Some Aspects and Examples of Innity Notions
J. W. Degen
July 4, 1994
Abstract
Our main contribution is a formal denition of what could be called a T -notion of
innity, for set theories T extending ZF . Around this denition we organize some old and
new notions of innity; we also indicate some easy independence proofs.
1
Introduction
The investigation of dierent denitions of innity constitutes a signicant part of the development of axiomatic set theory. Tarski [15], Mostowski [11], Levy [9], and many other authors
have devoted research papers to the theme of niteness denitions (which is the most often
used term for this subject). The niteness denitions one usually comes across are all mutually
equivalent if the full Axiom of Choice (AC ) is adopted | and this lies, in fact, in the very
nature of a niteness denition, as has been observed several times and will receive an explanation by an explicit theorem once the concept of a niteness denition has been formally xed;
see Theorem 6 below.
In this paper I shall propose a denition of what could be called a notion of innity, and with
this denition of "notion of innity" as a systematic point of reference I shall discuss several
already known notions of innity and also several new notions of innity.
2
What is a Notion of Innity?
Our set-theoretic notions and notations are the usual ones. Small greek letters range over
ordinals, < is the same as 2 on ordinals. ! is the rst innite ordinal. stands for the relation
of equipollence, i.e. two sets are equipollent i there is a bijection between them. a b means
that the set a can be injected into the set b, and a b means that a is empty or b can be
surjected onto a. p(a) is the powerset of a. The disjoint union of the sets a and b is denoted
by a + b, their cartesian product by a b and their set-theoretic dierence by a n b.
Denition 1 Let T be a set theory at least as strong as ZF. Then a T-notion of innity is a
formula '(x) with exactly the free variable x such that the theory T proves all of the following:
(1) 8 ! '()
(2) 8n < ! :'(n)
(3) 8x; y (x y ^ '(x): ! '(y))
(4) 8x; y (x y ^ '(x): ! '(y))
1
In the presence of clause 4 clause 1 reduces to '(!). But below we shall encounter cases
where 8 ! '() is provable whereas clause 4 is not provable.
The axiom of innity is usually stated as 9x '(x) where '(x) : ! 9y (y 2 x) ^ 8y(y 2
x ! y [ fyg 2 x). This formula '(x) however is not a T -notion of innity, for any extension T
of ZF , since it is neither invariant under equipollence nor invariant under supersets.
We next dene the two most commonly used notions of innity, viz. (simple) innity (inf)
and Dedekind-innity (D-inf), and also two related notions, viz. weak Dedekind-innity (wDinf) and dual Dedekind-innity (dD-inf).
Denition 2
inf(x) :() 8n < ! : n x
D ? inf(x) :() ! x
wD ? inf(x) :() ! x
dD ? inf(x) :() there exists a noninjective surjection from x onto x
ZF proves that a set x is Dedekind-innite i there is a nonsurjective injection from x into x.
This is Dedekind's original denition. We have perhaps given an unusual denition of (simple)
innity. But ZF proves that the following are equivalent: (1) inf(x) , (2) :9 n < ! : x n
and (3) 8 n < ! : n x. If : inf(x) then we say that x is nite, and if : D ? inf(x) we say
that x is Dedekind-nite, and similarily for other innity - niteness pairs. Def. 1 and Def. 2
yield immediately
Proposition 3 All four notions in Def. 2 are ZF -notions of innity, and therefore also T-
notions of innity for every supertheory T of ZF . In ZF Dedekind-innity implies weak
Dedekind-innity, and this implies simple innity.
The next theorem is also fairly easy but more interesting.
Theorem 4 Let T be any set theory at least as strong as ZF , and let '(x) be a T -notion of
innity. Then T proves 8x (D ? inf(x) ! '(x): ^ :'(x) ! inf(x))
Proof: Let x be a Dedekind-innite set. Then on the basis of ZF the set x can be represented
as a disjoint union x = y + z , where y !. Since '(x) is a T -notion of innity, we have by
the rst clause of Def. 1 that T ` '(!), hence by the third clause that T ` '(y). By clause
4 of Def. 1 it follows that we have in T that '(x). Next suppose '(x). We argue in T . Thus
by the third clause of Def. 1 : If n x for some n < ! then '(n). But this contradicts the
second clause of Def. 1. Thus we have in T , that if '(x) then for all n < ! it is not the case
that x n. From this we deduce that n x for all n < !, i.e. inf(x).
2
Since (by Prop. 3) dual Dedekind-innity is a ZF -notion of innity, Thm. 4 yields immediately that in ZF Dedekind-innity implies dual Dedekind-innity. |We want to remark
without proof that the converse of Thm. 4 does not hold:
Theorem 5 There is a (somewhat articial) formula '(x) such that ZF proves 8x(D ?
inf(x) ! '(x): ^ :'(x) ! inf(x)) but '(x) is not a ZF -notion of innity.
