’s own free-variable Consistency formula
8 O O O
8 Con = Con (k) = ¬Prov (k, pfalseq ) : N → 2, k ∈ N free.
3 πOR πOR def πOR
. Here PR predicate Prov (k,u) says, for an arithmetical theory T : number
9 T
0 k ∈ N is a T-Proof code proving internally T-formula code u, arithmetised
8
Proof in G¨odel’s sense.
0
: As to expect from classiccal setting, Self-Consistency of π R gives (uncon-
v O
i ditioned) Objective Soundness. Eventually we show Termination-Conditioned
X
Soundness “already” for PR . But it turns out that present derivation of Self-
A
r
a Consistency, and already that of Consistency formula of PR from this con-
A
ditioned Soundness “needs” schema (π˜) of non-infinit descent in Ordinal N[ω],
which is presumably not derived by PR itself.
A
0 LegendofLOGO:εforConstructiveevaluation,C forSelf-Consistency tobe derivedfor
suitable theories πOR, πO•R strengthening in a “mild” way the (categorical) Free-Variables
Theory PRA of Primitive Recursion with predicate abstraction
∗Consideration of implicational version (π•) of Descent axiom added
O
†TU Berlin, Mathematik, [email protected]
‡last revised January 30, 2009
1
1 Summary
G¨odel’s first Incompleteness Theorem for Principia Mathematica and “ver-
wandte Systeme”, on which in particular is based the second one, on non-
provability of PM’s own Consistency formula Con , exhibits a (closed) PM
PM
formula ϕ with property that
PM ⊢ [ϕ ⇐⇒ ¬(∃k ∈ N)Prov (k, pϕq )], in words:
PM
Theory PM derives ϕ to be equivalent to its “own” coded, arithmetised non-
Provability.
Since this equivalence needs already for its statement “full” formal, “not
testable” quantification, theConsistency Provability issue isnotsettledforFree-
Variables Primitive Recursive Arithmetic and its strengthenings – Theories T
which express (formalised, “internal”) Consistency as free-variable formula
Con = Con (k) = ¬Prov (k, pfalseq ) : N → 2 :
T T T
“No k ∈ N is a Proof code proving pfalseq .”
This is the point of depart for investigation of “suitable” strengthenings
π R = PR +(π )ofcategoricalTheoryPR ofPrimitiveRecursion, enriched
O A O A
with predicate abstraction Objects {A|χ} = {a ∈ A|χ(a)} : Plausibel axiom
schema (π ), more presisely: its contraposition π˜ , states “weak” impossibility
O O
of infinite descending chains in any Ordinal O extending polynomial semiring
N[ω], with its canonical, lexicographical order.
Central Non-Infinite Descent Schema, Descent Schema for short:
We need an axiom-schema for expressing – in free variables – Finite de-
scent (endo-driven) chains, descending in complexity value out of Ordinal
O (cid:23) N[ω], a schema called (π ), which gives the “name” to Descent1 Theory
O
π R = PR +(π ) : This theory is a pure strengthening of PR , it has the
O A O A
same language.
Easier to interprete logically is (π )’s equivalent, Free-Variables contraposi-
O
0extended Poster Abstract “Arithmetical Consistency via Constructive evaluation”, Con-
ference celebrating Kurt G¨odel’s 100th birthday, Vienna april 28, 29, 2006
1notion added 2 JAN 2009
2
tion, on “absurdity” of infinite descending chains, namely:
c = c(a) : A → O PR (complexity),
p = p(a) : A → A PR (predecessor endo),
PR ⊢ c(a) > 0 =⇒ cp(a) < c(a) (descent),
A . O .
PR ⊢ c(a) = 0 =⇒ p(a) = a (stationarity at zero)
A O
ψ = ψ(a) : A → 2 absurdity test predicate,
PR ⊢ ψ(a) =⇒ cpn(a) > 0 ,
A O
with quantifier decoration:
PR ⊢ ∀a[ψ(a) =⇒ ∀ncpn(a) > 0 ]
A O
the latter statement: “infinit descent”, is felt absurd,
and “therefore” so “must be”, by axiom,
condition ψ implying this “absurdity”:
(π˜ )
O .
π R ⊢ ψ(a) = false : A → 2, intuitively:
O
π R ⊢ ∀a ¬ψ(a).
