Natural Internal Forcing Schemata extending ZFC.

arXiv:math/9209209v1 [math.LO] 25 Sep 1992Natural Internal Forcing Schemata extending ZFC.A Crack in the Armor surrounding CH?Garvin Melles∗Hebrew University of JerusalemSeptember 6, 2018Mathematicians are one over on the physicists in that they already havea unified theory of mathematics, namely set theory.Unfortunately theplethora of independence results since the invention of forcing has takenaway some of the luster of set theory in the eyes of many mathematicians.Will man’s knowledge of mathematical truth be forever limited to thosetheorems derivable from the standard axioms of set theory, ZFC? Thisauthor does not think so, and in fact he feels there is a schema concerningnon-constructible sets which is a very natural candidate for being consideredas part of the axioms of set theory.

To understand the motivation why, letus take a very short look back at the history of the development of math-ematics. Mathematics began with the study of mathematical objects veryphysical and concrete in nature and has progressed to the study of thingscompletely imaginary and abstract.

Most mathematicians now accept theseobjects as as mathematically legitimate as any of their more concrete coun-terparts. It is enough that these objects are consistently imaginable, i.e.,exist in the world of set theory.

Applying the same intuition to set theoryitself, we should accept as sets as many that we can whose existence areconsistent with ZFC. Of course this is only a vague notion, but knowl-edge of set theory so far, namely of the existence of L provides a goodstarting point.

What sets can we consistently imagine beyond L? Since byforcing one can prove the consistency of ZFC with the existence of non-constructible sets and as L is absolute, with these forcing extensions of Lyou have consistently imagined more sets in a way which satisfies the vaguenotion mentioned above.

The problem is which forcing extensions should∗Would like to thank Ehud Hrushovski for supporting him with funds from NSF GrantDMS 89595111

you consider as part of the universe? But there is no problem, because if youprove the consistency of the existence of some L generic subset of a par-tially ordered set P ∈L with ZFC, then P must be describable and wecan easily prove the consistency of ZFC with the existence of L genericsubsets of P for every P definible in L. Namely, the axiom schema IFSL(For internal forcing schema over L) defined below is consistent with ZFC.Definition 0.1 IFSL is the axiom schema which says for every formulaφ(x), if L |= there is a unique partial order P such that φ(P), then thereis a L generic subset of P in the universe V.IFSL is a natural closure condition on a universe of set theory.Givena class model of ZFC which has no inner class model of the form L[G]for some partial order P definable in L, we can (by forcing) consistentlyimagine expanding the model to include such a class.

Conversely, no classmodel of ZFC + IFSL can be contained in a class model of ZFC whichdoes not satisfy IFSL.Theorem 0.2 If there is a sequence ⟨Mn | n < ω⟩of transitive modelswith Mn |= ZFCn where ZFC = Sn∈ω ZFCn, then Con(ZFC + IFSL)proofBy the compactness theorem and forcing.Theorem 0.3 If Vis a model of ZFC, then V |= IFSL if and only ifV |= every set definable in L is countable.proofCertainly if every set definable in L is countable, then every par-tially ordered set definable in L is countable, so therefore is the set of densesubsets of P in L countable and so P has generic subsets over L in theuniverse. In the other direction, if s is a set definable in L, then so is thepartially ordered set consisting of maps from distinct finite subsets of s todistinct finite subsets of ω, so a L generic subset over the partial orderingis a witness to |s| = ω.Perhaps IFSL is not surprising since ZFC + 0# ⊢IFSL.

But the samereasoning as led to IFSL leads to the following stronger schema, IFSAb L[r](For internal forcing schema for absolute class models of ZFC constructibleover an absolutely definable real) which implies that if 0# exists, then allsets definable in L[0#] are countable.2

Definition 0.4 A subset r of ω is said to be absolutely definable if forsome Π1 formula θ(x),1. V |= θ(r)2.

