rudimentary recursion, provident sets and forcing mathias

7/29/2019 Rudimentary Recursion, Provident Sets and Forcing Mathias

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Rudimentary recursion, provident sets and forcing

A. R. D. MATHIAS

ERMIT, Universite de la Reunion

This is the corrected and slightly expanded text of the talk given by me on July 6th 2009 at theEuropean Set Theory Meeting held at Bedlewo in honour of Ronald Jensen.

Pages i - iii give the definition of rudimentary recursion, with numerous examples; pages iii andiv establish some fundamental properties of rud rec functions; pages v and vi give the definitionsof provident set and canonical progress; pages vii - xi sketch a proof that a set-generic extension ofa provident set is provident; finally pages xi-xiii comment on the problem of forcing over a modelof Zermelo or of Mac Lane set theory, and culminate in a commuting diagram. More detaileddefinitions and proofs will be found in my papers [M3, 4, 5, 6]

Rudimentary recursion

The 1 recursion theorem of Kripke-Platek set theory KP proves for G a 1 function that if G is total, so isthe function F given by the recursion

F(x) = G(Fx),

and further F is provably equal to a 1 function.If the defining function G is rudimentary in the sense of Jensen, we shall speak of F as given by a

rudimentary recursion, or, more briefly, that F is rud rec.In favourable cases we may also use this terminology when F is intended to be a function defined on

On rather than on V, or defined by recursion on other well-founded relations related to the epsilon relation.

A set of generators for the class of rudimentary functions

We define rudimentary functions R0, . . . R8 and certain auxiliary functions which we show to be gener-ated by R0, . . . R7 under composition.

R0(x, y) = {x, y}A0(x) = {x} [ = R0(x, x)]

R1(x, y) = x \ y

A4(x, y) = x y [ = x \ (x \ y)]

A5(x) = [ = x \ x]

R2(x) =

x

R3(x) = Dom(x)

R4(x, y) = x yR5(x) = x {(a, b)2 |a,b a b}

R6(x) = {(b,a,c)3 |a,b,c (a,b,c)3 x}

R7(x) = {(b,c ,a)3 |a,b,c (a,b,c)3 x}

A9(x) = x1 [ = Dom

R7(R7(R6(R7

{} x

)))

]

A14(x, y) = x{y} [ = Dom((x ([

x] {y}))1)]

R8(x, y) = {x{w} |w w y}

Some rudimentary recursions

EXAMPLE The definition of rank:

(x) =

{ (y) + 1 |y y x}EXAMPLE The definition of transitive closure:

tcl(x) = x

{tcl(y) |y y x}

EXAMPLE Let S(x) be the set of finite subsets ofx. Restricted to ordinals, this has a rudimentarily recursivedefinition:

S(0) = {}; S( + 1) = S() {x {} |x x S()}; S() =


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Rudimentary recursion from parameters

Let p be a set. We call a unary function F p-rud rec if there is a binary rudimentary G such that forall x,

F(x) = G(p, Fx).

EXAMPLE Ordinal addition is given by the recursion

A(, 0) = ; A(, + 1) = A(, ) + 1; A(, ) =


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Bounding rudimentary functions in a finite progress

DEFINITION A -progress is a sequence P | of transitive sets such that for each < , T(P) P+1and for each limit ordinal ,


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Provident sets

DEFINITION A set A is p-provident, where p is a set, if it is non-empty, transitive, closed under pairing andfor all p-rud rec F and all x in A, F(x) A.

REMARK If A is p-provident, p A.

EXAMPLE The Jensen fragment J is -provident for all 1.DEFINITION A is provident if it is p-provident for every p A.

EXAMPLE Each J is provident.

REMARK For provident sets, it is unnecessary to demand that they be closed under pairing, for if x A,the function y {x, y} is x-rud rec, being given by the recursion F(y) = {x, Dom Fy}.

A new symbol

DEFINITION F u =df {Fx | x u}

Bounding rudimentarily recursive functions in a single canonical progress

THEOREM Let F be p-rud rec, given by G. Then there exist sF and gF in such that for any transitive c

and any ordinal 0 with p Pc0 , any non-successor ordinal and any k ,

(i) F Pc Pc0+

; (ii) FPc+k Pc0++sF+kgF

.

