THE HARTOGS–LINDENBAUM SPECTRUM OF SYMMETRIC EXTENSIONS

. We expand the classic result that AC WO is equivalent to the statement “For all X , ℵ ( X ) = ℵ ∗ ( X )” by proving the equivalence of many more related statements. Then, we introduce the Hartogs–Lindenbaum spectrum of a model of ZF , and inspect the structure of these spectra in models that are obtained by a symmetric extension of a model of ZFC . We prove that all such spectra fall into a very rigid pattern.


Introduction
Perhaps one of the most powerful consequences of the axiom of choice is a straightforward classification of size for all sets: Given any set X, there is a minimum ordinal α such that X has the same cardinality as α.Indeed, grappling with how to compare sizes of infinite objects (or appropriate abstractions of this notion) is an inalienable aspect of modern mathematical foundations.Even without the axiom of choice, it is still understood that if Y is a superset of X, then Y is still 'at least as big as X' in some way.Taking this further, we are able to compare the cardinalities of sets through functions that map between them.If there is a surjection from X onto ω, then we can still describe X as being 'at least as big as ω'.Even when we lose the straightforward description of cardinality classes that is obtained from the axiom of choice, we can still consider comparisons between sets and ordinals in this way.Definition 1.1.Let X be a set.The Hartogs number of X is ℵ(X) . .= min{α ∈ Ord | There is no injection f : α → X}.
If X is well-orderable, then ℵ(X) = ℵ * (X) = |X| + , and so in the case that AC holds, these descriptions of sets tell us no more than cardinality already did.However, if the axiom of choice does not hold and X is not well-orderable then such descriptions still provide insight into the cardinality of X. Indeed the existence of ℵ(X) and ℵ * (X) is a theorem of ZF, in the former case by Hartogs's lemma and in the latter To take Theorem 1.2 further, given that AC WO is not a consequence of ZF, it is quite possible to build models of ZF in which we have sets X such that ℵ(X) = ℵ * (X).Let us produce a classification tool for such objects.Definition 1.3 (Hartogs-Lindenbaum Spectrum).Given a model M of ZF, the Hartogs-Lindenbaum spectrum (or simply spectrum) of M , denoted by Spec ℵ (M ), is the class Spec ℵ (M ) . .= { λ, κ | (∃X)ℵ(X) = λ, ℵ * (X) = κ}.This paper explores the possible spectra of models of ZF that arise as symmetric extensions of models of AC.This behaviour is captured internally to a model by small violations of choice, or SVC.This axiom, introduced in [1], is the statement "There is a set S such that for all X there is an ordinal η and a surjection f : S × η → X", but is also surprisingly a precise description of being a symmetric extension of a model of AC, by [11].Indeed, SVC is equivalent to several statements relating to symmetric extensions and AC, and is introduced more thoroughly in Section 2. 1.4.In any model of SVC, the Hartogs-Lindenbaum spectrum is broken down into four parts. 1 Hartogs's lemma is proved in [2].Lindenbaum's theorem is stated in [8,Théorème 82.A 6 ] with the first published proof in [10]. 2 For symmetry, if nothing else.
Main Theorem.Let M SVC.Then there are cardinals φ ψ χ 0 Ω, a cardinal ψ * ψ, a cardinal χ ∈ [χ 0 , χ + 0 ], and a set C ⊆ [φ, χ 0 ) such that That is, there is a necessary core to the spectrum { λ + , λ + | λ ∈ Card} that every model of ZF contains, since ℵ(λ) = ℵ * (λ) = λ + for all cardinals λ; there is a bounded chaotic part of the spectrum containing those λ, κ that have no restrictions other than ψ λ κ χ, and ψ * κ; there is a bounded but potentially irregular part of the spectrum in which the only values are λ, λ + for those λ such that cf(λ) ∈ C, with λ < Ω; and there is an unbounded, controlled tail of the spectrum containing precisely λ, λ + for all λ such that cf(λ) ∈ C.
1.1.Structure of the paper.Section 2 establishes preliminaries for the paper.No knowledge of forcing will be needed for the results in Section 3 or Section 4.2; the results can be understood entirely using the framework of transitive nested models of ZF.However, Section 4.1 uses the forcing framework, so we briefly introduce our standard treatment of forcing in Section 2.2.In particular, we introduce all necessary concepts for our use of SVC.
In Section 3 we investigate the equivalence between AC WO , Spec ℵ = SC, and several other equivalent conditions using core machinery that allows an 'upwards transfer' of eccentricity.In Section 4 we investigate the spectrum of models of SVC, providing bounds for the behaviour of such models and some instances of sharpness on those bounds.

