From Hilbert's 10th Problem to slim, Undecidable Fragments of Set Theory

Undecidable Fragments of Set Theory

^?

Domenico Cantone[0000−0002−1306−1166]1,

Eugenio G. Omodeo[0000−0003−3917−1942]2, and Mattia Panettiere¹

1 Dept. of Mathematics and Computer Science, University of Catania, Italy [email protected], [email protected]

2 Dept. of Mathematics and Earth Sciences, University of Trieste, Italy [email protected]

Abstract. We revisit, deepen, and make more systematic, the study—

undertaken ca. 1990—of reductions of Hilbert’s tenth problem to frag- ments of set theory, whereby sublanguages of the Zermelo-Fraenkel ax- iomatic theory are shown to have an undecidable satisfiability problem.

Key words:Hilbert’s 10th problem; Undecidability;ZFset theory; Sat- isfiability problem; Cartesian product; (∀∃)0 formulas; Proof-checkers.

Introduction

We show that the satisfiability problems for various fragments of ZF—the Zermelo-Fraenkel first-order set theory with regularity axiom—areundecidable.

To wit, if Φ is among those sublanguages of ZF, no algorithm can establish whether or not any given formula ψ in Φ becomes true under suitable assign- ments of sets to its free variables.

For eachΦtaken into account, the undecidability result stems from a uniform translation method which turns every instanceD= 0 (withD∈Z[x₁, . . . , x_m]) of Hilbert’s 10^thproblem into a formulaψofΦso thatψis satisfiable if and only if the polynomial equation D(x₁, . . . , x_m) = 0 has natural solutions. Through this translation, the algorithmic unsolvability of Hilbert’s 10^th problem carries over to the satisfiability problem forΦ.

Some of the undecidableΦ’s are slight extensions of a core language consist- ing of all conjunctions of literals of the formsx=y∪z,x=y⊗z,x∩y=∅, and

|x|=|y|, wherex, y, z stand for variables,y⊗z is a variant of Cartesian prod- uct consisting of singletons and unordered doubletons, and |x|=|y|designates equinumerosity betweenxandy. Additional conjuncts entering into play can be, e.g.: one literal of the formFinite(x), stating thatxhas finitely many elements,

?Copyright c2020 for this paper by its authors. Use permitted under Creative Com- mons License Attribution 4.0 International (CC BY 4.0). We gratefully acknowledge partial support from project “STORAGE—Universit`a degli Studi di Catania, Piano della Ricerca 2020/2022, Linea di intervento 2”, and from INdAM-GNCS 2019 and 2020 research funds.

taken along with one literal of the form x 6= ∅. Another option would be to extend the said syntactic core by allowing threenegated equinumerosity literals of the form |x| 6= |y| to appear in the conjunction. Our interest in these, and similar, slim undecidable set-theoretic languages is increased by the fact that they lie extremely close to fragments ofZFwhich are either known [3, Sec.11.1], or conjectured [4, p. 239], to be decidable.

One can recast the undecidable fragments ofZFunder study in a set-theoretic language entirelydevoidof function symbols (such as∪,⊗, etc.), which only in- volves the relators ∈and =, propositional connectives, and also the constructs

∀x ∈ y ϕ and ∃x ∈ y ϕ involving bounded quantifiers, subject to a very re- strained use. The fact that somewhat sophisticated notions like “being an or- dinal”, “being the first limit ordinalω”, “having a finite cardinality”, “being a hereditarily finite set”, can be specified inside this collection of formulas, dubbed (∀∃)₀ formulas, gives evidence of the high expressive power of bounded quantifi- cation in the context ofZF.

In a full-fledged language for set theory, (∀∃)0specifications are only seldom used in order to define mathematical notions. Hence, to support the correct- ness of the proposed (∀∃)0-specifications ofω, equinumerosity, and finitude, we have proved that they are equivalent to more direct and practical characteriza- tions of the same notions. As we document at thehttp://aetnanova.units.

it/scenarios/SetTheoreticH10/, this formal accomplishment was carried out with the aid of a proof-checker embodying a computational version ofZF.

