Consistency Decision

# Consistency Decision

Michael Pfender111michael.pfender@alumni.tu-berlin.de
April 2014
last revised July 3, 2019
###### Abstract

The consistency formula for set theory can be stated in terms of the free-variables theory of primitive recursive maps. Free-variable p. r. predicates are decidable by set theory, main result here, built on recursive evaluation of p. r. map codes and soundness of that evaluation in set theoretical frame: internal p. r. map code equality is evaluated into set theoretical equality. So the free-variable consistency predicate of set theory is decided by set theory, -consistency assumed. By Gödel’s second incompleteness theorem on undecidability of set theory’s consistency formula by set theory under assumption of this -consistency, classical set theory turns out to be -inconsistent.

## 1 Primitive recursive maps

Define the theory of objects and p. r. maps as follows recursively as a subsystem of theory

• the objects

• the map constants

(zero),  (successor), (identities), (terminal maps), (left and right projections);

• closure against (associative) map composition,

• closure against forming the induced map into a product, for given components

• closure against forming the iterated map

 f\lx@sectionsign=f\lx@sectionsign(a,n)=fn(a):A×\mathbbmN→A, f0(a)=idA(a)=a, fsn(a)=f\lx@sectionsign(a,sn)=(f∘f\lx@sectionsign)(a,n)=f(fn(a,n)).

Furthermore is to inherit from uniqueness of the initialised iterated, in order to inherit uniqueness in the following full schema of primitive recursion:

 g=g(a):A→B (initialisation), h=h((a,n),b):(A×\mathbbmN)×B→B (step) (pr) f=f(a,n):A×\mathbbmN→B, f(a,0)=g(a) f(a,sn)=h((a,n),f(a)) +uniqueness of such p. r. defined map f.

This schema allows in particular construction of for loops,

for to dood

as for verification if a given text (code) is an (arithmetised) of a given coded assertion, Gödel’s p. r. formula 45.  ist von

(Formel 46.  is provable, is not p. r.)

## 2 PR code sets and evaluation

The map code set—set of gödel numbers—we want to evaluate is in the set of p. r. map codes from to

Together with evaluation on suitable arguments it is recursively defined as follows:

• Basic map constants in

• (zero),

 ev(┌0┐,0)=0,

(successor),

 ev(┌s┐,n)=s(n)=n+1,
• For an object (identity),

 ev(┌idA┐,a)=idA(a)=a,

(terminal map),

 ev(┌ΠA┐,a)=ΠA(a)=0.
• for objects (left projection),

 ev(┌lA,B,(a,b)┐=lA,B(a,b)=a,

(right projection),

 ev(┌rA,B,(a,b)┐=rA,B(a,b)=b.
• For

(internal composition),

 ev(v⊙u,a)=ev(v,ev(u,a)).
• For

(induced map code into a product),

 ev(⟨u;v⟩,c)=(ev(u,c),ev(v,c)).
• For (iterated map code),

 ev(u$,0)=idA(a)=a, ev(u$,sn)=ev(u,ev(u$,n)) (double recursion) This recursion terminates in set theory with correct results: Objectivity Theorem: Evaluation is objective, i. e. for in we have  ev(┌f┐,a)=f(a). Proof by substitution of codes of maps into code variables in the above double recursive definition of evaluation, in particular: • composition  ev(┌g┐⊙┌f┐,a)=ev(┌g┐,ev(┌f┐,a)), =g(f(a))=(g∘f)(a) recursively, and • iteration  ev(┌f┐$,⟨a;sn⟩)=ev(┌f┐,ev(┌f┐$,⟨a;n⟩)) =f(f\lx@sectionsign(a,n))=f(fn(a))=fsn(a) recursively. ## 3 PR soundness within set theory Notion of p. r. maps is externally p. r. enumerated, by complexity of (binary) deduction trees. Internalising—formalising—gives an internal notion of PR equality,  uˇ=kv∈PR×PR coming by th internal equation by th internal deduction tree PR evaluation soundness theorem framed by set theory For p. r. theory with its internal notion of equality ‘’ we have: 1. to evaluation soundness:  T⊢ uˇ=v⟹ev(u,x)=ev(v,x) (∙) Substituting in the above “concrete” codes into resp. we get, by objectivity of evaluation 2. -framed objective soundness of For p. r.maps  T⊢ ┌f┐ˇ=┌g┐⟹f(a)=g(a). 3. Specialising to case a p. r. predicate, and to we get -framed logical soundness of  T⊢ ∃kProvPR(k,┌χ┐)⟹∀xχ(x): If a p. r. predicate is—within -internally provable, then it holds in for all of its arguments. Proof by primitive recursion on the th deduction tree of the theory, proving its root equation These (argument-free) deduction trees are counted in lexicographical order. Super Case of equational internal axioms, in particular • associativity of (internal) composition:  ev(⟨w⊙v⟩⊙u,a)=ev(⟨w⊙v⟩,ev(u,a)) =ev(w,ev(v,ev(u,a))) =ev(w,ev(⟨v⊙u⟩,a))=ev(w⊙⟨v⊙u⟩,a). This proves assertion in present associativity-of-composition case. • Analogous proof for the other flat, equational cases, namely reflexivity of equality, left and right neutrality of identities, all substitution equations for the map constants, Godement’s equations for the induced map:  l⊙⟨u;v⟩ˇ=u, r⊙⟨u;v⟩ˇ=v, as well as surjective pairing  ⟨l⊙w;r⊙w⟩ˇ=w and distributivity equation  ⟨u;v⟩⊙wˇ=⟨u⊙w;v⊙w⟩ for composition with an induced. • proof of for the last equational case, the Iteration step, case of genuine iteration equation the internal cartesian product of map codes:  T⊢ ev(u$⊙⟨┌id┐#┌s┐⟩,⟨a;n⟩) (1) =ev(u$,ev(⟨┌id┐#┌s┐⟩,⟨a;n⟩)) =ev(u$,⟨a;sn⟩) =ev(u,ev(u$,⟨a;n⟩) =ev(u⊙u$,⟨a;n⟩). (2)

Proof of termination-conditioned inner soundness for the remaining genuine Horn case axioms, of form

 uˇ=iv∧u′ˇ=jv′⟹wˇ=kw′, i,j

Transitivity-of-equality case

 uˇ=iv∧vˇ=jw⟹uˇ=kw:

Evaluate at argument and get in fact

 T⊢ uˇ=kw ⟹ev(u,a)=ev(v,a)∧ev(v,a)=ev(w,a) (by hypothesis on u,v) ⟹ev(u,a)=ev(w,a): transitivity export q. e. d. in this case.

Compatibility case of composition with equality,

 uˇ=u′⟹⟨v⊙u⟩ˇ=⟨v⊙u′⟩: ev(v⊙u,a)=ev(v,ev(u,a))=ev(v,ev(u′,a)) =ev(v⊙u′,x),

by hypothesis on and by Leibniz’ substitutivity in q. e. d. in this first compatibility case.

Case of composition with equality in second composition factor,

 vˇ=iv′⟹⟨v⊙u⟩ˇ=k⟨v′⊙u⟩: ev(⟨v⊙u⟩,x)=ev(v,ev(u,x))=ev(v′,ev(u,x)) (∗) =ev(⟨v′⊙u⟩,x).

holds by induction hypothesis on and Leibniz’ substitutivity: same argument put into equal maps.

This proves soundness assertion in this 2nd compatibility case.

(Redundant) Case of compatibility of forming the induced map, with equality, is analogous to compatibilities above, even easier, since the two map codes concerned are independent from each other what concerns their domains.

(Final) Case of Freyd’s (internal) uniqueness of the initialised iterated, is case

 ⟨w⊙⟨┌id┐;┌0┐⊙┌Π┐⟩ˇ=iu⟩ ∧⟨w⊙⟨┌id┐#┌s┐⟩ˇ=j⟨v⊙w⟩⟩

Comment: is here an internal comparison candidate fullfilling the same internal p. r. equations as the initialised iterated It should be – is: soundness – evaluated equal to the latter, on

Soundness assertion for the present Freyd’s uniqueness case recurs on turned into predicative equations ‘’, these being already deduced, by hypothesis on Further ingredients are transitivity of ‘’ and established properties of evaluation

So here is the remaining – inductive – proof, prepared by

 T⊢ ev(w,⟨a;0⟩)=ev(u;a) (¯0) as well as ev(w,⟨a;sn⟩)=ev(w⊙⟨┌id┐#┌s┐⟩,⟨a;n⟩) =ev(v⊙w,⟨a;n⟩), (¯s)

the same being true for in place of once more by (characteristic) double recursive equations for this time with respect to the initialised internal iterated itself.

and put together for both then show, by induction on iteration count —all other free variables together form the passive parameter for this induction—soundness assertion for this Freyd’s uniqueness case, namely

 T⊢ ev(w,⟨a;n⟩)=ev(v\$⊙⟨u#┌id┐⟩,⟨a;n⟩).