Preliminaries
Throughout this paper we work in ZF.Given a set X, we denote by |X| its cardinal number.If X can be well-ordered, then |X| is simply the least ordinal α such that a bijection between α and X exists.Otherwise, we use the Scott cardinal of X, the set {Y ∈ V α | ∃f : X → Y a bijection} with α taken minimal such that the set is non-empty.Greek letters, when used as cardinals, always refer to well-ordered cardinals.We call an ordinal α a cardinal if |α| = α, and we shall denote by Card the class of all well-ordered cardinals.
We write |X| |Y | to mean that there is an injection from Y to X, and |X| * |Y | to mean that there is a surjection from Y to X or that X is empty.2.1.Choice-like axioms.Throughout, we shall use several axioms that can be considered to be partial fulfilments of the full strength of AC.Recall that if X is a set of non-empty sets, a choice function for X is a function c : X → X such that for all x ∈ X, f (x) ∈ x.Proposition 2.1.W λ is equivalent to the statement "for all X, either X is wellorderable or ℵ(X) λ + ".Likewise, W * λ is equivalent to the statement "for all X, either X is well-orderable or ℵ * (X) λ + ".Definition 2.2.Since (∀λ)DC λ , (∀λ)W λ , and (∀λ)W * λ are all equivalent to AC, whenever M is a model of ZF + ¬AC we shall denote by λ DC (respectively λ W , λ * W ) the least cardinal λ such that DC λ (respectively W λ , W * λ ) does not hold.2.1.4.Small violations of choice.In [1], the author introduces a choice-like axiom called small violations of choice, also written SVC.At its inception, it was defined by setting SVC(S) to be "for all X there is an ordinal η and a surjection f : η×S → X", where S is a set (known as the seed).We then use SVC to mean (∃S)SVC(S).However, this is equivalent to several other statements.

Fact 2.3 ([1]
, [11]).The following are equivalent: (1) M SVC; (2) M "There is a set A such that for all X there is an ordinal η and an injection f : X → A × η"; (3) there is an inner model V ⊆ M such that V ZFC and there is a symmetric system P, G , F ∈ V such that M = HS G F for some V -generic G ⊆ P; (4) there is an inner model V ⊆ M such that V ZFC and there is x ∈ M such that M = V (x); and (5) there is a notion of forcing P ∈ M such that 1 P AC.Definition 2.4 (Injective Seed).We shall say that a set A is an injective seed for M ZF if it satisfies Condition (2).That is, for all X ∈ M there is an ordinal η and an injection f : Proof.Suppose that A is an injective seed for M .Then M SVC(A).
On the other hand, suppose that S is a seed for M .We claim that P(S) is an injective seed for M .Indeed, suppose that X ∈ M , and let f : η × S → X be a surjection.For x ∈ X, let Then whenever g(x) = g(y), we have α x = α y , and thus for all s such that f (α x , s) = x, we have f (α y , s) = y.However, f (α x , s) = f (α y , s), so x = y as required.
2.2.Forcing.By a notion of forcing we mean a preordered set P with maximum element denoted 1 P , or with the subscript omitted when clear from context.We write q p to mean that q extends p. Two conditions p, p ′ are said to be compatible, written p p ′ , if they have a common extension.We follow Goldstern's alphabet convention so p is never a stronger condition than q, etc.
When given a collection of P-names, { ẋi | i ∈ I}, we will denote by { ẋi | i ∈ I} • the canonical name this class generates: { 1, ẋi | i ∈ I}.The notation extends naturally to ordered pairs and functions with domains in the ground model.Given a set x, the check name for x is defined inductively as x = {y | y ∈ x} • .