1 From Hilbert’s 10

^th

problem to unsolvable satisfiability problems in set theory

1.1 How to flatten instances of Hilbert’s 10^th problem Consider a polynomial Diophantine equation

D(x₁, . . . , x_m) = 0

to be solved inN. By pulling out subterms of the polynomialD, we can flatten this equation into a system (viz., a conjunction) of equations of the forms

x=y+z , x=y·z , x= 1, x=y ,

where x, y, z stand for variables, to be regarded—the new ones as well as the original ones, x1, . . . , xm—as unknowns inN(cf. [10,1]). We can then eliminate each equation of the formx=y by rewriting it asx=y+ζ, whereζ is forced to assume the value 0 (see equations (5) below).

The flattening process can easily enforce that x, y, z are distinct variables when they appear together in the same equationx=y ? z (with?∈ {‘+’,‘·’});

moreover, we will keep a sole equation, o= 1, involving 1. The equi-solvability between the system∆ so obtained and the original equation will be obvious.

Example 1. The equation (from [8, p. 4])

4x³₁x₂−2x²₁x³₃ − 3x²₂x₁ + 5x₃ = 0 (1) can be flattened into the following system ∆1 consisting of 26 equations in 28 unknowns, 3 of which are x1, x2, x3, namely the original unknowns of (1):

ζ=ζ1+ζ2, ζ1=ζ2+ζ , ζ2=ζ+ζ1, o= 1, u₂=u₁+o , u₄=u₃+o , u₁=o+ζ , u₃=u₂+o , u₅=u₄+o , P_x¹₁ =x1+ζ , P_x²₁=P_x¹₁·x1, P_x³₁ =P_x²₁·x1, P_x¹₂ =x2+ζ , P_x²₂=P_x¹₂·x2,

P_x¹

3 =x₃+ζ , P_x²

3=P_x¹

3·x₃, P_x³

3 =P_x²

3·x₃, M₁¹=u4·P_x³₁, M1=M₁¹·x2, M4=u5·x3, M₂¹=u2·P_x²₁, M2=M₂¹·P_x³₃, L=M1+M4, M₃¹=u3·P_x²₂, M3=M₃¹·x1, L=M2+M3.

By then replacing the equation o= 1 in∆1 by the constraintso 6= ζ and

o=o·o⁰, o⁰=o+ζ , (2)

(which are equisatisfiable with o=o² & o6=ζ), we get a system ∆ which is equisolvable—in N—with the initial equation (1) and consists of: 28 equations in 29 unknowns, of the forms

x=y+z , x=y·z , (3)

(where all compound terms y ? zinvolve distinct variables) and one inequality

o6=ζ . (4)

a Definition 1. Systems that consist of flat equations of the forms (3) conjoined with the equations

ζ=ζ₁+ζ₂, ζ₁=ζ₂+ζ , ζ₂=ζ+ζ₁ (5) and (2) and with the inequality (4), and inside which all terms of typey+zand y·zinvolve distinct variables, are namedcanonical Diophantine systems. Remark 1. One can also express multiplication in terms of the squaringoper- ation (see [6, p. 230]). Notice, in fact, that any equation of the form x=y·z is equisolvable inNwith the flattened system of equations

p=y+z , x⁰=x+ζ ,

q=p², f =y², g=z², k=x+x⁰, h=f+g , q=k+h ,

as can be seen from the identity

q

z }| { (y+z

| {z }

p

)²=

k

z }| { y·z

|{z}x

+y·z

|{z}x⁰

+

h

z }| { y²

|{z}

f

+ z²

|{z}g

. a

From a canonical Diophantine systems∆, we will next get a conjunction∆b of set-theoretic constraints of the forms:

= ∪ union (ternary relation)

= × Cartesian product (ternary relation)

∩ = ∅ disjointness (dyadic relation)

| | = | | equinumerosity (dyadic relation)

Finite( ) finitude (property)

={ } singleton formation (dyadic relation) 6= ∅ non-emptyness (property)

| | 6= | | non-equinumerosity (dyadic relation)

Here, in light of the replaceability of multiplication by the squaring operation (as pointed out in Remark 1), we might only employ Cartesian squarey×y, without ever resorting to the producty×zwithydistinct fromz.