O
[The first four lines of the antecedent constitute (p,c) as (the data of) a
CCI : of a Complexity Controlled Iteration, with (stepwise) descending order
O
values inOrdinal O.Central example: General Recursive, Acckermann type
PR-code evaluation ε will be resolved into such a CCI , O := N[ω] ⊂ N.]
O
My Thesis then is that these theories π R, weaker than PM, set theories
O
and even Peano Arithmetic PA (when given its quantified form), derive their
own internal (Free-Variable) Consistency formula Con (k) : N → 2, see
πOR
above.
Notions and Arguments for Self-Consistency of π R : In order to
O
obtain constructive Theories – candidates for self-Consistency – we introduce
first, into fundamental Theory PR of (categorical) Free-Variables Primitive
Recursion, predicate abstraction of PR maps χ = χ(a) : A → 2 (A a finite
power of NNO N), into defined Objects {A|χ}, and then strengthen Theory
PR obtained this way, by a free-variables, (inferential) schema (π ) of “on”-
A O
terminating descent,intoTheorie(s)π R,on-terminatingdescent ofComplexity
O
Controlled Iterations (CCI ’s, see above), with (descending) complexity values
O
in Ordinal O (cid:23) N[ω].
Strengthened Theory π R = PR +(π ), with its language equal to that
O A O
of PR , is asserted to derive the (Free-Variable) formula Con (k) which
A πOR
expresses internally: within π R itself, Consistency of Theory π R, see above.
O O
Proof is by CCIN[ω] (descent) property of a suitable, atomic PR evaluation
step e applied to PR-map-code/argument pairs (u,x) ∈ PR × .
A
X
[Here ⊂ N denotes the Universal Object of all (codes of) singletons and
X
(nested) pairs of natural numbers, enriched by a shymbol ⊥ equally coded in
N, to designate undefined values, of defined partially defined PR maps. Objects
A of PR , π R admit a natural embedding A ⊏ into this this universal
A O
X
Object.]
3
Iteration ε, of step e, is in fact controlled by a syntactic complexity c (u) ∈
PR
N[ω], descending with each application of e as long as minimum complexity
0 = c ( pidq ) is not “yet” reached.
PR
Strengthening of PR by schema (π ) – cf. its free-variables contraposition
A O
(π˜ ) above – into Theory π R = PR+(π ), is “just” to allow for a so to say
O O O
sound, canonical evaluation “algorithm” for π R :
O
On one hand it is proved straight forward that evaluation ε above has the
expected recursive properties of an evaluation, this within (categorical, Free-
Variables) Theory µR of µ-Recursion.
Onthe other hand, π R has the same Language asPR ,so that this ε is a
O A
natural candidate for likewise – sound – evaluation of internal version of theory
π R, and for being totally defined in a suitable Free-Variables sense, techni-
O
cally: to on-terminate, this just by its property to be a Complexity Controlled
Iteration, with order values in N[ω].
In fact, by schema (π ) itself (O extending N[ω]), ε preserves the extra
O
equation instances inserted by internalisation of (π ).
O
Dangerous bound: is there a good reason that this evaluation is not a
self-evaluation for Theory π R?
O
Answer: ε is – by definition – not PR: If you take the diagonal
diag(n) = ε(enum (n),cantor (n)) : N → N,
def PR
X
enum an internal PR count of all PR map codes, and cantor : N −∼=→
PR
“the” Cantor’s count of ⊂ N, then you get Ackermann’s origXinal diagonXal
X
function2 which grows faster than any PR function: but π R has only PR maps
O
as its maps, it is a (pure) strengthening of PR .
A
Ontheotherhand,εisintuitively total,since,intuitively, complexitycem(u,x)
“must” reach 0 in finitely many e-steps. The latter intuition can be, in free
variables (!), expressed formally by π R’s schema (π˜ ) : Free-Variables con-
O O
traposition of (π ). Schema (π˜ ) says that a condition which implies infinite
O O
descent of such a chain (on all x), must be false (on all x), “absurd”.
Complexity Controlled Iteration ε of e extends canonically into a Com-
plexity Controlled evaluation ε , of argumented deduction trees, ε again
d d
defined by CCIN[ω] : this time by iteration of a tree evaluation step ed suitably
extending basic evaluation step e to argumented deduction trees.