Induction runs as follows:

Anchor

step:

 ev(w,⟨a;n⟩)=ev(w′,⟨a;n⟩)⟹ ev(w,⟨a;sn⟩)=ev(v,ev(w,⟨a;n⟩)) =ev(v,ev(w′,⟨a;n⟩))=ev(w′,⟨a;sn⟩),

q. e. d.

## 4 PR-predicate decision

We consider here predicates for decidability by set theorie(s) Basic tool is -framed soundness of just above, namely

 χ=χ(a):A→\mathbbm2  PR predicate T⊢ ∃kProvPR(k,┌χ┐)⟹∀aχ(a).

Within define for out of a partially defined (alleged, individual) -recursive decision by first fixing decision domain

 D=Dχ:={k∈\mathbbmN:¬χ(ctA(k))∨ProvPR(k,┌χ┐)},

(retractive) Cantor count of and then, with (partial) recursive within

 ∇χ =def ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩false % if ¬χ(ctA(μD))(\emphcounterexample),true if ProvPR(μD,┌χ┐)(\emphinternalproof),⊥ (\emphundefined) otherwise, i.\thinspace e.if ∀aχ(a)∧∀k¬ProvPR(k,┌χ┐).

[ This (alleged) decision is apparently -recursive within even if apriori only partially defined.]

There is a first consistency problem with this definition: are the defined cases disjoint?

Yes, within frame theory which soundly frames theory

 T⊢ (∃k∈\mathbbmN)ProvPR(k,┌χ┐)⟹∀aχ(a).

We show now, that decision is totally defined, the undefined case does not arise, this for -consistent in Gödel’s sense.

We have the following complete – metamathematical – case distinction on

• 1st case, termination: has at least one (“total”) PR point and hence

 t=t∇χ =bydef μD=minD:\mathbbm1→D

is a (total) p. r. point.

Subcases:

• 1.1, negative (total) subcase:

[ Then ]

• 1.2, positive (total) subcase:

[ Then

by -framed objective soundness of ]

These two subcases are disjoint, disjoint here by -framed soundness of theory which reads

 T⊢ ProvPR(k,┌χ┐)⟹∀aχ(a), k free,

here in particular – substitute into free:

 πR⊢ ProvPR(t,┌χ┐)⟹∀aχ(a).

So furthermore, by this framed soundness, in present subcase:

 T⊢∀aχ(a)∧ProvPR(t,┌χ┐).(∙)
• 2nd case, derived non-termination:

[ then in particular

so in this case ],

and furthermore

 T⊢ ∀k¬ProvPR(k,┌χ┐), so T⊢ ∀aχ(a)∧∀k¬ProvPR(k,┌χ┐)(∗)

in this case.

• 3rd, remaining, ill case is:

(metamathematically) has no (total) points but is nevertheless not empty.

Take in the above the (disjoint) union of 2nd subcase of 1st case, and of 2nd case, as new case. And formalise last, remaining case. Arrive at the following

Quasi-Decidability Theorem: each p. r. predicate gives rise within set theory to the following complete (metamathematical) case distinction:

1. or else

2. (defined counterexample), or else

3. non-empty, pointless, formally: in this case we would have within

 T⊢ ∃^a∈D, {and} nevertheless'' {for each} \text{p.\thinspace r.}\ % point p:\mathbbm1→\mathbbmN T⊢ p∉D.

We rule out the latter – general – possibility of a non-empty predicate without p. r. points, for frame theory by gödelian assumption of -consistency. In fact it rules out above instance of -inconsistency: all numerals are p. r. points. Hence it rules out – in quasi-decidability above – possibility (c) for decision domain of decision operator for predicate and we get

Decidability theorem: Each free-variable p. r. predicate gives rise to the following complete case distinction by set theory

Under assumption of -consistency for

• (theorem) or

•  (counterexample)

Now take here for predicate ’s own free-variable p. r. consistency formula

 ConT=¬ProvT(k,┌false┐):\mathbbmN→\mathbbm2,

and get, under assumption of -consistency for a consistency decision for by

This contradiction to (the postcedent of) Gödel’s 2nd Incompleteness theorem shows that the assumption of -consistency for set theories must fail:

Set theories are -inconsistent.

This concerns all classical set theories as in particular and The reason is ubiquity of formal quantification within these (arithmetical) theories.

Problem: Does it concern Peano Arithmetic either?

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