ZFC ⊢∃xθ(x) →∃! xθ(x)Definition 0.5 IFSAb L[r] is the axiom schema of set theory which saysif r is an absolutely definable real then all definable elements of L[r] arecountable (equivalently, every partial order P definable in L[r] has an L[r]generic subset.

)The following theorem is a formal justification of IFSAb L[r].Theorem 0.6 Suppose V is a countable transitive model of ZFC and let{θi(x) | i < ω} be the list of all formulas defining absolute reals such thatV |= Vi<ω∃xθi(x). Suppose that the supremum of the ordinals definable inVis in V. Then there is a countable transitive extension V ′ of Vwiththe same ordinals such thatV ′ |= ZFC + IFSAb L[r] +^i<ω∃xθi(x)proofLet α∗be the sup of all the ordinals definable in L. Let P bethe set of finite partial one to one functions from α∗to ω.

Let V ′ = V [G]where G is a Vgeneric subset of P. To finish the proof it is enough toprove the following claim.Claim: If ψ(x) defines a real in M[G] then it is in M.proofSince P is separative, if p ∈P and π is an automorphism of P,then for every formula ϕ(v1, . .

. , vn) and names x1, .

. .

, xn∗p ⊩ϕ(x1, . .

. , xn) iffπp ⊩ϕ(πx1, .

. .

, πxn)Let ϕ(x) = ∃Y (ψ(Y ) ∧x ∈Y ). Let n ∈ω.

If for no p ∈Pdoesp ⊩||ϕ(ˇn)|| then ||ϕ(ˇn)|| = 0. So let p ∈Psuch that p ⊩||ϕ(ˇn)||.By ∗if π is an automorphism of Pthen πp ⊩||ϕ(ˇn)||.

Let π be apermutation of ω.π induces a permutation of P by letting for p ∈P,dom πp = dom p and letting πp(α) = π(p(α)). By letting π vary over the3

permutations of ω it follows that ||ϕ(ˇn)|| = 1. Let ˙r be the name withdomain {ˇn | n < ω} and such that˙r(ˇn) = ||ϕ(ˇn)||iG( ˙r) = r, but then r = {n | ||ϕ(ˇn)|| = 1} which means it is in M.Corollary 0.7 ZFC + IFSAb L[r]+ ’there are no absolutely definable non-constructible reals’ is consistent.

(Relative to the assumption of a countabletransitive model of L with its definable ordinals having a supremum in themodel. )Since classes of the form L[r] are absolute if r is an absolutely definablereal, they provide reference points from which to measure the size of theuniverse.

We can extend the schema IFSAb L[r] by exploiting the similaritybetween a class such as L(R) and a class of the form L[r] where r is anabsolutely definable real. We can argue that if P is a partial order definablein L(R), and if a V generic subset of P cannot add any reals to V, thenan L(R) generic subset of P should exist in V.L(R) is concrete in thesense the interpretation of L(R) is absolute in any class model containingR , and thereby like classes of the form L[r] where r is an absolutelydefinable real, L(R) provides a reference point from which to measure thesize of the universe.

This leads to the following natural strengthening ofIFSL and IFSAb L[r].Definition 0.8 x ∈Vis said to be weakly absolutely definable of the formVα if for some formula ψ(v) which provably defines an ordinal and whichis provably ∆1 from ZF,V |= ∃!αψ(α) ∧∀y(y ∈x ↔ρ(x) ≤α)Let θ(x) denote ∃!αψ(α) ∧∀y(y ∈x ↔ρ(x) ≤α)and let ZFθ be afinite part of ZF which proves ψ(v) is ∆1 and proves ψ(v) defines anordinal. θ(x) is said to define a weakly absolutely definable set of the formVα.