THEOREM Let be indecomposable andc a transitive set. Then Pc is provident.

Proof : Let p Pc ; choose 0 < with p Pc0

. Let F be p-rud rec. Then for each limit < ,

F Pc Pc0+ P

c . So F

Pc Pc , as required.

PROPOSITION Let c be a transitive set and an indecomposable ordinal. Then

Pc = Pc =


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PROPOSITION Let be an indecomposable ordinal, and let (Q) be a -progress with Q =


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Propagation of =

We have defined the progress Pc for c a transitive set. We could continue to work with progresses of theabove kind, but a problem would then arise at the end of the paper, in the proof that a set-generic extensionof a provident set is provident.

Here it is better to work with other progresses, which might be called construction from e as a set and= as a predicate, with the definition of = evolving during the construction.

DEFINITION Let e be a transitive set of which P is a member; let = (P). We define by a p-rudimentaryrecursion a sequence ((e, P

e;= ,

e)3) of triples, thus obtaining a new progress (P

e;= ).

For every , e will be defined as before; for we set Pe;= = P

e; for < , we set

e = but at

, we set e = =Pe , which will be a set by the last Corollary. Thereafter we set

e+1 = e {x | x e} e =


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No new ordinals !

REMARK In section 6 of The strength of Mac Lane set theory, a forcing construction is done over a non-standard model N, and it was there blithely stated without proof that the generic extension would bringno new ordinals. Fortunately the model N was power-admissible, and therefore certainly a model ofPROVI, which is a sub-theory ofKP, so that that blithe statement was correct, as forcing over provident sets

indeed adds no new ordinals. The reader should be warned, though, that forcing over certain, admittedlypathological, transitive models presented in [M6] leads to generic extensions with, in some cases, more and,in other cases, fewer ordinals than in the ground model.

Construction of rudimentarily recursive Cohen terms for rank and transitive closure

Rank and transitive closure are pure rud rec; we show here that P-rud rec Cohen terms exist for them.Let S() be the basic function z z {z}.

LEMMA There is a rud function SP() such that valG(SP(x)) = S(valG(x)).

DEFINITION P(x) =df

P{(p, SP( P(y)) | (p, y) x & p IP}

REMARK P is rud rec in the parameter P.

LEMMA

Let A be provident, andP

A. For all x A, valG (

P

(x)) = (valG(x)).REMARK That all makes sense: if x is in A, the name P(x) is in A. Note that (valG(x)) is evaluated inthe universe. At present we do not know that the evaluation can be carried out in AP[G].

Proof : (valG(x)) =

{ (y) + 1 |y y valG(x)} definition of

=

{ (valG(w)) + 1 |w p :G (p, w) x} definition of valG(x)

=

{valG (P(w)) + 1 |w p :G (p, w) x} induction hypothesis

=

{valG(SP( P(w))) |w p : G (p, w) x} property of SP

=

valG({(p, SP( P(w))) |p,w (p, w) x}) by Lemma 60

= valG(

P({(p, SP( P(w))) |p,w (p, w) x})) property of

P

= valG (P(x)), by the definition of P.

DEFINITION tclP(x) =df x P P

({(p, tclP(z)) | (p, z) x}).

REMARK tclP is rud rec in the parameter P.

LEMMA Let A be provident, andP A. For all x A, valG(tclP(x)) = tcl(valG(x)).

Proof : by similar reasoning.

Forcing over primitive-recursively closed sets

Jensen and Karp give, following Gandy, this definition of the class of primitive recursive set func-tions: there are some initial functions, which are all rudimentary; two versions of substitution: F(x, y) =G(x, H(x), y) and F(x, y) = G(H(x), y); and this recursion schema:

F(z, x) = G(

{F(u, x) |u u z}, z, x).

LEMMA Let A be transitive and primitive recursively closed, and let F be primitive recursive. Then

valG({(p, F(y)) |p,y (p, y) x}) = {valG(F(y)) |y p : G (p, y) x}.