Deduction-tree evaluation starts on trees of form dtree /x, obtained as fol-
k
lowsfromk andx :Calldtree the(first)deduction tree which(internally)proves
k
kth internal equation u=ˇ v of theory π R, enumeration of proved equations
k O
being (lexicographically) by code of (first) Proof. This argument-free deduction
tree dtree then is provided – node-wise top down from given x ∈ – with
k
its spread down arguments in = ∪˙ {(cid:3)} = ∪˙ {hi} ⊂ N; (eXmpty list
(cid:3) def
X X X
(cid:3) = hi refers to a not yet known argument, not “yet” ata given time ofstepwise
evaluation e .)
d
2 for a two-parameter,simple genuine Ackermann function cf. Eilenberg/Elgot1970
4
Spreading down arguments this way eventually converts argument-free kth
deduction tree dtree into (partially non-dummy) argumented deduction tree
k
dtree /x.
k
Iteration ε , of tree evaluation step e , again is Complexity Controlled de-
d d
scending in Ordinal N[ω],when controlled by deduction tree complexity c . This
d
complexity is defined essentially as the (polynomial) sum of all (syntactical)
complexities c (u) of map codes appearing in the deduction tree.
PR
So, as it does to basic evaluation ε, schema π˜ applies to complexity con-
N[ω]
trolled evaluation ε of argumented deduction-trees as well, and gives
d
Deduction-Tree Evaluation non-infinit Descent: Infinit strict descent
of endo map e – with respect to complexity c – is absurd.
d d
This deduction-tree evaluation ε externalises, as far as terminating, kth in-
d
.
ternalequation u=ˇ v oftheory π Rinto complete evaluation ε(u,x) = ε(v,x) :
k O
Termination-Conditioned Inner Soundness, our Main Theorem.
For a given PR predicate χ = χ(x) : → 2, the Main Theorem reads:
X
Theoryπ Rderives: If fork ∈ Nandforx ∈ r{⊥}given,Prov (k, pχq )
O X πOR
“holds”, and if argumented π R deduction tree dtree /x admits complete eval-
O k
uation by m (“say”) deduction-tree evaluation-steps e ,
d
Then the pair (k,x) is a Soundness-Instance, i.e. then kth given (inter-
nal) π R-Provability Prov (k, pχq ) implies χ(x), for the given argument
O πOR
x ∈ r{⊥}. All this within Theory π R itself.
O
X
Corollary: Self-Consistency Derivability for Theory π R :
O
π R ⊢ Con , i.e. “necessarily” in Free-Variables form:
O πOR
π R ⊢ ¬Prov (k, pfalseq ) : N → 2, i.e. equationally:
O πOR
π R ⊢ ¬[ pfalseq =ˇ ptrueq ] : N → 2, k ∈ N free :
O k
Theory π R derives that no k ∈ N is the internal π R-Proof for pfalseq .
O O
Proof of this Corollary to Termination-Conditioned Soundness:
By the last assertion of the Theorem, with χ = χ(x) := false (x) : → 2,
X X
and x := h0i ∈ , we get:
X
Evaluation-effective internal inconsistency of π R, i.e. availability of an
O
evaluation-terminating internal deduction tree of pfalseq , implies false :
.
π R ⊢ pfalseq =ˇ ptrueq ∧ c em(dtree /h0i) = 0 =⇒ false (h0i).
O k d d k
X
Contraposition to this, still with k,m ∈ N free:
π R ⊢ true (h0i) =⇒ ¬[ pfalseq =ˇ ptrueq ] ∨ c em(dtree /h0i) > 0,
O k d d k
X
i.e. by Free-Variables (Boolean) tautology:
π R ⊢ pfalseq =ˇ ptrueq =⇒ c em(dtree /h0i) > 0 : N2 → 2.
O k d d k
5
This πOR derivative invites to apply schema (π˜N[ω]) of πOR :
“infinite endo-driven descent with order values in N[ω] is absurd.”
We apply this schema to deduction tree evaluation ε given by step e and
d d
complexity c which descends – this is Argumented-Tree Evaluation Descent –
d
with each application of e , as long as complexity 0 is not (“yet”) reached. We
d
combine this with choice of “overall” absurdity condition
ψ = ψ(k) := [ pfalseq =ˇ ptrueq ] : N → 2, k ∈ N free (!)
k
and get, by schema (π˜N[ω]), overall negation of this (overall) “absurd” predicate
ψ, namely
π R ⊢ ¬[ pfalseq =ˇ ptrueq ] : N → 2, k ∈ N free.