Symmetric extensions. It is key to the role of forcing that if V
ZFC, and G is V -generic for some notion of forcing P ∈ V , then V [G] ZFC.However, this demands additional techniques for trying to establish results that are inconsistent with AC.Symmetric extensions extend the technique of forcing in this very way by constructing an intermediate model between Given a notion of forcing P, we shall denote by Aut(P) the collection of automorphisms of P. Let P be a notion of forcing and π ∈ Aut(P).Then π extends naturally to act on P-names by recursion: Such automorphisms extend to the forcing relation in the following way, proved in [4, Lemma 14.37].
Note in particular that for all π ∈ Aut(P) we have π1 = 1.Therefore, πx = x for all ground model sets x, and Given a group G , a filter of subgroups of G is a set F of subgroups of G that is closed under supergroups and finite intersections.We say that F is normal if whenever H ∈ F and π ∈ G , then πHπ −1 ∈ F .
A symmetric system is a triple P, G , F such that P is a notion of forcing, G is a group of automorphisms of P, and F is a normal filter of subgroups of G .Given such a symmetric system, we say that a P-name ẋ is this notion holds for every P-name hereditarily appearing in ẋ.We denote by HS F the class of hereditarily F -symmetric names.When clear from context, we will omit subscripts and simply write sym( ẋ) or HS.The following theorem, [4,Lemma 15.51], is then key to the study of symmetric extensions.
Theorem 2.7.Let P, G , F be a symmetric system, G ⊆ P a V -generic filter, and let M denote the class Finally, we have a forcing relation for symmetric extensions HS defined by relativising the forcing relation to the class HS.This relation has the same properties and behaviour of the standard forcing relation .Moreover, when π ∈ G , the Symmetry Lemma holds for HS .

AC WO
AC WO , the axiom of choice for all well-orderable families of non-empty sets, is known to be equivalent to the statement (∀X)ℵ(X) = ℵ * (X), and the proof make use of the idea of transferring eccentricity upwards.This idea is best explained by proving the theorem.
Proof. ( =⇒ ).Let X be a set.We always have that ℵ(X) ℵ * (X), so it is sufficient to prove that ℵ * (X) ℵ(X).Let λ < ℵ * (X), and let f : X → λ be a surjection.Since f is a surjection, if we set C = {f −1 (α) | α < λ} then C is a well-ordered family of non-empty sets, and so by AC WO , there is a choice function c : λ → X.However, c must be an injection since ( ⇐= ).We shall prove AC ℵ δ for all ordinals δ by induction.Suppose that we have established AC <ℵ δ (indeed, this is a theorem of ZF for δ = 0), and let Define by induction on α < ℵ δ the cardinal κ α and the set D α in the following way: By projection to its second co-ordinate, there is a surjection D → λ, and so ℵ * (D) λ + .By assumption, we must also have that ℵ(D) λ + .Let f : λ → D be an injection.Since λ > ℵ( β<α D β ) for all α < δ, it cannot be the case that f "λ ⊆ β<α D β for any α < δ.Therefore, by projection to its first co-ordinate, f "λ gives a well-ordered set of partial choice functions for X of unbounded domain.Setting c(α) = 1 st (f (γ))(α), where 1 st is the projection a, b → a and γ is minimal such that α ∈ dom(1 st (f (γ))), produces a choice function c ∈ X as desired.
Inspired by this proof, we produce a general framework for taking a set X and producing a set D of larger Lindenbaum number with some control over the Hartogs and Lindenbaum numbers produced.Definition 3.2.Let κ be a cardinal, δ > 0 a limit ordinal, and X = {X α | α < δ} be such that for all α < δ, Y α . .= β<α X β = ∅.Inductively define the cardinals κ α and sets D α for α < δ as follows: We then define the upwards transfer construction D = D(X, κ) as α<δ D α and λ = λ(X, κ) as sup{κ α | α < δ}.We observe that λ > κ and that λ is a limit cardinal.
Finally, suppose that ℵ(D) λ + , so there is an injection f : λ → D. Note that since λ > ℵ( β<α D β ) for all α < δ, we cannot have that f "λ ⊆ β<α D β for any α < δ.Hence, f "λ intersects D α for unboundedly many αs and so, by projection to the first co-ordinate and the well-order of f "λ, we may select some y α ∈ Y α for unboundedly many αs.Putting these partial choice functions together yields c ∈ X as desired.Explicitly, c(α With upwards transfer construction in hand, we may produce a great many new statements that are all equivalent to AC WO through the general framework of Theorem 3.1.
If we instead assume Condition (5), then since λ is a limit cardinal we have ℵ(D) λ + , and thus X = ∅.
Hence, in each case, we can conclude Condition (4).
Remark.Note that, by combining the techniques exhibited in the proof of Theorem 3.4, one can produce a vast collection of conditions that are equivalent to AC WO .For example, AC WO is equivalent to the statement "there is κ such that for all X, if ℵ * (X) κ then either ℵ(X) is a successor or ℵ(X) is regular".