1.2 How to translate a canonical Diophantine system ∆ into unquantified fragments of set theory

Let∆be a Diophantine canonical system.

We translate each conjunct of∆ according to the following rules:

x=y+z u^y,z=y∪z & y∩z = ∅ ^& |u^y,z|=|x| , x=y·z w^y,z=y×z & |w^y,z|=|x| ,

o6=ζ o6=∅,

where eachu^y,zand eachw^y,zis a new variable. By also adding, for each variable uin∆, the conjunctFinite(u) (meaning that|u| ∈N), we obtain the set-theoretic counterpart∆bof∆.

Next we prove that the canonical system∆ is satisfiable inN if and only if its set-theoretic counterpart ∆b is satisfiable in the universe of all sets. In view of the unsolvability of Hilbert’s Tenth problem (see [8, Chapter 5]), we readily obtain the algorithmic unsolvability of the satisfiability problem for set-theoretic conjunctions of positive literals of the forms

x=y∪z , x=y×z , x∩y=∅, |x|=|y| , Finite(x), plus a single inequality of the formx6=∅.

To carry out the first half of the proof, we observe that any set-assignment v 7→ M v over the variables of ∆b that satisfies ∆b induces naturally the corre- spondingN-assignment

v7→ |M v| (6)

over the variables of ∆, and it is a simple matter to check that (6) satisfies∆.

For instance, ifx=y+zis in ∆, then the following conjuncts are in∆b u^y,z = y∪z , y∩z = ∅, |u^y,z| = |x| , Finite(x) , Finite(y) , Finite(z) . Hence, we have:

M u^y,z = M y∪M z , M y∩M z = ∅, |M u^y,z| = |M x| ,

|M x| ∈N, |M y| ∈N, |M z| ∈N, so that

|M x|=|M u^y,z|=|M y∪M z|=|M y|+|M z|, proving that theN-assignment (6) satisfies the equationx = y+z.

Similarly, one can show that theN-assignment (6) also satisfies all equations of the formx = y·z in∆ as well as the inequalityo6= 0.

For the converse proof, we will make use of the following notation, for every k∈NandA⊆N:

[k] :={1,2, . . . , k} and k+A:={k+a:a∈A}.

Suppose that v 7→ vvv is a solution to ∆ in N, and let v1, v2, . . . , v` be the distinct variables in ∆, in any fixed order. We define a set-assignment over the variables in ∆bby putting:

– M vj:=kj+ [vvvj], forj= 1, . . . , `, where k_j :=Pj−1

r=1vvv_r (so thatk1= 0);

– M u^y,z:=M y∪M z, for every literalx = y+z in∆; – M w^y,z:=M y×M z, for every literalx = y·z in∆.

We prove that the set-assignmentM so defined over the variables of∆bsatisfies all of the conjuncts of ∆b.

– Letx = y+zbe in∆. By the definition ofM, we have M u^y,z=M y∪M z ,

so thatM satisfies the conjunctu^y,z=y∪z.

Lety and z be the variables vi andvj, respectively, where without loss of generality we are assumingi < j. Also, letxbe the variablevh. Preliminarily, we show that

M vi∩M vj=∅. (7)

If∅ ∈ {M v_i, M v_j}, then (7) plainly holds. Otherwise, recalling that M vi:=ki+ [vvvi] and M vj :=kj+ [vvvj],

we get

maxM vi = ki+vvvi =

i−1

X

r=1

vvvr+vvvi =

i

X

r=1

vv vr 6

j−1

X

r=1

vvvr < minM vj, whence (7) follows in this case too. From (7), it follows that

|M u^y,z| = |M y∪M z| = |M y|+|M z| = |M v_i|+|M v_j|

= vvvi+vvvj = vvvh = |M vh| = |M x| ,

proving that the set-assignmentM satisfies the conjunct|M u^y,z| = |M x|.