( ρ(x) denotes the foundation rank. )Definition 0.9 IFSWAb L(Vα) is the axiom schema of set theory which saysfor every weakly absolutely definable set of the form Vα for every partialorder P definable in L(Vα), if||V V [G]α= V Vα ||(r.o.P )V = 14

then there exists an L(Vα) generic subset G of P.Theorem 0.10 If there is a sequence ⟨Mn | n < ω⟩of transitive modelswith Mn |= ZFCn where ZFC = Sn∈ω ZFCn then Con(ZFC+IFSWAbL(Vα))proofLet ⟨θi | i < n⟩be a list of formulas defining weakly absolute setsof the form Vα. Let {ϕij(x) | i < n, j < m, } be a set of formulas.

It isenough to show the consistency with ZFC of^i

. .

, αn−1⟩be the increasing sequence of ordinals suchthatM |= θi(Vαi)for i < n. We define by induction on (i, j) ∈n × m sets Gij. SupposePij is a partial order definable in L(V M[{Gh,l|h≤i,l

(If not, letGij = ∅. ) LetN = M[{Gij|i < n, j < m}]N has the property that if Pij is a partial order definable in L(Vαi) byϕij(x) and G is an N generic subset of Pij such that||V N[G]αi= V Nαi ||(r.o.P )N = 1then an L(V Nαi ) generic subset of Pij exists in N.Theorem 0.11 If ⟨Mn | n < ω⟩is a sequence of transitive models withMn |= ZFCn where ZFC = Sn∈ω ZFCn, then Con(ZFC +IFSWAb L[Vα] +IFSAb L[r])5

proofSame as the last theorem except we start with a model of enoughof ZFC + IFSAb L[r].Theorem 0.12 V [G] has no functions f : κ →κ not in the ground modelif and only if r.o.P is (κ, κ)-distributive.proofSee [Jech1].Corollary 0.13 IFSWAb L(Vα)is equivalent to the axiom schema of settheory which says for every weakly absolutely definable set of the form Vα,for every partial order P definable in L(Vα), if(r.o.P)V is (κ, κ)-distributivefor each κ such that for some β < α, κ ≤|Vβ|, then there exists an L(Vα)generic subset G of P.Theorem 0.14 ZFC + IFSWAb L(Vα) ⊢CHproofLet P = the set of bijections from countable ordinals into R. SinceP is σ closed, ω1 = ωL(R)1, and P is a definable element of L(R), thereis an L(R) generic subset of P in V. If α is an ordinal less than ω1 andr is a real, let Dα = {p ∈P | α ∈dom p} and Dr = {p ∈P | r ∈ran p}.For each α < ω1,G ∩Dα ̸= ∅and for each r ∈R,G ∩Dr ̸= ∅, so S Gis a bijection from ω1 to R.Perhaps the following is a better illustration of the kind of result obtain-able from ZFC + IFSWAb L(Vα).Definition 0.15 A Ramsey ultrafilter on ω is an Ultrafilter on ω suchthat every coloring of ω with two colors has a homogenous set in the ultra-filter.Theorem 0.16 ZFC + IFSWAb L(Vα) ⊢there is a Ramsey ultrafilter onω.proofLet P be the partial order (P(ω), ⊆∗) where P(ω) is the powerset of ω and a ⊆∗b means a is a subset of b except for finitely manyelements.P is definable is L(R) and is ω closed. The generic object is6

an Ramsey ultrafilter over L(R), and since all colorings of ω are in L(R),it is a Ramsey ultrafilter over V.One can argue that IFSWAb L(Vα)is not a natural axiom since amongthe definable sets X with the property that L(X) is absolute when notincreasing X, why should you choose only those of the form Vα? But it isnatural in the sense it is a way of forcing the universe as large as possible withrespect to the existence of generics by first fixing the height of the modelsunder consideration and then by fixing more and more of their widths.