The next theorem shows that if we adopt ZFC , we have a certain collapse of innity notions.
2
Theorem 6 Let T ; T be any set theories such that ZF Ti ZFC; i = 1; 2. For i = 1; 2
let 'i (x) be a Ti -notion of innity. Then ZFC proves 8x(' (x) ! ' (x)).
Proof: By the assumption in the Thm. to be proved and by Thm. 4 we can prove in ZFC for
i = 1; 2 : 8x(D ? inf(x) ! 'i (x): ^ :'i (x) ! inf(x)). However, in ZFC we can also prove
8 x(inf(x) ! D ? inf(x)).
2
Remark: In order to prove the implication "innity =) Dedekind-innity", full AC is not
1
2
1
2
needed; the axiom of countable choices is sucient. That means that the interval between simple
innity and Dedekind-innity shrinks to one point in the presence of some proper consequences
of AC . It is of course possible to consider notions of innity that are not ZFC -notions of
innity, as the example '(x) : ! inf(x) ^ GCH . We shall discuss below ZFC - but not
ZF -notions of innity which are more natural than the example just given in that they concern
the inner structure of the sets falling under them.
3 Nonequivalent ZF-Notions of Innity
How many dierent, i.e. nonequivalent ZF -notions of innity are there ? First we give
Denition 7 A set x is called amorphous if x is is innite but cannot be partitioned into two
innite sets. We call a set x superamorphous (see e.g.[1]) if x is innite and if every relation
on x is denable by a rst-order formula that contains besides the logical signs only the equality
symbol = and parameters from x, i.e. individual constants that are interpreted as elements of
x.
Clearly, every superamorphous set is amorphous.
Theorem 8 The existence of superamorphous sets is consistent with ZF .
Proof: Start with ZFA, i.e. with ZF admitting a countably innite set of atoms At. Take
the full symmetric group on At whereby you get At as a superamorphous set in the resulting
permutation model. By the method of the Jech-Sochor transfer one gets a model of ZF with
a superamorphous set. One may consult the book [7] for details.
2
Theorem 9 There are innitely many nonequivalent ZF -notions of innity.
Proof: In order to prove this we dene an innite sequence (x); (x); : : : of ZF -notions of
1
2
innity such that
(1) If 1 m < n then ZF ` 8 x ( n (x) ! m (x)),
(2) The theory ZF + f9 x ( m (x) ^ : n (x)) : 1 m < n g is consistent.
(3) For each n 1 the theories ZF + 9 x (inf(x) ^ : n (x)) and ZF + 9 x ( n(x) ^ : D ?
inf(x)) are consistent.
For n 1 we dene n(x) :() x can be partitioned into n+1 (simply) innite sets . Then
each such n (x) is a ZF -notion of innity. Let y be an amorphous set; taking m + 1 disjoint
copies of y we form x : = y1 + : : : + ym+1 . Then we have, consistently with ZF : m (x) and,
if n > m; : n (x). Observe also, that a Dedekind-innite set can be split into innitely many
innite sets.
2
3
Thm. 9 is quite trivial, and the ZF -notions of innity we supplied in its proof are not very
interesting. However, the use of (super)amorphous sets is interesting and very important for
our whole subject. The next theorem being a simple application of superamorphous sets has
nevertheless many useful consequences in the theory of ZF -notions of innity.
Theorem 10 Let be any closed rst-order formula, and suppose that ZF proves that every
innite set is the underlying universe of a model of . Then has also nite models.
Proof: Let A be a superamorphous set in some model of ZF . Since A is innite it is, according
to our assumption, the underlying universe of a model of . This means that there are relations
R1; : : :; Rm and A-elements a1 ; : : :; an such that hA; R1; : : :; Rm; a1; : : :; ani j= . (We construe
functions as special relations.) Since A is superamorphous we can dene every relation Ri by
a formula containing only constants and the equality symbol (besides the logical symbols).
Now let be the formula that arises from if we replace each relation symbol by its dening
formula. For the interpretation of we may need some extra elements from A, say an+1; : : :; ak .
Then hA; a1; : : :; ak i j= . Now, every model of an equality-and-constants formula possesses
a nite submodel. Therefore we have a nite subset B of A such that hB; a1 ; : : :; ak i j= .
If S1 ; : : :; Sm are those relations on B which are dened by the same equality-and-constants
formulae contained as parts in as the respective relations R1; : : :; Rm on A, then we get
hB; S1 ; : : :; Sm ; a1 : : :; ani as a nite model of our original formula .
2
Remark: We want to point out some limitations of Thm. 10. First, if we take instead of
a single rst-order sentence , innite sets of rst-order sentences then the Thm. 10 becomes
false. A simple counterexample is the set fthere are at least n things : n 1g. Second,
Thm. 10 becomes false for third-order formulae : In ZF a set x is simply innite if and
only if there is a nonsurjective injection f : p(p(x)) ! p(p(x)): This last condition is
expressible by a third-order formula over x, which can be chosen even third-order monadic.