– It is even easier to prove thatM satisfies also the literals in∆bof the forms w^y,z = y×z , |w^y,z| = |x| ,

resulting from the translation of equations of the formx = y·z, and the only literal of the formx6=∅in∆b.

– To end, for each variableuin∆, the assignmentM also satisfiesFinite(u) . We have thus obtained that the set-assignmentM satisfies the conjunction

∆, and hence conclude that the translationb ∆ 7→∆bis satisfiability-preserving, namely:

Theorem 1. A canonical Diophantine system∆is solvable in N if and only if the corresponding conjunction∆bis satisfied by some set-assignment.

Consequently, from the undecidability of Hilbert’s tenth problem we get:

Lemma 1. The satisfiability problem for set-theoretic conjunctions of any num- ber of positive literals of the forms

x=y∪z , x=y×z , x∩y=∅, |x|=|y| (‡) and of the form Finite(x), plus one negative literal of the form x6=∅, is algo- rithmically unsolvable.

Since finitude is⊆-hereditary, any conjunction V

i∈IFinite(xi) of finitude constraints can be replaced, without affecting satisfiability, by the conjunction Finite(F) & V

i∈IF =F∪xi containing a single finitude constraint, whereF is a newly introduced variable. The preceding undecidability result can hence be slightly strengthened into:

Lemma 2. The satisfiability problem for set-theoretic conjunctions of any num- ber of positive literals of the forms (‡), plusonepositive finitude literal,Finite(x), and onenegative literal of the form x6=∅is algorithmically unsolvable.

A fortiori, we also have the following negative result:

Lemma 3. The satisfiability problem for set-theoretic conjunctions of any num- ber of positive literals of the forms (‡), plusonepositive finitude literal,Finite(x), and onenegative literal of the form |x| 6=|y|is algorithmically unsolvable.

The following claim gives us a way of expressing the finitude of sets:

Proposition 1. A set with at least two members is finite if and only if it is the union of two sets whose cardinalities differ from its own cardinality.

Proof. It suffices to note that, for an infinite setsand any decompositions=s1∪ s2, we have (independently of whethers1∩s2=∅or not)|s|= max(|s1|,|s2|).

Thus, without affecting satisfiability, any conjunction V

i∈IFinite(xi) of finitude literals can be replaced by the conjunction

F =F1∪F2 & |F1| 6=|F| & |F2| 6=|F| & ^

i∈I

F =F∪xi

containing no finitude literal, whereF, F₁, F₂ are newly introduced variables.

In view of the preceding remark, Lemma 3 can be restated as follows.

Lemma 4. The satisfiability problem for set-theoretic conjunctions of any num- ber of positive literals of the forms (‡), plus at most threenegative literals of the form|x| 6=|y|, is algorithmically unsolvable.

One can express finitude also in the following way:

Proposition 2. A nonempty set is finite if and only if, by removing a member from it, one obtains a set of different cardinality.

Therefore, we have:

Lemma 5. The satisfiability problem for set-theoretic conjunctions of any num- ber of positive literals of the forms (‡), plus two positive literals of the form x = {y}, and one negative literal of the form |x| 6= |y|, is unsolvable.

Proof. In view of Lemma 2, it is enough to observe that:

– a negative literal of the formx 6= ∅can be expressed via a conjunction of the formy⁰ = {x⁰} & x = x∪y⁰ where x⁰, y⁰ are brand new variables;

– any conjunction V

i∈IFinite(xi) of finitude literals can be expressed by means of the conjunction

F^∗={f^∗} &F =F∪F^∗ &F⁻=F\F^∗ &

F⁻

6=|F| & ^

i∈I

F=F∪xi

containing no finitude literal, whereF, F^∗, F⁻ are new variables;

– a literal of the formx=y\zcan be expressed by means of the conjunction y=x∪x⁰ & x∩x⁰=∅ ^& x∩z=∅ ^& x⁰∩y=∅.

1.3 Undecidability results regarding unordered Cartesian product Much the same reductions can be carried out when the operation⊗defined by

s⊗t :=

{u, v}:u∈s, v∈t .

(for setss, twhatsoever) supersedes the standard Cartesian product operator×.