Inany case we should consider the strengthenings of IFSWAb L(Vα) definedbelow.Definition 0.17 x ∈V is said to be weakly absolutely definable if for someformula ψ(x) which is provably ∆1 from ZF,V |= ∀y(y ∈x ↔ψ(y))Definition 0.18 IFS is the axiom schema of set theory which says forevery weakly absolutely definable set X, for every partial order P definablein L(X), if||XV [G] = XV ||(r.o.P )V = 1then there exists an L(X) generic subset G of P.If Xis an weakly absolutely definable set and Pis a partial orderingdefinable in L(X) such that||XV [G] = XV ||(r.o.P )V = 1and if there is no L(X) generic subset of P in V, we say that Vhas agap. IFS says there are no gaps.

The intuition that such gaps should notoccur in Vleads to the following:Conjecture 1 ZFC + IFS is consistent.If ZFC + IFS is consistent, then this means that it is consistent thatthe universe is complete with respect to the natural yardstick classes, (theclasses of the form L(X) where X is weakly absolutely definable.) In myview, confirming the consistency of ZFC + IFS would be strong evidencethat the universe of set theory conforms to the axioms of IFS.

One reasonfor this opinion is that there is no apriori reason for the consistency ofZFC + IFS, so if ZFC + IFS is consistent, it seems that confirmimg itsconsistency would involve some deep mathematics implying IFS should betaken seriously.7

1Formalizing the arguments in favor of IFSL andthe other schemataIn this section we try to formalize the vague notion that IFSL is a naturalclosure condition on the universe, and that gaps in general are estheticallyundesirable. For simplicity we concentrate on IFSL.Definition 1.1 Let Tbe a recursive theory in the language of set theoryextending ZFC.

Let Pbe a unary predicate. If ϕ is a formula of settheory then ϕ∗is ϕ with all its quantifiers restricted to P, i.e., if ∃xoccurs in ϕ then it is replaced by ∃x(P(x) ∧.

. .) and ∀x is replaced by∀x(P(x) →.

. .

). The theory majorizing T,T ′, is the recursive theory inthe language {ε, P(x)} such that1.

ϕ ∈T →ϕ∗∈T ′2. P(x) is transitive ∈T ′3.

∀x(x ∈ORD →P(x)) ∈T ′4. ZFC ⊆T ′If θ(x) = ∀y(y ∈x ↔ψ(x)) is a formula defining a weakly absolutelydefinable set then the theory majorizing T with respect to θ(x) is T ′ plusall the axioms of the formϕ1 ∧.

. .

∧ϕn →(ψ(y) ↔∃zψ0(y, z) ↔∀zψ1(y, z))−→∀y(ψ(y) →P(y))where ϕ1, . .

. , ϕn ∈ZF and ψ0(y, z) and ψ1(y, z) are ∆0 formulas.Theorem 1.2 Let Tbe a recursive extension of ZFC.

Let T = Sn∈ω Tnwhere for some recursive function F, for each n,F(n) = Tn, a finitesubset of T and the Tn are increasing. If there is a sequence ⟨Mn | n < ω⟩of countable transitive models such thatMn |= Tnthen T ′ + IFSL ( T ′ is the theory majorizing T ) is consistent and thereis a sequence ⟨Nn | n ∈ω⟩of countable transitive models such thatNn |= T ′nwhere T ′ =Sn∈ω T ′n and for some recursive function H, for each n ∈ω,H(n) = T ′n a finite subset of T ′.8

proofLet IFSL =Sn∈ω(IFSL)n where for each n ∈ω, (IFSL)n isfinite.We can find a subsequence ⟨Nn | n ∈ω⟩of the ⟨Mn | n ∈ω⟩and Nn -generic sets Gn such that Nn[Gn] |= (IFSL)n, Nn |= Tn. LetNn[Gn]∗be the model in the language {ε, P(x)} obtained by letting theinterpretation of P(x) to be Nn.