However, there is also a second-order sentence which makes Thm. 10 fail, viz. the sentence
? () 6 9 R (RandR?1 are well-orderings), where R?1 is the converse of R. This ? has no nite
models although ZF proves that all simply innite sets satisfy ?. [ Reason in ZF as follows: If
X has well-orderings R; R?1 then there is an ordinal such that hX; Ri is isomorphic to h; 2i.
But then the innity of would cause in h; 2?1i a contradiction to the axiom of regularity. ]
It is easy to check that the proof of Thm. 10 goes through also for monadic second-order
logic ( over any rst-order signature). Thus we pose the
Problem 1 For which extensions of rst-order logic does Thm. 10 hold?
Let us give an application of Thm. 10:
Denition 11 A set x is called Russell-innite i x carries an irreexive, transitive and unbounded relation r. (r is unbounded on x means 8 u 2 x 9 v 2 x : u r v )
Let denote the following rst-order formula of which we can recognize that it has no nite
models. : () 8 x:R(x; x) ^ 8x; y; z (R(x; y) ^ R(y; z ) ! R(x; z )) ^ 8x9yR(x; y) . Taking
the contrapositive of Thm. 10 we infer that ZF cannot prove that all (simply) innite sets are
the underlying universe of a model of the formula | in other words it is consistent with ZF
that there is an innite set which is not Russell-innite.
4
We show that, w.r.t. ZF , Russell-innity is properly weaker than Dedekind-innity. We
shall do this in a slightly indirect way as this will yield in addition some other useful results.
Theorem 12 ZF does not prove that 8x(dD ? inf(x) ! D ? inf(x)).
Proof: Truss proved in [16] that it is consistent with ZF that there is a full innite binary tree
which does not possess any innite branch. So let such a tree be given, and denote by x the
set of its points. If we assign the root to the root and to each other point its father the we get
in this way a noninjective surjection of x onto x . Hence the set x is dually Dedekind-innite.
Now we show that x cannot be Dedekind-innite. Let us suppose the contradictory; i.e. we
suppose that we can inject the set ! into x. To facilitate the argument, we may assume that
! x. Then we dene | in ZF | an innite path through : We let the path begin with
the root. If we have dened the path up to a certain height we must decide for just one of two
possible continuations. Let a be the point reached thus far, and let a1 and a2 be the sons of
a. We consider the subtrees spanned by ai; i = 1; 2. Either the span of a1 or the span of a2 or
both contain innitely many elements of !. If one of them has only nitely many elements of !
we go in the direction of the other span in order to extend our path. If both spanned trees have
innitely many elements from ! then we take that direction in which the minimal !-element is
strictly smaller than the minimal !-element of the other spanned tree.
2
Proposition 13 ZF proves that 8x(dD ? inf(x) ! wD ? inf(x)).
Proof: Let f be a noninjective surjection from x into x. Furthermore, let a be a point that
witnesses the noninjectivity of f . That means: 9x; y (x =
6 y ^ f (x) = f (y) = a). Now we
dene a function g from x onto ! as follows: g(x) = minfn 2 ! : f n (x) = ag if there is such
an n, and g(x) = 0 otherwise. Let us see that g is indeed a surjection onto !. There are two
relevant cases. In the rst case we have for all elements n 2 ! n f0g : f n (a) 6= a. Then we get
the surjectivity of g immediately from the surjectivity of f . In the other case we have f n (a) = a
for some n 1. Choose n minimal. Thus a lies on a cycle of length n. Since a witnesses the
noninjectivity of f there must be an element u not on that cycle such that f (u) = a. Again the
surjectivity of g follows from that of f .
2
Proposition 14 ZF proves that 8 x(wD ? inf (x) ! Russell ? inf (x)).
Proof: If f is a function of x onto ! dene for a; b 2 x the relation ab : $ f (a) < f (b). 2
Summa summarum we have proved the promised result that it is consistent with ZF that
there is a Russell-innite set that is not Dedekind-innite. | We have also proved:
Corollary 15 It is consistent with ZF that there is a weakly Dedekind-innite but Dedekindnite set.
Another Proof of Corollary 15: In the basic Cohen model (see [7], p.81) there exists an
innite Dedekind-nite set of real numbers. On the other hand, one can show already in ZF
that every innite set of real numbers can be surjected onto !. A short proof of this runs as
follows: Identify the set of real numbers with the Cantor space 2! of innite 0; 1-sequences.
Let A 2! be given. We call an a 2 2! an accumulation point of A if there are arbitrarily
large n's such that there are x 2 A with a(n) 6= x(n) ^ 8 m < n : a(m) = x(m). Now, Konig's
5
Lemma for binary trees with nite 0; 1-sequences as nodes is provable in ZF and yields the
theorem that every innite set A 2! possesses an accumulation point a, which may or may
not belong to A. Given an accumulation point a of A dene s : A ! ! by the condition
s(x) = minfn : a(n) 6= x(n)g for all x 6= a and as s(a) = 0, if a 2 A. The range of s is an
innite subset of !, i.e. a set which stands in bijective relation with ! itself.