O k
This is π R-derivation of the free-variable Consistency Formula of π R itself.
O O
From this Self-Consistency of Theorie(s) π R, which is equivalent to in-
O
jectivity of (special) internal numeralisation ν : 2 ֌ [ ,2] , we get im-
mediately injectivity of all these numeralisatio2ns ν = ν1(a)πO:RA ֌ [ ,A] =
A A
1
⌈ ,A⌉/=ˇ , and from this, with naturality of this family, “full” objective Sound-
1
ness of Theory π R which reads:
O
Formalised π R-Provability of (code of) PR predicate χ : → 2 implies –
O
X
within Theory π R – “validity” χ(x) of χ at “each” of χ’s arguments x ∈ .
O
X
But for derivation ofSelf-Consistency fromTermination-conditioned Sound-
ness, a suitable strengthening of PR , here by schema (π˜) = (π˜ ), stating
A N[ω]
absurdity of infinite descent in Ordinal N[ω], seems to be necessary: my guess
is that Theories PRA as well as PR and hence PR , are not strong enough
A
to derive their own (internal) Consistency. On the other hand, we know from
G¨odel’s work that Principia Mathematica “und verwandte Systeme” are too
strong for being self-consistent. This is true for any (formally) quantified Arith-
metical Theory Q, in particular for the (classical, quantified) version PA of
Peano Arithmetic: Such theory Q has all ingredients for G¨odel’s Proof of his
two Incompleteness Theorems.
In section 7 We discuss3 a formally stronger, implicational, “local” variant
(π•) of inferential Descent axiom (π ), with respect to Self-Consistency and
O O
(Objective) Soundness: In particular, Self-Consistency Proof becomes techni-
cally easier for corresponding theory π•R.
O
The final section 84 gives a proof of (Objective) Consistency for Theorie(s)
π•R (hence π R) relative to basic Theory PR of Primitive Recursion and
O O A
hence relative to fundamental Theory PR of Primitive Recursion “itself”.
For proof of this (relative) Consistency, we use a schema, (ρ ), of recur-
O
sive reduction for predicate validity, reduction along a Complexity Controlled
Iteration (CCI ), admitted by Theory PR (and its strengthenings.)
O A
3
insertion ? JAN 2009
4inserted 2 JAN 2009
6
2 Iterative Evaluation of PR Map Codes
Object- andmaptermsofallour theoriesarecodedstraight ahead, inparticular
since formally we have no (individual) variables on the Object Language level:
We code all our terms just as prime power products “over” the LATEXsource
codes describing these terms, this externally in naive numbers, out of N as well
as into the NNO N of the (categorical) arithmetical theory itself.
Equality Enumeration: As “any” theories, fundamental Theory PR of
Primitive Recursion as well as basic Theory PR = PR+(abstr), definitional
A
enrichement of PR by the schema of predicate abstraction: hχ : A → 2i 7→
{A|χ}, a “virtual”, abstracted Object in PR , admit an (external) primitive
A
recursive enumeration of their respective theorems, ordered by length (more
precisely: by lexicographical order) of the first proofs of these (equational)
Theorems, here:
=PR (k) : N → PR×PR ⊂ N×N and
=PRA (k) : N → PR ×PR ⊂ N×N
A A
respectively.
BythePR Representation Theorem5.3of Roma`n1989,theseenumerations
give rise to their internal versions
=ˇPR : N → PR×PR ⊂ N2 and
k
=ˇPRA : N → PR ×PR ⊂ N2,
k A A
with internalisation (representation) property
PR ⊢ =ˇ = num(=PR) : → PR×PR ⊂ N2 and
num(k) k
1
PR ⊢ =ˇ = num(=PRA) : → PR ×PR ⊂ N2.
num(k) k 1 A A
Here (external) numeralisation is given externally PR as
num(n) = sn : −→0 N −→s ... −→s N,
1
num(m,n) = (num(m),num(n)) : → N×N, m,n (“meta”) free in N,
1
PR = {N|PR}isthepredicative, PR decidablesubset ofN“ofallPRcodes” (a
PR -Object), internalisation of PR ⊂ N of all PR-terms on Object Language
A
level. Analogeous meaning for internalisation PR ⊂ N of PR ⊂ N.