Let D be an ultrafilter on ω. ThenYNn[Gn]∗/Dis a model for T ′ + IFSL.Definition 1.3 A theory extending ZFC is ω−complete if whenever ϕ(x)is a formula of set theory and if for each natural number n,T ⊢ϕ(n)then T ⊢∀n ∈ωϕ(n).Theorem 1.4 Let T be a recursive extension of ZFC and suppose it hasa consistent, complete and ω−complete extension T ∗. Then T ′ +IFSL isconsistent.proofBy reflection in T ∗, by its ω−completeness and by the axiom ofchoice in T ∗,T ∗⊢∃⟨Nn | n ∈ω⟩with the ⟨Nn | n ∈ω⟩having the same properties as in the previoustheorem.

As in the previous theorem since ZFC ⊂T,T ∗⊢Con(T ′ +IFSL). Since T ∗is ω−complete, (by the omitting types theorem) it hasan model M with the standard set of integers.

Since M |= T ∗,M |= Con(T ′ + IFSL)and as Con(T ′+IFSL) is an arithmetical statement, it must really be true.Certainly if the hypothesis of the theorem fails, then T cannot be a suitableaxiom system for set theory.Definition 1.5 If θ(x) is a formula defining an weakly absolutely definableset, then IFS ↾θ(x) is IFS restricted to the set defined by θ(x), i.e.,it says for all partial orders Pdefinable in L(X) were X is defined byθ(x) such that||XV [G] = XV ||(r.o.P )V = 1there is an L(X) generic subset of P.9

Theorem 1.6 Let θ(x) be a formula defining an weakly absolutely defin-able set. Let T be a recursive extension of ZFC and suppose it has a con-sistent, complete and ω−complete extension T ∗.

Then T ′θ(x) + IFS ↾θ(x)is consistent.proofSame as above.Theorem 1.7 Let θ(x) be a formula defining an weakly absolutely defin-able set. Let T be a recursive extension of ZFC +IFS ↾θ(x) and supposeT ′θ(x) majorizes T with respect to θ(x).

Then T ′θ(x) ⊢IFS ↾θ(x).proofWorking in T ′ the generics in the inner model are still generic overL(X) since the inner model is a transitive class containing all the ordinals.The theorems in this section are meant as the formalization of the notionthat we can ’consistently imagine’ a class model of ZFC not satisfyingIFSL as being contained in a larger class satisfying ZFC + IFSL, andthat models of ZFC not satisfying IFS have a gap.2ConclusionThese axiom schemata lead to many questions, among them1. Are there models of IFS or IFSWAb L[Vα] which are forcing exten-sions of L ?2.

Are there similar natural schema’s making the universe large, but con-tradicting IFS or IFSWAb L[Vα]?3. What are the consequences for ordinary mathematics of these axioms?The conventional view of the history of set theory says that Godel in 1938proved that the consistency of ZF implies the consistency of ZFC andof ZFC + GCH, and that Cohen with the invention of forcing proved thatCon(ZF) implies Con(ZF + ¬AC) and Con(ZFC + ¬GCH) but fromthe point of view of IFSL a better way to state the history would be tosay that Godel discovered L and Cohen proved there are many generic setsover L.I think confirming the consistency of IFS with ZFC would be a vin-dication of the idea that generics over partial orders definable in L(X) withX an weakly absolutely definable set exist, and thereby put a crack in the10

armor surrounding the continuim hypothesis as ZFC +IFS ↾R ⊢CH. Onthe other hand, if ZFC +IFS is not consistent, it would show the universemust have some gaps, i.e., incomplete with respect to some concrete set, anesthetically unpleasing result.

It is ironic that although mathematics andespecially mathematical logic is an art noted for its precise and formalizedreasoning, it seems that in order to solve problems at the frontiers of logic’sfoundations we must tackle questions of an esthetic nature.11

REFERENCES1. [CK] C. C. Chang and J. Keisler, Model Theory, North Holland PublishingCo.2.

[Jech1] T. Jech, Multiple Forcing, Cambridge University Press.3. [Jech2] T. Jech, Set Theory, Academic Press.12

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