2
Conjecture 2 We conjecture that there is a weakly Dedekind-nite set that is not dually
Dedekind-innite.
The following are some scattered results about Russell-innity. It can be shown (by the
proof of Thm. 10) that no superamorphous set is Russell-innite. M. Goldstern observed a
stronger fact, which we state without proof:
Proposition 16 ZF proves: Let x be Russell-innite and let n > 0 be a natural number. Then
x can be partitioned into n innite sets. A fortiori there is no set which is both amorphous and
Russell-innite.
It is not dicult to prove:
Proposition 17 ZF proves that every linearly ordered innite set is Russell-innite.
Proposition 18 (1) It is consistent with ZF that there is a Russell-innite set that cannot be
linearly ordered. (2) ZF proves that the power set of every innite set is Russell-innite.
Proof of Prop. 18:
Ad(1): In one of the Cohen models (see [7]) p(R), i.e. the power set of the set of the reals,
cannot be linearly ordered. We want to show that p(R) is Russell-innite.| Let A be the set
of nite subsets of R and B the set of innite subsets of R . On A dene x y as 9 z (z 2
A ^ : z x ^ y = x [ z ). And on B dene x y as 9 z (z 2 A ^ : z y ^ x = y [ z ).
Finally, if x 2 A and y 2 B then set x y. Then p(R)is Russell-innite via .
Ad(2): The argument in Ad(1) works for every innite set instead of R
2
Since it is consistent with ZF that there is an innite set x such that the power set of x is
Dedekind-nite (see [7], p. 95), Prop. 18 yields a second proof of the fact that the existence of
a Russell-innite but Dedekind-nite set is consistent with ZF .
As another application of Thm. 10 one may also prove the perhaps oldest and most famous
independence result in the realm of niteness denitions, viz. that it is consistent with ZF that
there is an innite Dedekind-nite set.
One may consider innity notions which are dened by reference to innite sets of rstorder sentences; in the next denition we will restrict ourselves to sentences with just a unary
function symbol as nonlogical symbol.
Denition 19 We abbreviate the sentence 8x; y(fx = fy ! x = y) by injec(f ) and the
sentence 8x9y fy = x by surjec(f ). Further, let [9nonper] denote the set f9x fx 6= x, 9x ffx 6=
x, 9x fffx =
6 x; : : :g of sentences, and let [8nonper] denote the set f8x fx =6 x, 8x ffx =6 x,
8x fffx =6 x; : : :g of sentences. Then we can dene the following ve sets of sentences:
(1) ? consists of the sentences : injec(f ), and those in [9nonper],
(2) ? consists of the sentences : injec(f ), surjec(f ), and those in [9nonper],
1
2
6
(3) ?3 consists of the sentences in [8nonper],
(4) ?4 consists of the sentences : injec(f ), and those in [8nonper],
(5) ?5 consists of the sentences: injec(f ), surjec(f ), and those in [8nonper]
Let s be a set. If s is the underlying universe of a model of the set ?i then we call s
period-innite of level (i) for i = 1; 2; 3; 4; 5, or [i]-per-inf(s).
None of the ?1 ; : : :; ?5 has a nite model. But each has hZ; i as a model where Z are the
integers, and is the successor function on them. Every nite subset of each ?i has also a nite
model.
Theorem 20 In ZF we can prove the following implications:
[5]-per-inf =) [4]-per-inf=) [3]-per-inf =) D-inf =) [2]-per-inf =) [1]-per-inf =) wD-inf
Proof: We show only the last implication. Assume [1]-per-inf(s) and let f : s ! s be injective
and satisfy the set [9nonper]. We have to dene a surjection g : s ! ! .
Case I. There is an element a 2 s such that for all positive numbers n : f n (a) =
6 a. Then g
n
sends f (a) 7! n (therefore a 7! 0) and b 7! 0 for every b 2 s such that there is no n 2 ! with
f n (a) = b.
Case II. For all a 2 s there is a positive number n such that f n (a) = a. We dene the
order ord(a) of a as the minimal such positive number. By recursion we dene : ! ! ! n 1:
(0) := minford(a) : a 2 sg and (n + 1) := minford(a) : a 2 s ^ ord(a) > (n)g. Then for
each n 2 ! the nth slice Sn is the set fx 2 s : (n) = ord(x)g. Finally set g(x) = n ! x 2 Sn .
This function g surjects our set s onto !.
2
It is also easily shown that [2]-per-innity and therefore also [1]-per-innity are ZF -notions
of innity. In the next section we show that [3]-per-innity and therefore [4]-per-innity and
[5]-per-innity are not ZF -notions of innity.