A A
For discussion of “constructive” evaluation, we need representation of all
PR maps within one PR endo map monoid, namely within PR( , ),
A ⊥ ⊥
where ⊂ N, = {N| : N → 2} is the (predicative) UniversalXObjeXct of
N-singlXetons {hnXi|n ∈ N}X, possibly nested N-pairs {ha;bi|a,b ∈ }, and
X
.
= ∪˙ {⊥} = (a) ∨˙ a = ⊥ : N → 2
⊥ def
X X X
7
isX augmentedbysymbol(code)⊥ : → N,⊥ takingcareofdefinedundefined
1
arguments of defined partial maps.5
Here we view (formally) = (a), = (a) : N → N as PR-
⊥ ⊥
predicates, not “yet” as abstrXacted XObjectXs = X{N| }, = {N| },
⊥ ⊥
X X X X
of Theory PR = PR+(abstr).
A
.
We allow us to write “a ∈ ” instead of (a) = true : N → N, and
. X X
“a ∈ ” for (a) = true, and similarly for other predicates.
⊥ ⊥
X X
This way we introduce – `a la Reiter – “Object” 2 just as target for predi-
cates χ : A → 2, meaning χ : A → N to be a predicate in the exact sense that
χ : → N satisfies
A
χ◦sign = χ◦¬◦¬ = χ : N −χ→ N −si→gn N, “still” A fundamental.
bydef
We define, within endo map set PR(N,N) a subTheory PR externally PR
X
as follows, by mimikry of schema (abstr) for the special case of predicate =
(a) : N → N, but without introduction of a coarser notion of equality, aXs in
X
case of schema of abstraction constituting Theory PR = PR+(abstr).
A
So Theory PR ⊂ PR(N,N) comes in, by external PR enumeration of its
X
Object and map terms as follows:
Objects of PR are predicates χ : → 2, i.e. PR-predicates χ : N → 2
X X
such that
PR ⊢ χ(a) =⇒ (a) : N → 2, i.e. such that
X
PR ⊢ χ(a) =⇒ (a) ∧ a 6= ⊥ : N → 2.
⊥
X
PR -maps in PR (χ,ψ) are PR-maps f : N → N such that
X X
.
¬ (a) =⇒ f(a) = ⊥, and χ(a) =⇒ ψ ◦f(a) : N → 2,
X
observe the “truncated” parallelism to definition of PR -maps
A
f : {A|χ} → {B|ψ}.
⊏
Then “assignment” I : PR −→ PR is defined as follows externally PR:
X
I = ˙ = {h0i} : N ⊃ ⊃ → 2,
def ⊥
1 1 X X
IN = N˙ = hNi = {hni|n ∈ N} : N ⊃ ⊃ → 2,
def def ⊥
X X
and further recursively:
I(A×B) = hA×Bi = {ha;bi|(a,b) ∈ (A×B)} : N ⊃ → 2,
def def
X
Functorial definition of I on PR maps:
I ˙
PR(A,B) ∋ f 7→ If = f ∈ PR
X
5
cf. Ch. 1, final section
X
8
then is “canonical”, by external PR on the structure of PR-map f : A → B,
in particualar by mapping all “arguments” in NrA˙ = NrIA into ⊥ ∈ :
⊥
X
one waste basket outside all Objects of PR .6
X
Interesting now is that we can extend embedding I above into an embedding
I : PR −→ PR , by the following
A
X
Definition: For a (general) PR Object, of form {A|χ}, define
A
I{A|χ} = {A˙ |χ˙} = {IA|Iχ}
def bydef
.
= {a ∈ IA|Iχ(a) = htruei} : N ⊃ → 2.
bydef ⊥
X
We replace here “don’t-worry arguments” in the complement ¬χ of PR -
A
Object {A|χ} by cutting them out in the definition of replacing PR -Object
I{A|χ} = {A˙ |χ˙}. “Coarser” notion =PRA (coarser then =PR) isXthen re-
placed by original notion of equality, =PR itself, notion of map-equality of roof
PR “⊂”PR(N,N) : This formal “sameness” of PR equality was the goal of
X
the considerations above: The new version PR replacing PR isomorphically,
XA A
is a subTheory of PR with notion of equality – objectively as well as (then)
internally – inherited from fundamental Theory PR.