Spectra
Theorem 3.4 gives the very strong conclusion that AC WO is not just equivalent to (∀X)ℵ(X) = ℵ * (X), but also that ℵ(X) is a successor cardinal for all X.In this way, AC WO gives us that the class of all pairs ℵ(X), ℵ * (X) is minimal, that is just { λ + , λ + | λ ∈ Card}.However, it is possible to violate AC WO , and we inspect here the various ways in which this can be violated in models of SVC.Definition 4.1 (Hartogs-Lindenbaum Spectrum).For a model M of ZF, the (Hartogs-Lindenbaum) spectrum of M , denoted by Spec ℵ (M ), is the class of all possible pairs λ, κ such that there is X ∈ M with ℵ(X) = λ and ℵ * (X) = κ.
In [7], the authors show that it is consistent with ZF to have a model However, this was achieved with a class-length iteration of symmetric extensions and (by the main theorem) cannot be optimised further.Recall that M SVC if M is a symmetric extension of a ground model V AC.We shall show that in this case Spec ℵ (M ) is controlled on a tail of cardinals.In particular, there is Ω ∈ Card such that for all λ Ω, if ℵ(X) = λ then ℵ * (X) λ + .
4.1.The spectrum of Cohen's first model.Let us first establish the methods that will be employed in our favourite test model of ZF + SVC + ¬AC: Cohen's first model.In fact, Cohen's first model is not even a model of AC ω , and so in particular is not a model of AC WO .This will be a consequence of the existence of a set A such that ℵ(A) = ℵ 0 ; by [3, Section 2.4.1], if AC ℵ0 holds then every infinite set has a countably infinite subset.
We begin with a quick reminder of how Cohen's first model is defined.Let V be a model of ZFC and, working in V , let P = Add(ω, ω), that is the forcing whose conditions are finite partial functions p : ω × ω → 2 with the ordering q p if q ⊇ p.The group G is the finitary permutations of ω, those π ∈ S ω such that {n < ω | πn = n} is finite.This has group action on P given by πp(πn, m) = p(n, m).For E ∈ [ω] <ω , let fix(E) G be the subgroup {π ∈ G | π ↾ E = id}, and let F be the filter of subgroups of G generated by the fix(E) as E varies over [ω] <ω .Let M be the symmetric extension of V by this symmetric system, and let V [G] be the full extension by P. Note that, since P is c.c.c., V , M , and V [G] will agree on the cardinalities and cofinalities of ordinals, and V and V [G] will agree on the cardinalities of all sets in V .

Fact 4.2 ([3, §5.5]). The following hold in M :
(1) ℵ(A) = ℵ 0 ; (2) for every infinite X there is a surjection f : X → ω; and (3) for every X there is an ordinal η and an injection f : An immediate corollary of this fact is the following.
We shall use these facts alongside the techniques laid out in Section 3 to produce a complete picture of the spectrum of Cohen's first model.
To translate this to the notation of the main theorem, we have that φ = ψ = ℵ 0 , Therefore, both D and C are subsets of { ℵ 0 , ℵ 1 }.However, we have already seen by Fact 4.  Proof.We may assume that X ⊆ [A] <ω × η for some ordinal η.For each a ∈ so µ is the union of countably many sets (note that A is countable in V [G], and so [A] <ω is as well).If µ has uncountable cofinality then there is a ∈ Since M and V [G] agree on the cardinalities of sets of ordinals, and X a is well-orderable, we have that |X a | µ in M as well.However, this contradicts µ ℵ(X).Hence we must have cf(µ) = ℵ 0 .Lemma 4.6.For all X, ℵ * (X) is a successor.
Proof.As before, we may assume that X ⊆ [A] <ω × η for some ordinal η, and again we shall denote by X a the set X ∩ ({a} × η) for all a ∈ [A] <ω .Let κ be a limit cardinal such that ℵ * (X) κ.Then we must show that ℵ * (X) > κ, that is we must show that there is a surjection X → κ.Let µ < κ be infinite.Since κ is a limit, we still have that µ + < κ, and µ + is regular in both M and V [G].Since µ + < κ, there is a surjection f : However, X a is well-orderable, and so |X a | µ + in M as well.Therefore, for all µ < κ, there is a ∈ [A] <ω such that |X a | > µ.Hence the projection of X onto its second co-ordinate is a surjection onto a subset of η of cardinality at least κ, and so this can be turned into a surjection X → κ.Corollary 4.7.Both "for all X, ℵ * (X) is regular" and "for all X, ℵ * (X) is a successor" are strictly weaker than AC WO .Lemma 4.9.For all λ such that cf(λ) = ℵ 0 there is a set X such that ℵ(X) = λ and ℵ * (X) = λ + .
Proof.By Corollary 4.8, it will be sufficient to prove that there is a set X such that ℵ(X) = λ.By Fact 4.2, suppose that λ > ℵ 0 .Let λ n | n < ω be a strictly increasing sequence of cardinals such that sup{λ n | n < ω} = λ.For each n < ω, let X n = λ n × Inj(n, A), and let X = n<ω X n .We certainly have that ℵ(X) λ, as for all n < ω there is an injection f n : λ n → X by taking arbitrary c ∈ Inj(n, A), and having f n (α) = α, c .Suppose now that there were an injection f : λ → X.Note that in V [G], |Inj(n, A)| = ℵ 0 , and so m<n λ m × Inj(m, A) < λ for all n < ω.Therefore, f "λ is not a subset of m<n λ m × Inj(m, A) for any n < ω, so f must give us a well-ordered collection of injections n → A for arbitrarily large n.We may now put these injections together to produce an injection ω → A, contradicting ℵ(A) = ℵ 0 .Hence an injection λ → X cannot exist and ℵ(X) = λ as desired.