4 Innity Notions that are not ZF-Notions of Innity
Theorem 21 It is consistent with ZF that there exists a Dedekind-innite set s that does not
satisfy [3]-per-inf(s).
Proof: Let a be an amorphous set such that a \ ! = ;. If we set s := a + ! then trivially a
D ? inf (s). Suppose that [3]-per-inf(s). Then we have a function f : s ! s such that hs; f i
satises the set [8nonper]. We distinguish two cases:
Case I. There exists an element r 2 a such that f n (r) 2 a for all n 2 !. Then our amorphous
set a would be Dedekind-innite, which is impossible.
Case II. For every r 2 a there is a positive number n such that f n (r) 2 !. Dene g : a !
! as r 7! f n (r) where n = minfm : f m (r) 2 !g. This g is injective, hence we could via g?1 on
the range of g give an enumeration of our amorphous set, which is again a contradiction. 2
Thm. 21 together with Thm. 4 implies that [3]-per-innity is not a ZF -notion of innity.
We have however immediately:
Proposition 22 [5]-per-innity and therefore also [4]-per-innity and [3]-per-innity are ZF +
[split ? !]-notion of innity, where [split ? !] is the statement that for every simply innite set
y we have y y !.
7
It is known that [split ? !] does not imply AC . For it is shown in [6] that (in ZF ) [split ? !]
is equivalent to 8y (inf (y) ! y 2 y); and the latter statement has been shown by Sageev
in [12] not to imply AC .
Call a set of rst-order sentences with equality and just one unary function symbol f a
monounarian theory. E.g. the ?i 's in Def. 19 are monounarian theories. If is a consistent
monounarian theory having no nite models, then we dene a set s to be -innite i the set s
is the underlying universe of a model for . We call such innity notions monounarian innity
notions. The following theorem implies, together with the result of Sageev just quoted, that
there is no monounarian innity notion '(x) such that ZF + 8 x (inf (x) ?! '(x)) implies AC .
Theorem 23 ZF proves: Let be a consistent monounarian theory without nite models. If
x is equipollent to y ! for some nonempty set y, then there exist a function f : x ?! x such
that hx; f i is a model of .
We omit the (long) proof. See however the paper [10] for some pertinent techniques.
On the other hand there is a single rst-order sentence over a binary function symbol such
that we have for the innity notion dened w.r.t. this sentence that every set theory T with
the property that this notion is a T -notion of innity must prove AC . The sentence in question
is 9x; y x 6= y ^ 8x; y; u; v(g(x; y) = g(u; v) ! x = u ^ y = v) ^ 8z 9x; y g(x; y) = z . A
set s is the underlying universe of a model of this sentence i s has more than one element
and s s s, which we abbreviate to '(s); and we express this by saying that the set s is
pairing-innite. Let T be an extension of ZF such that '(s) is a T -notion of innity. Thm. 4
implies that T proves 8s (D ? inf (s) ?! '(s)).
Claim: In T we can prove that all simply innite sets are Dedekind-innite.
Then by the Claim we deduce that T proves: for all simply innite sets s we have: s s s.
Recall that the last statement is equivalent to AC .
Proof of the Claim: Let s be a simply innite set, assumed to be disjoint from !. Form
the Dedekind-innite set t := s + !. Then by a result just proven we have in T that '(t), i.e.
that t t t. We have s s + s ! + ! s + !. A fortiori we have s s s + ! via an
injection f . Consider the image f [s s]. If innitely many elements of f [s s] lie in ! then
we have a countably innite subset of s s, in other words, ! can be injected into s s and
therefore into s and we are done. Thus we may assume that only nitely many elements of
f [s s] lie in !. Let p1; : : :; pn be the ordered pairs which these !-elements stem from. Then
there are at least two (in fact innitely many ) elements of s, say a and b, which do not occur
as the right coordinate of any of the pi's. The two disjoint sets s fag and s fbg must be
injected into s via f . But from this we infer that there is also a nonsurjective injection from s
into s. Thus, in all events, our simply innite set s is even Dedekind-innite.
2
The innity notion '(x) just considered, i.e. pairing-innity, forces every set theory T which
proves that '(x) is a T -notion of innity also to prove 8x (inf (x) ?! '(x)). This is a contrast
to Dedekind-innity which is a ZF -notion of innity without ZF being able to prove that all
innite sets are Dedekind-innite.
Denition 24 We call a set x splittable i x is nonempty and there are two disjoint subsets
u and v of x such that u x and v x.
ZF proves the rst three clauses of Def. 1
8
Theorem 25 It is consistent with ZF that there is a Dedekind-innite set that is not splittable.