Proof : as before.

For notational simplicity there is only one x in the following, but it could easily be replaced by a finitesequence.

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PROPOSITION Let A be transitive and primitive recursively closed. Let P A, and let G be (A,P)-generic.Suppose that G(f , z , x) is primitive recursive in the parameterP, and that there is a primitive recursive GP

such that for all f , z , x in A,

valG(GP(f , z , x)) = G(valG(f), valG(z), valG(x)).

Suppose that F(z, x) = G(

{F(u, x) | u z}, z , x). Define FP

by

FP(z, x) = GP(

P({(p, FP(u, x)) |p,u (p, u) z}), z , x).

Then FP is primitive recursive in the parameterP, and for all z, x in A,

valG(FP(z, x)) = F(valG(z), valG(x)).

Proof : for fixed x by recursion on z:

F(valG(z), valG(x)) = G

{F(w, valG(x)) |w w valG(z)}, valG(z), valG(x)

= G

{F(valG(u), valG(x)) |u p : G (p, u) z}, valG(z), valG(x)

= G

{valG(FP

(u, x)) |u p :G (p, u) z}, valG(z), valG(x)

= G

valG({(p, FP(u, x)) |p,u (p, u) z}), valG(z), valG(x)

= G

valG(

P({(p, FP(u, x)) |p,u (p, u) z})), valG(z), valG(x)

= GP

P({(p, FP(u, x)) |p,u (p, u) z}), z , x

= valG(FP(z, x))

The above confirms an observation made some years ago by Jensen:

COROLLARY A set-generic extension of a primitive recursively closed set is primitive recursively closed.

Construction of Cohen terms for the stages of a progress.

Let e be a transitive set in the ground model of which P is a member, and let be indecomposable,exceeding the rank of e. Pe is provident. Let d be the Cohen term e {G}

P, so that valG(d) will be the

transitive set d = e {G}.

REMARK d will be a member of Pe (P)+k for some (small) k, given the definition of G, our convention that

1P = 1 and the fact that is 1P-rud rec.

Our task is to build for each < a name N() for the stage Pd of the progress towards d.

A simplified progress

Now (G) (IP) < (P), so that for , d = e {G}. It might be that (G) < (IP); to avoidbuilding names which make allowance for that uncertainty, we shall build names for the terms of a slightlydifferent progress (Qd).

DEFINITION for < , Qd = Pe; Q

d = P

e {G};

for , Qd+1 = T(Qe) {d} d+1; Q

d =

.

PROPOSITION If is indecomposable, then Qd is provident and equals Pd .

Generic extensions of provident sets and of Jensen fragments

THEOREM Let be an indecomposable ordinal strictly greater than the rank of a transitive set e which

contains the notion of forcing, P. Let G be (Pe ,P)- generic. Then (Pe )P[G] = P

e{G} and hence is provident.

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Proof : (Pe )P[G] contains P

e{G} , as we have for each < built a name in P

e that evaluates under G to

Qe{G} , and we know by the previous Proposition that Q

e{G} equals P

e{G} .

For the converse direction, we know that Pe{G} is provident, and has G as a member and hence can

support the G-rudimentary recursion defining valG(). Further Pe{G} includes (P

e), which is defined by

an e-rudimentary recursion, and so includes (Pe )P[G].

REMARK Thus, in this special case, a generic extension of a model of PROVI is a model ofPROVI. We shalluse this result to establish it more generally.

REMARK The theorem remains true if the hypothesis on is weakened to requiring that > (P).

Proof that a generic extension of a provident set is provident.

THEOREM Let A be provident, P A and G (A,P)-generic. Then AP[G] is provident.

Proof : Let =df On A and let

T = {c | c A & c is transitive & P c}.

Then

A =

{Pc | c T},

since the union on the right contains each element of A and is contained in A. It follows that

AP[G] =cT

(Pc )P[G]

By the previous theorem, as each Pc is provident and contains P,

AP[G] =cT

Pc{G}

and each P

c{G}

is provident. Now in [M4, Proposition 552] we proved theLEMMA If is indecomposable and D is a collection of transitive sets each of rank less than and such thatthe pair of any two is a member of a third, then

dD P

d is provident.