In order that Lemmas 1, 2, 3, 4, and 5 retain their validity with thisunordered product operator⊗in place of×, it suffices that the above-proposed translation rule for arithmetical constraints of the formx=y·z gets retouched as follows:

x=y·z w^y,z=y⊗z & y∩z=∅ ^& |w^y,z|=|x| .

2 Undecidability of a restrained collection of bounded-quantifier formulas in set theory

So far we have been designating various set-theoretic operations (e.g., dyadic union, Cartesian product, cardinality) and properties and relations (finitude, disjointness, etc.) by means ofad hocsigns ( ∪ , × ,Finite( ), ∩ =∅, etc.) which, if adopted beforehand as primitives in the language supporting set theory, would make the undecidable fragments reviewed in Sec. 1 devoid of quantifiers.

Most often, set theories get formalized in a first-order language devoid of constants and function symbols, whose only relators are∈and =. How complex then becomes the syntactic structure of the formulas lying in the undecidable fragments? As we will see next, a very modest usage of quantification is needed to state them: only bounded quantifiers, and just one quantifier alternation are needed, with (bounded) universal quantifiers in leading position.

2.1 The syntax of (∀∃)0 formulas

Consider the first-order languageL_∈ endowed with:

– an infinite supplyν₀, ν₁, ν₂, . . . of set variables;

– dyadic relators∈,= designating membership and equality;

– the familiar propositional connectives¬(monadic) and &, ∨,→,↔(dyadic);

– associated with each set variableν_i, the familiar quantifiers ∀ν_i and∃ν_i. We enhance the usual syntax of formulas with two handy shortening devices:

Definition 2. Universal and existentialbounded quantifiersare introduced as follows:

(∀x∈y)ϕ ↔_Def (∀x)(x∈y→ϕ) ; (∃x∈y)ϕ ↔_Def (∃x)(x∈y &ϕ).

Definition 3. We dub (∀∃)₀-formulaany conjunctionΦof the form

M

^

j=0

(∀yj1∈y_j1⁰ )· · ·(∀yjp_j ∈y⁰_jpj) (∃xj1∈x⁰_j1)· · ·(∃xjq_j ∈x⁰_jqj)ϕj

where, for each j, the formula ϕj is devoid of quantifiers and either pj > 0, qj>0 orpj =qj = 0 holds.

2.2 (∀∃)0 specifications

Definition 4. We dub (∀∃)0 specificationof anm-place relationshipRover sets a (∀∃)0 formulaΦ with free variablesa1, . . . , am, x1, . . . , xκ (where κ>0) such that, under the axioms of set theory (see below), one can prove:

R(a1, . . . , am) ↔ (∃x1, . . . , xκ)Φ . E.g., the right-hand sides of

a=b\c ↔ (∀t∈a)(t∈b&t /∈c)&(∀t∈b)(t∈c∨ t∈a), Sngl(a) ↔ (∃x) x∈a&(∀y∈a)(y=x)

are (∀∃)₀ specifications of the 3-place relationshipa=b\cand, respectively, of the property “being a singleton set”.

In the ongoing, our set-theoretic framework will be the theoryZF, axiom of regularity included. First, in order to design a (∀∃)₀specification of the property

“being a finite set” (see Sec. 2.4), we will extend temporarily the signature ofL_∈ with a constantωwhich is meant to designate the first limit ordinal; then we will figure out a (∀∃)0specification of the property “being the first limit ordinal” (see Sec. 2.5), thus ending in an impeccable (∀∃)0specification offinitude. Along the way, we will also specify the important equinumerositypredicate, which relates two sets when they have the same cardinality (see Sec. 2.4).

2.3 (∀∃)0 specifications referring to disjointness, unionset, and weak Cartesian product

x∩y=∅↔(∀v∈x) v /∈y ,

x⊆

∪

^{y↔(∀x0∈x)(∃y0 ∈y) x0 ∈y0,}

∪

^{f ⊆v↔(∀p∈f)(∀w∈p)(w∈v),}

Mapw(f)↔(∀p∈f)(∀x1, x2, x3∈p) x1=x2∨x2=x3∨x3=x1 ,

f ⊆x⊗y↔Map_w(f) &

(∀p∈f)(∃x⁰∈x)(∃y⁰∈y) x⁰∈p&y⁰ ∈p) &

(∀p∈f)(∀w∈p) w∈x∨ w∈y).