4.2.
The spectrum of a model of SVC.When dealing with Cohen's first model, having an outer model of ZFC that agrees on the cardinalities and cofinalities of ordinals was an important fact that appeared in almost every proof of the previous section.Fortunately, this is not unique to Cohen's first model, and is very close to the conclusions that can be drawn from SVC.Let M be a model of SVC + ¬AC, witnessed by an inner model V of ZFC, a symmetric system P, G , F , and an injective seed A. Let V [G] be the outer forcing extension of M , so " |X| = |δ| and δ is a cardinal".Unlike in Cohen's model, we may not have that ⌈κ⌉ = κ for all cardinals κ ∈ M since P may collapse some cardinals, but by appealing to large enough cardinals we are able to overcome this obstacle.
The following is an immediate corollary of Fact 4.10.

Corollary 4.11.
There is a cardinal λ such that V , M , and V [G] agree on cardinalities above λ and cofinalities greater than λ.
Let τ be the least cardinal such that M and V [G] agree on cardinalities at least τ and cofinalities greater than or equal to τ .Finally, fix an injective seed A for M and let ν = ⌈A⌉.
Throughout this section, all sets and statements about sets are understood to be in the context of M unless stated otherwise.The proof of the following is effectively identical to the proof of Lemma 4.5.
We may assume that X ⊆ A × η for some η ∈ Ord.For each a ∈ A, let X a = X ∩({a}×η), so each X a is well-orderable.We aim to show that, if π : X → η is the projection of X to its second co-ordinate, then |π"X| κ and thus ℵ * (X) > κ as required.
Proof.Suppose that ℵ(X) λ, and let λ α | α < µ be a continuous strictly increasing cofinal sequence in λ with λ 0 = 0. Define the tree T to be the set α<µ Inj(λ α , X), with f g if f ⊆ g.Since ℵ(X) λ, T has no maximal nodes and, by taking unions, we see that T is µ-closed.By DC µ , T has a maximal branch b.Let f = b.Then f is an injection κ → X for some κ, and κ is either λ α for some α < µ or is λ.In the former case, we may extend f to a function from λ α+1 by inspecting any node at level α + 1 (ℵ(X) > λ α+1 so it is nonempty) and removing any duplicate entries.We'll remove fewer than λ α+1 entries, so adjoining the two will still yield an extension in T .This contradicts the maximality of b.Therefore, we are in the latter case and f is an injection λ → X, so ℵ(X) > λ.
The underlying pattern of Proposition 4.16 is that once we have dealt with the chaos of ℵ * (X) max(ν + , τ + ) and the inevitability of ℵ(X) = ℵ * (X) = λ + , all that we have are sets X with ℵ(X) = λ and ℵ * (X) = λ + for some cardinal λ.Indeed, this scenario is the only one in which we may have an eccentric set of arbitrarily large Hartogs or Lindenbaum number.Definition 4.17 (Oblate cardinal).An oblate cardinal is a cardinal λ such that there is a set X with ℵ(X) = λ and ℵ * (X) = λ + .As we have observed, the only candidates for oblate cardinals that are at least as large as max(ν + , τ ) are those singular cardinals λ with cf(λ) ∈ [λ DC , max(ν + , τ )).However, this does not tell us which of those cardinals will be oblate.Fortunately, when we begin to produce a lower bound for the spectrum, we will have very strong results for transferring eccentricity that force cardinals to be oblate.