Proof: The heart of the proof is again the use of an amorphous set. So let x be an amorphous
set such that x \ ! = ;. Then we may dene: y := x + !. Then show that the Dedekind-innite
set y is not splittable. Suppose y was splittable. Then there are disjoint subsets y1 and y2 of
y such that y1 y and y2 y. It follows that yi ; i = 1; 2 contains innitely many elements of
x. For suppose that yi contained only nitely many x1 ; : : : ; xn 2 x then it would follow that
yi ! and hence we would have ! ! [ x, and x would be Dedekind-innite. However, it is
easily seen that an amorphous set cannot be Dedekind-innite. If we dene zi := x \ yi; i = 1; 2
then (z1 ; z2) is a partition of x into two innite sets. And this is a contradiction to the fact
that x is amorphous.
2
It follows that splittability is not a ZF -notion of innity, since Thm. 25 together with Thm.
4 imply that we cannot prove in ZF that splittability fullls clause 4 of Def. 1, i.e. the invariance
under supersets.
Proposition 26 Let T be an extension of ZF such that splittability is a T -notion of innity.
Then T proves that all simply innite sets are splittable.
Proof: The argument is like that for pairing-innity and nally boils down to showing (in
ZF ) that an arbitrary set a is Dedekind-innite, if a is innite, disjoint from ! and such that
a f0; 1g can be injected into a + !.
2
We now turn to a notion of innity that occurs | at least implicitly | in many philosophical
and therefore popular discussions about the foundations of mathematics and exact knowledge
in general. The underlying idea is the idea of inexhaustibility, of which we like to give the
following formal
Denition 27 We call a set x inexhaustible i x has at least two elements and we have for
all subsets y of x : if y x then x (x n y). Here a b means a b ^ :b a.
Theorem 28 It is consistent with ZF that there is a Dedekind-innite set that is not inexhaustible.
Proof: Like the proof of Thm. 25, using an amorphous set.
On the other hand, since every aleph is inexhaustible and under AC every innite set is
equipollent to an aleph, inexhaustibility is a ZFC -notion of innity.
Problem 3 Let T be an extension of ZF such that inexhaustibility is a T -notion of innity.
Does this imply that T proves AC ?
We wish to append a list of some further ZFC - but not ZF -notions of innity; each of them
implies Dedekind-innity.
Denition 29
(1) The set x is division-innite i x has at least three elements and 8y x(y 6= ; ?!
9u x y u)
(2) The set x is even-odd-innite if there is an element a 2 x such that both x and x n fag
can be represented as the disjoint union of two-element sets.
(3) Let n 2 ! n f0; 1g; then the set x is n-selnjective i x has at least two elements and
8f : x ! n 9i 2 n : x f ?1 [i].
9
One can prove in ZF that if n; m 2 then: x is n-selnjective i x is m-selnjective i x
is inexhaustible.
5 More ZF-Notions of Innity
We are now going to give a somewhat rhapsodic list of ZF -notions of innity.
Denition 30
(1) The set x is combinator-innite i x has at least two elements and there exists an
f : x2 ! x such that 9a 8b; c : f (f (a; b); c) = b
(2) The set x is tree-innite i there is an f : x ! x such that the sentence 9u8y9z (f (z ) =
y ^ : u = z ) holds in hx; f i.
(3) The set x is inversion-innite i there is an f : x ! x such that there is no permutation
g of x such that f g f = f where denotes functional composition.
(4) The set x is strongly Tarski-innite i there is a -chain (yi )i2! in the powerset of x
without a -maximal element. (Note: this is only one of the notions of innity considered by
Tarski, see [15]).
(5) The set x is almost Dedekind-innite i p(x) is Dedekind-innite.
(6) The set x is star-innite i there is an f : p<! (x) ! p<! (x) (where p<! (x) is the set
of nite subsets of x) such that for all y 2 p<! (x) : f (y) 6 y.
(7) The set x is strongly star-innite i there is an f : p<! (x) ! x such that for all
y 2 p<! (x) : f (y) 62 y.
In order to show, that a proposed denition is indeed a ZF -notion of innity we have, of course,
to show all four clauses of Def. 1 in ZF . In general, it is the fourth clause that requires some
signicant eort. So, to take just one example, it is not obvious that inversion-innity satises
ZF -provably the invariance under supersets.
Proposition 31 ZF proves: If x is inversion-innite and if x y, then y is also inversioninnite.
Proof: It is easier to prove the contrapositive. So let the set y be inversion-nite. If in addition
we have x y, then x must be inversion-nite. Let f : x ! x be an arbitrary function. Dene
f : y ! y as f on x and as identity on y n x. The inversion-niteness of y implies that
there exists a permutation g : y ! y such that f g f = f . By the choice of f the
permutation g must be the identity map on y n x. If we let g be the restriction of g to the
set x then it follows f g f = f ; hence x is inversion-nite.
2
Admittedly, inversion-innity is somewhat articial; it is implied by the perhaps less articial
tree-innity. Tree-innity in turn is, w.r.t. ZF , equivalent to dual Dedekind-innity. A set is
strongly Tarski-innite i it is almost Dedekind-innite i it is weakly Dedekind-innite. See
[8] for proofs of these equivalences. It is also easy to show that if a set is star-innite then
it is weakly Dedekind-innite, and strong star-innity implies star-innity but not the other
way around, as will be shown presently. Strong star-innity is equivalent to Dedekind-innity.