The explanation of the last of these is straightforward; here it is:

No member of f has more than two members: (∀p∈f) |p|62 , where

|p|62↔(∀x1, x₂, x₃∈p) x₁=x₂ ∨x₂=x₃ ∨x₃=x₁ . Each member of f has the form {x⁰, y⁰} with x⁰ ∈xand y⁰∈y:

(∀p∈f)(∃x⁰∈x)(∃y⁰∈y) x⁰∈p&y⁰∈p).

No member of a member of f lies outside x∪y:

∪

^{f ⊆x∪y .}

2.4 (∀∃)0 specifications of equinumerosity, squaring, and finitude

1-1_w(x, f, y)↔x∩y=∅^&f ⊆x⊗y &x⊆

∪

^{f &y⊆}

∪

^{f &}

(∀p , q∈f)(∀v∈p) v∈q → p=q .

Meaning:

1-1w(x, f, y) can only hold ifxandyaredisjoint sets, in which casef models a one-to-one mapping betweenxand y by means of unordered pairs, in the sense that:

f consists of doubletons proper;

each doubleton inf pairs up a member ofxwith one ofy;

f pairs up exactly one member ofywith each member ofx;

f pairs up exactly one member ofxwith each member ofy.

Stating that setsx, y are of the same cardinality amounts to the statement that there is a sety⁰ such that

y⁰can be put in one-to-one correspondence withx, y⁰can be put in one-to-one correspondence withy, y⁰andxare disjoint, andy⁰andyare disjoint.

Summing up, we have:

|x|=|y| ↔(∃y⁰, g , h) 1-1_w(x, g, y⁰) &1-1_w(y, h, y⁰) . Clue: One way of concretizingy⁰, here, is by puttingy⁰=y⊗ {y∪x}.

Likewise, stating that the cardinalities|x|,|y|are such that|x|=|y|²amounts to the statement that there are setsy⁰, x⁰ such that

y⁰can be put in one-to-one correspondence withy,

x⁰=y⊗y⁰ andx⁰ can be put in one-to-one correspondence withx, x⁰ andxare disjoint, andy⁰ andyare disjoint.

Summing up, we have:

|x|=|y|²↔(∃x⁰, y⁰, g , h) Mapw(x⁰) & 1-1w(x⁰, g, x) &

1-1w(y⁰, h, y) &

(∀p∈x⁰)(∃u∈y)(∃v∈y⁰)(u∈p & v∈p) &

(∀u∈y)(∀v∈y⁰)(∃p∈x⁰)(u∈p & v∈p) . Clue:The equalityx⁰=y⊗y⁰follows from the conjunction ofMapw(x⁰) with the last two conditions, thanks to the disjointness constrainty⁰∩y=∅hidden inside 1-1w(y⁰, h, y). A convenient way to instantiatey⁰is, again, to puty⁰=y⊗{y∪x}.

A finite set is one that can be put in one-to-one correspondence with a cardinal preceding (i.e., belonging to) the first infinite ordinal, ω:

Finite(x)↔(∃o) o∈ω & |x|=|o|

. More explicitly:

Finite(x)↔(∃o , x⁰, g , h) o∈ω &1-1w(x⁰, g, x) &1-1w(x⁰, h, o) .

2.5 (∀∃)0 specification of ordinals and of the first limit ordinal A sett is said to be transitive ift⊆P(t); equivalently, if

∪

^t⊆t. After John von Neumann and Raphael M. Robinson, ordinal numbers are those transitive sets within which any two different elements can be compared by membership:

Ord(o)↔(∀y∈o)(∀y⁰ ∈y)y⁰ ∈o &

(∀o1∈o)(∀o2∈o)(o1=o2∨ o1∈o2 ∨o2∈o1)

Those ordinal numbersosuch thato6=∅ando=

∪

^{oare called}limit ordinals.