Take D = {c {G} | c T} to complete the proof.

Forcing over models of Mac Lane set or Zermelo set theory

The present treatment of set forcing does not work satisfactorily over a transitive but improvident modelof, say, Zermelo set theory, as is shown in [M6]; the difficulties would be removed by passing first to theprovident closure of the given model.

The main technical tool of an earlier paper, [M2], was a construction that starting from any modelM of M0 yields a model, which in [M6] we call Lune(M), ofM1 + every well founded extensional relation

is isomorphic to a transitive set; it follows that if M models the well-ordering principle, then Lune(M) willmodel KP.

Further, M, if it models M1, will be interpretable as a submodel of Lune(M); and it is shown in [M6]that if M models M, Lune(M) will model PROVI; so that we then have a second way of extending M to amodel supporting our presentation of forcing.

The passage from M to Lune(M) can be considered in two ways. If M is (a transitive set and) a-model in the sense that every binary relation (a, r) M that is considered by M to be extensional andwell-founded is genuinely well-founded, we can form the larger model by taking every such ( a, r), transitisingit, and taking the union of the resulting transitisations.

EXAMPLE Provably in ZF, Lune(V+) = H .

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If M is not a -model, the construction is still possible: we summarise the method in [M2].

HISTORICAL NOTE A similar construction, though differing in various details, is presented in Hinnionsthesis [H] which starting from models ofNF yields models of subsystems ofZF.

Construction of the lune

We note first that the existence of cartesian products is provable in M0:PROPOSITION M0 the Cartesian product of two sets is a set.

DEFINITION PutF1 =df {(a, r) | r is an extensional well-founded relation on a}

andW1 =df {(,a,r) | (a, r) F1 and a}.

The elements ofW1 will be the ingredients of our model: we think of (a, r) as denoting, in the model weare building, a transitive set of which names a member. We must say when two names denote the sameelement. To assist our intuition we write r for r{}, i.e. , r, when (a, r) F1 and and are in a.

DEFINITION Given (a, r), (b, s) F1, a map from a subset of a to a subset of b will be called a partialisomorphism from (a, r) to (b, s) if

(i) rdom dom ;(ii) for all dom , () = {() | r}s; i.e., for all r, () s (), and for all s () there

is a r with () = .

Thus the condition (ii) on is that { | s ()} = {() | r{}}; or, more succinctly, thats{()} = r{}. Note that, provably in M0, the property of being a partial isomorphism is 0 .

LEMMA The class of partial isomorphisms from (a, r) to (b, s) is a 0(a,r,b,s) subset of P(a b).

LEMMA A partial isomorphism is 1-1 as far as it goes.

LEMMA Given (a, r) and(b, s) in F1, any two partial isomorphisms from (a, r) to(b, s) agree on their commondomain.

LEMMA There is a largest partial isomorphism from (a, r) to (b, s).

Proof : The union of all partial isomorphisms from a to b is a set by Lemma 29; its domain is closed underr, and, by the last lemma, that union is also a partial isomorphism hence maximal and unique.

We write arbs for the maximal partial isomorphism from (a, r) to (b, s). The following properties areeasily checked.

LEMMA arar = ida; 1arbs = bsar; arbsbsct arct.

Now define two relations on W1:

(,a,r) 1 ( ,b,s)df dom arbs & arbs() = ,

and(,a,r)E1( ,b,s)df dom arbs & arbs()s.

REMARK If (a, r) and (c, t) are two members of F1 in a set A, then any partial isomorphism f between them

is a member of P(3

A 3

A), and hence locally the relations 1 and E1 are sets.

The above Lemma may be applied to prove the

PROPOSITION 1 is an equivalence relation, and a congruence with respect to E1.

PROPOSITION E1 is set-like in the sense that given any x in W1, there is a set y with x y such thatz(zE1x = w :y z 1 w).

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rudimentary recursion, provident sets and forcing mathias

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