Finally, combinator-innity is equivalent to Dedekind-innity.
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Proposition 32 It is consistent with ZF that there exists a star-nite set which is not strongly
star-innite.
Proof: We already remarked that strong star-innity is equivalent with Dedekind-innity.
Now, the set x used in the proof of Thm. 12 is not Dedekind-innite, as we showed there. On
the other hand, we have a function g : x ! x (constructed from the binary tree on x), such
that the xed point of g has itself and its two proper sons as pre-images, and every other point
has exactly two pre-images. So let us dene a function f : p<! (x) ! p<! (x) as follows : f (;) =
the singleton of the xed point of g, otherwise we set f (y) = g?1[y]. It follows that for all
y 2 p<! : f (y) 6 y
2
Theorem 33 ZF proves 8 x (x is Dedekind-innite ! x is combinator-innite ).
Proof: We write the function f occurring in the denition of combinator-innity simply as an
inxed point and introduce an extra individual constant a. Then a set A will be combinatorinnite if and only if one can have on A a nontrivial model of the equation (a x) y = x. Recall
that this equation is one of two equations of combinatorial logic, hence our nomenclature. |
The function a (x) = a x is obviously injective and we show now that it is also nonsurjective if
the domain A has more than one element. If a was surjective then we would have an element
b 2 A such that a (b) = ab = a: But then we would have for all x 2 A: a (x) = ax = (ab)x = b
and the function a would not be injective if A has more than one element. Thus we have shown
that combinator-innity implies Dedekind-innity.
To prove the other direction we have to show that we can dene on a given Dedekind-innite
set A a model for the equation (a x) y = x. Without loss of generality we may assume that
A = ! + U . Then we dene: a := 0 ; 0 n := n + 1 and (n + 1) z := n for n 2 !; z 2 A and
nally 0 u := u; u z := u for u 2 U; z 2 A.
2
There is another curious innity notion that is also implied by tree-innity, viz. trivolutioninnity.
Denition 34 Call a function f such that f f f = f a trivolution. A set x is called
trivolution-innite i there is a function f : x ! x such that there are no functions g; h : x ! x
with f = g h; g g g = g; h h h = h.
Concerning trivolution-innity only the third clause of Def. 1, viz. invariance under equipollence is obvious. Clause 2 rests upon the following
Theorem 35 Every function on a nite set is the composition of two trivolutions.
Proof: Following Sperber's proof [13] we are showing something even stronger, viz. that if x is
a nite set and if f is a function from x into x then there are functions g; h : x ! x such that
g g = id; h h h = h and f = g h (rst h then g). | So let the function f be given. The
function f , as a graph, divides up into components, i.e. maximally connected parts. Each such
component possesses just one cycle (xed points, as cycles of length 1, included), onto which
several (possibly none) tree-like connected parts of the function f are planted. Call these parts
simply trees, or f -trees. For the proof it is sucient to show how one can construct the required
functions g and h on a single component, say, C.
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Let the cycle of C have the length n, and let c be an arbitrary but xed element on that
cycle. Then we have f n (c) = c ^ 8i : (0 < i < n ! :f i (c) = c). Using the chosen element
c we dene the functions g and h on the cycle of our component C as follows:
g : f i (c) 7! f n?i (c) and h : f i (c) 7! f n?1?i (c) 8 i < n.
Thus we have on the cycle of C already satised the equations g g = h h = id. It
remains to treat the f -trees of our component C. We can treat these trees separately; so x
one of them, say . We choose a leaf d of and dene the functions g and h on the path
to d from the root of by which is planted onto the cycle of C. For the root of , i.e. for
f n (d) the functions g and h are already dened as the root is an element of the cycle. For the
remaining points of the path from the root to the leaf d we dene the functions g and h as
follows: g : f i (d) 7! f n?i?1 (d) ; h : f i (d) 7! f n?i (d) 8 i < n . In this way we proceed with
all paths of the tree . If a new leaf lands per iteration of f in a point p of an already treated
path such that p does not lie on the cycle then we take this point p as the root of the new path.
On the trees of our component the equations g g = id and h h h = h are valid.
2
Remark: Thm. 35 is an adaptation to nite sets and general functions of the fact that each
permutation f on any set (nite or innite) can be written as a product of two reections
a; b, f = a b, where a reection is a permutation of order 2. This is not provable without
some use of AC , see [2].
On the other hand, we deduce clause 1 of Def. 1 as a corollary to the following
Proposition 36 ZF proves that dual Dedekind-innity implies trivolution-innity.