Plainly, the property “being a limit ordinal” is (∀∃)₀-specifiable.

To characterize uniquely the first limit ordinal, ω, among all sets, it will suffice to conjoin together the following conditions, whereZ, a, andsare meant to represent, respectively:ω itself, an element ofω, and the successor function (modeled as a set of doubletons, each one implicitly ordered by membership):

Z is a non-null ordinal:

a∈Z & Ord(Z) (8)

No member of s has more than two members:

Map_w(s) (9)

Each member of sis a doubleton {x, y} with x∈y∈Z: (∀p∈s)(∃x, y∈p) x∈y &y∈Z

(10)

sis single-valued:

(∀p, q∈s)(∀x, y∈p)(∀y⁰∈q) (x∈y &x∈y⁰ &x∈q)→p=q (11) The domain of sincludes Z:

(∀x∈Z)(∃p∈s)(∃y∈p) x∈p&x∈y

(12) The multi-image of sincludes Z\ {∅}:

(∀y∈Z)(∀e∈y)(∃p∈s)(∃x∈p) y∈p&x∈y

(13)

Thus, one can prove (inZFwith regularity):

(∀Z) Z=ω↔(∃a)(∃s) (8)& · · · &(13) . In fact:

by (10) and (12), the domain ofsequalsZ; (12) and (13) yield thatZhas no largest element;

consequently, by (8),Zis an ordinal such thatZ∈/ω;

ifZwere such thatω∈Zthen, by (13), there should exist somex∈Zsuch that{x, ω} ∈s,

which is untenable: for, sincesrepresents an increasing function, the imme- diate successorx∪ {x}ofx would not be in the multi-image ofs. From

Z∈/ωandω /∈Z, we getZ=ω.

Concluding remarks and open problems

We have revisited, deepened, and made more systematic, the study undertaken long ago with [2] (see also [3, pp. 161–165]).

Can we do more along the directions envisioned in the following excerpt from a historical paper?

[· · ·]the translation of a theorem of the appropriate form in some part of math- ematics shows that the corresponding Diophantine equation has no solution.

Hence whatever methods went into proving the theorem can in fact be used to show that a particular Diophantine equation has no solution. It is possible that the same methods can be used to show that a class of equations includ- ing perhaps an equation of interest in itself are unsolvable. Such an example providing a new tool for solving Diophantine equations would be a consider- able breakthrough. In any case, any mathematical method that has been used to prove a theorem of the appropriate form has in fact been used to show that a particular Diophantine equation has no solution. Thus all mathematical meth- ods can be tools in the theory of Diophantine equations and perhaps we should consciously attempt to exploit them.” [5, pp. 338–339]

The quest is open. . . It may be rewarding to:

– translate back into number theory decidability results regarding fragments of set theory;

– mimic the proofs of the theorems, dubbed DPR and DPRM in [7], concern- ing the exponential Diophantine representability and the polynomial Dio- phantine representability of any r.e. set, directly inside set theory (possibly making some technical aspects of those proofs more transparent).

Open problems

LetBSTC⊗(Boolean Set Theory with Cardinality comparison and the unordered Cartesian product) be the collection of conjunctions of any number of positive literals of the forms

x=y∪z , x=y⊗z , x∩y=∅, |x|=|y| .

In sight of the still in progress decidability result forMLS⊗(multilevel syllo- gistic with the unordered Cartesian product), concerning a positive solution to the satisfiability problem for conjunctions of literals of the forms

x=y∪z , x=y⊗z , x∩y=∅, x∈y ,

to precisely locate the boundary between decidability and undecidability, one should attempt to find the decidability status of the satisfiability problem for the following collections of equalities:

– BSTC⊗-conjunctions plus one negative literal of the formx6=∅, – BSTC⊗-conjunctions plus twoliterals of any of the following forms:

|x| 6=|y| and x={y}. Acknowledgements

We are grateful to Martin Davis for his encouragement. We thank the anonymous referees for their careful comments.

References

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