Universal Embedding Theorem:7
(i) I : PR −→ PR ⊂ PR(N,N) above is an embedding which preserves
X
composition.
∼
=
(ii) (Enumerative) Restriction I : PR −→ PR = I[PR] of this embed-
X def
ding to its (enumerated) Image defines an isomorphism of categories. It
is defined above as
hf : A → Bi −→I hf˙ : A˙ → B˙ i,
by the “natural” (primitive) recursion on the structure of f as a map in
fundamental Theory PR of (Cartesian) Primitive Recursion.
(iii) PR embedding I “canonically” extends into an embedding (!)
I : PR −→ PR(N,N)
A
of Theory PR = PR+(abstr) – Theory PR with abstraction of predi-
A
cates into (“new”, “virtual”) Objects {A|χ : A → 2} – to the Set of PR
endomaps of N, of which – by the way – PR ( , ) is (formally) a
A ⊥ ⊥
X X
SubQuotient.
[Equality =PRA of (distinguished) PR endo maps when viewed as
PR endo maps on = {N| : N → 2}, is embedded to
A ⊥ ⊥
PR - (PR-)equalityXby I : PRX −→ PR “⊂”PR(N,N).]
A
X X
6for the details see Ch. 1, final section .
7from Ch. 1, final section X
X
9
(iv) Mainassertion: Embedding Iabovedefinesanisomorphismofcategories
∼
=
I : PR −→ PR
A XA
ontoa “naturallychoosen” (emumerated) categoryPR ofPRpredicates
XA
on Universal Object (PR-predicate) : N → N, with canonical maps
⊥
X
in between (see above), and whith composition inherited from that of
PR(N,N). This isomorphism is defined (naturally) by
˙ ˙
I(f : {A|χ} → {B|ψ}) = hf : χ˙ → ψi,
χ˙ : N ⊃ ⊃ ⊃ A˙ → 2,
⊥
X X
ψ˙ : N ⊃ ⊃ ⊃ B˙ → 2,
⊥
X X
f˙ = I (f) : N ⊃ A˙ → B˙ ⊂ N above.
bydef PR
Bythis isomorphism of categories, PR inherits fromcategoryPR allof
XA A
its (categorically described) structure: the isomorphism transports Carte-
sian PR structure, equality predicates on all Objects, schema of predicate
abstraction, equalisers, and – trivially – the whole algebraic, logic and
order structure on NNO N and truth Object 2.
We have furthermore:
˙
(v) For each fundamental Object A, embedded Object A = IA ⊂ comes
⊥
X
with a retraction retr : → A˙ ∪˙ {⊥}, defined by retr (a) = a for
XA X⊥ XA def
˙
a ∈ A, retr (a) = ⊥ otherwise.
XA def
This family of retractions clearly extends to a retraction family
retr : → {A˙ |χ˙}∪˙ {⊥} = I{A|χ}∪˙ {⊥}
X{A|χ} X⊥
for all PR -Objects {A|χ} : This is what ⊥ ∈ is good for.
A ⊥
X
(vi) ForeachObject{A|χ}ofPR ,inparticularforeachfundamental Object
A
A ≡ {A|true }, PR comes with the characteristic (predicative) subset
A A
χ˙ : I{A|χ} : → 2 of defined PR above, isomorphic to {A|χ}
⊥ ⊥
X X
within PR (!) via “canonical” PR -isomorphism
A A
∼= ˙
iso : {A|χ} −→ I{A|χ} = {A|χ˙},
X{A|χ}
the PR -isomorphism defined PR on the “structure” of {A|χ}, as re-
A
striction of iso : A → IA for fundamental Object A, in turn (exter-
XA
nally/internally) PR defined by
iso (0) = h0i : → I ⊂ ,
X def ⊥
1 1 1 X
iso (0) = h0i : → [ I ⊂ ] IN ⊂ ,
XN def ⊥
1 1 X
further externally PR:
∼=
iso (a,b) = hiso (a);iso (b)i : A×B −→ I(A×B) ⊂ .
X(A×B) def XA XB X⊥
∼=
We name the inverse isomorphism jso : I{A|χ} −→ {A|χ}.
X{A|χ}
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