Proof: Let a set x and a surjective function f : x ! x be given. Suppose we had g; h : x ! x
such that f = g h; g g g = g; h h h = h. The surjectivity of f implies the surjectivity of
h, hence by cancelling: h h = id . This implies that h is a bijection. Since f was surjective
and f = g h we have that g is surjective. Then cancelling yields g g = id. This implies that
g is a bijection. It follows that f is a bijection and hence injective. Therefore, every dually
Dedekind-innite set is also trivolution-innite.
2
Problem 4 Can one prove in ZF that trivolution-innity is invariant under supersets?
If the answer is YES then trivolution-innity is in indeed a ZF -notion of innity. If the answer
is NO then we would have a natural example for Thm. 5 | since ZF proves that trivolutioninnity lies (properly) between Dedekind-innity and simple innity.
Let us separate trivolution-innity from simple innity.
Denition 37 A set is called hyperamorphous i it is amorphous and cannot be partitioned
into simply innitely many sets with at least two elements.
By Thm. 8 the existence of a hyperamorphous set is consistent with ZF .
Theorem 38 It is consistent with ZF that there is an innite set that is trivolution-nite;
moreover, ZF proves that every hyperamorphous set is trivolution-nite.
Proof: Let x be hyperamorphous and f : x ! x arbitrary. At most one component of f can be
innite, and there are only nitely many nite components with more than one point. These
nite components are factored into g; h as in the proof of Thm. 35 | the treatment of the
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possibly innitely many xed points is trivial. Let us now consider the possibly present innite
component, say C . Since x is hyperamorphous the cycle of C must be nite, and the heights
of the trees of C must also be nite. The innity of C can have only one reason, viz. that
there is just one point p such that the pre-image f ?1 [p] is innite. The hyperamorphousness
of x implies however, that only nitely many elements from f ?1 [p] have pre-images in turn.
Such an innite component can be subjected to the same procedure as the nite component
in the proof of Thm. 35. Nothing beyond ZF is used for the construction of factors g; h with
g g = id ; h h h = h for f , given the hyperamorphousness of x.
2
6 Higher Dedekind sets
In Def. 2 of innity and Dedekind-innity the cardinal ! played a role that we now generalize
to every innite cardinal in the following
Denition 39 A set x is called -innite i 8 < : x; and -Dedekind-innite i
x. A set x is called a -Dedekind set if x is -innite but not -Dedekind-innite.
Then an @ -Dedekind set is a simply innite set which is not Dedekind-innite. Consider the
statement
0
(BigDed) 8 9x : x is a -Dedekind set
First, if is a successor cardinal, = + , then is trivially a -Dedekind set. Thus the
statement (BigDed) is equivalent to the statement that there are -Dedekind sets for every
limit cardinal (@0 included). Note that ZFC proves that there are no -Dedekind sets for all
limit cardinals . However we have the
Theorem 40 The statement (BigDed) is consistent with ZF
Proof ( Due to M. Goldstern and M. Gitik ) : M. Goldstern showed in ZF that the existence
of an @ -Dedekind set implies the existence of a -Dedekind set for all limit cardinals with
conality !. His proof runs as follows: If d is an !-Dedekind set we denote by dk all nite
sequence with terms from d without repetition. Then the innity of d implies that dk can be
surjected onto !, say via s. However, ! cannot be injected into dk. Let be a limit cardinal
0
of conality !. Then we have a strictly monotone sequence (i )i2! of cardinals such that
supi2! i = . Then dene:
D := f(x; ) : x 2 dk ^ 2 s(x)g. Show that D is a -Dedekind set.
On the other hand, M. Gitik in his letter [4] told me that although his model in [3] in
which every cardinal has conality ! does not contain an @0 -Dedekind set one can force such
an @0 -Dedekind set into it in such a way that all conalities are preserved.
2
Although in [3] Gitik uses a large cardinal assumption, one can probably obtain the consistency of (BigDed) without the assumption of large cardinals in the ground model. | We could
not dispose of the following
Problem 5 Call a set x a strong -Dedekind set if x is -innite but x cannot be surjected
onto . Then let (strongBigDed) be the statement that for every innite cardinal there exists
a strong -Dedekind set. How can one prove that (strongBigDed) is consistent with ZF ?
13
The ZF -notions obviously form a lattice w.r.t. conjunction and disjunction. However,
before we can begin the closer study of the structure of this lattice, we must solve the Main
Open Problem, which reads:
Problem 6 Are there incomparable ZF -notions of innity? That means: Are there two
ZF -notions of innity ' (x); ' (x) and a model M of ZF such that there is a set u 2 M with
M j= ' [u] ^ :' [u] and a set w 2 M with M j= ' [w] ^ :' [w]
Acknowledgement: I wish to thank Klaus Kuhnle who streamlined in his [8] several of
1
1
2
2
2
1
my denitions and proofs concerning the subject matter of this paper. Some ideas and results
arose from discussions with Klaus Leeb. Jan Johannsen discovered some mistakes in an earlier
version.
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[4] M. Gitik letter to the author, December 1992
[5] M. Goldstern private electronic communication, September 1992
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