A lower bound.
Having just seen the upper bound of the spectrum of a model of SVC, we now wish to exhibit a lower bound, which we shall construct entirely through controlling oblate cardinals.By Theorem 3.4, if any eccentric set exists then we must have eccentric sets of arbitrarily large Hartogs or Lindenbaum number.However, by Proposition 4.16, if ℵ(X) max(ν + , τ ) and X is eccentric, then we must have that ℵ(X) is an oblate cardinal.Therefore, there must be a proper class of oblate cardinals.In this section we will explore methods of lifting oblate cardinals to larger oblate cardinals with the same cofinality.We do this in two ways: The first lifts any set B with ℵ(B) a limit cardinal, and the other lifts any set B with ℵ(B) < ℵ * (B).Lemma 4.18.Let B be such that ℵ(B) = µ a limit cardinal, λ a cardinal with λ ν + τ + ⌈B <µ ⌉ + , and δ α | α < µ a strictly increasing sequence of ordinals such that sup{δ α | α < µ} = λ. 4 Then there is a set X such that ℵ(X) = λ and ℵ * (X) = λ + .

The future
Chief among unanswered questions in this field is as follows: Question 5.1.Precisely which spectra are possible to achieve with models of SVC? Throughout the work in Section 4.2, the spectre of the injective seed haunted our calculations.If we wished to decide precisely which spectra are achievable in models of SVC, then more information on the injective seeds of the models that we produce is required.It is also unclear to me how much control we can have over C, C and Ω.In [7], for each infinite λ κ, the authors construct a symmetric system P, G , F such that for all V ZFC, 1 P HS (∃X)ℵ(X) = λ, ℵ * (X) = κ.In fact, such models can be constructed so that the model and the outer model agree with the cardinality and cofinality of all ordinals, just as in the case of Cohen's model.Question 5.2.What are the spectra of the symmetric extensions produced in [7]?
The Bristol model, introduced in [5] and expanded upon in [6], is an inner model of L[c], where c is a single Cohen real.It therefore satisfies the same conditions that made manipulating Cohen's model so nice: There is an outer model of ZFC that agrees with the inner model on all cardinalities and cofinalities of ordinals.However, the Bristol model was constructed with explicit intention to violate SVC, and so many of the techniques used in this paper cannot be applied 'as is'.However, it does not seem too far-fetched that an amount of this work can be reclaimed.The Bristol model is, in a sense, a limit of models of SVC that approaches the final model, and it is reasonable to believe that one can look at intermediate models to obtain results about the spectrum.

Proposition 2 . 5 .
M SVC if and only if M has an injective seed.

4. 2 . 1 .
An upper bound.We first aim to create an upper bound on the Hartogs-Lindenbaum spectrum of M by showing scenarios in which combinations of Hartogs and Lindenbaum numbers are not possible.

Question 5 . 6 ([ 6 ,
Question 10.19]).Does the Bristol model satisfy AC WO ?If not, what is the spectrum of the Bristol model?

all Y , either |Y | * |X| or |X| * |Y |. We immediately observe the following:
2.1.1.The axiom of choice AC X .For any set X, we shall denote by AC X the statement that all families of non-empty sets indexed by X admit a choice function.If α is an ordinal or a cardinal, we shall denote by AC <α the statement that all families of non-empty sets indexed by some β < α admit a choice function.Finally, by AC WO , we mean the statement that all well-orderable families of non-empty sets admit a choice function; equivalently, this can be written (∀λ ∈ Card)AC λ .When the subscript is omitted, we mean the full axiom of choice: every family of non-empty sets admits a choice function.2.1.2.Dependent choice.We say that a partially ordered set (T, ) is a tree if T has a minimum element and, for all t ∈ T , the set {s ∈ T | s t} is well-ordered by .Given a cardinal λ, we say that T is λ-closed if all -chains in T of length less than λ have an upper bound in T .Finally, the statement DC λ , the principle of dependent choice (for λ), is the statement that every λ-closed tree has a maximal element or a chain of order type λ.When the subscript is omitted, we mean DC ω .While