We consider choice correspondences that assign a subset to every choice set of alternatives, where the total set of alternatives is an arbitrary finite or infinite set. We focus on the relations between several extensions of the condition of independence of irrelevant alternatives on one hand, and conditions on the revealed preference relation on sets, notably the weak axiom of revealed preference, on the other hand. We also establish the connection between the condition of (...) independence of irrelevant alternatives and so-called strong sets; the latter characterize a social choice correspondence satisfying independence of irrelevant alternatives. (shrink)

We interpret solution rules on a class of simple allocation problems as data on the choices of a policy maker. We analyze conditions under which the policy maker’s choices are (i) rational (ii) transitive-rational, and (iii) representable; that is, they coincide with maximization of a (i) binary relation, (ii) transitive binary relation, and (iii) numerical function on the allocation space. Our main results are as follows: (i) a well-known property, contraction independence (a.k.a. IIA) is equivalent to rationality; (ii) every contraction (...) independent and other-c monotonic rule is transitive-rational; and (iii) every contraction independent and other-c monotonic rule, if additionally continuous, can be represented by a numerical function. (shrink)

It is shown that a fundamental question of revealed preference theory, namely whether the weak axiom of revealed preference (WARP) implies the strong axiom of revealed preference (SARP), can be reduced to a Hamiltonian cycle problem: A set of bundles allows a preference cycle of irreducible length if and only if the convex monotonic hull of these bundles admits a Hamiltonian cycle. This leads to a new proof to show that (...) preference cycles can be of arbitrary length for more than two but not for two commodities. For this, it is shown that a set of bundles satisfying the given condition exists if and only if the dimension of the commodity space is at least three. Preference cycles can be constructed by embedding a cyclic $(L-1)$ -polytope into a facet of a convex monotonic hull in $L$ -space, because cyclic polytopes always admit Hamiltonian cycles. An immediate corollary is that WARP only implies SARP for two commodities. The proof is intuitively appealing as this gives a geometric interpretation of preference cycles. (shrink)

This article explores rationalizability issues for finite sets of observations of stochastic choice in the framework introduced by Bandyopadhyay et al. (Journal of Economic Theory, 84(1), 95–110, 1999). It is argued that a useful approach is to consider indirect preferences on budgets instead of direct preferences on commodity bundles. A new rationalizability condition for stochastic choices, “rationalizable in terms of stochastic orderings on the normalized price space” (rsop), is defined. rsop is satisfied if and only if there exists a solution (...) to a linear feasibility problem. The existence of a solution also implies rationalizability in terms of stochastic orderings on the commodity space. Furthermore it is shown that the problem of finding sufficiency conditions for binary choice probabilities to be rationalizable bears similarities to the problem considered here. (shrink)

The phenomenon of weakness of will – not doing what we perceive as the best action – is not recognized by neoclassical economics due to the axiomatic assumptions of the revealed preference theory (RPT) that people do what is best for them. However, present bias shows that people have different preferences over time. As they cannot be compared by the utility measurements, economists need to normatively decide between selves (short- versus long-term preferences). A problem is that neoclassical economists (...) perceive RPT as value-free and incorporate present bias within the economic framework. The axiomatic assumption that people do what is best for them leads to theoretical and practical dilemmas. This work examines weakness of will to resolve some shortcomings of RPT. The concept of intention is used to provide multiple self conception with the framework to decide between selves, which had not been done before. The paper concludes that individuals should not always follow their revealed preferences (desires) but the intentions (reason) because the latter indicates what people really want. (shrink)

In economics, it has often been claimed that testing choice data for violation of certain axioms-particularly if the choice data is observed under laboratory conditions-allows conclusions about the validity of certain preference axioms and the neoclassical maximization hypothesis. In this paper I argue that these conclusions are unfounded. In particular, it is unclear what exactly is tested, and the interpretation of the test results are ambiguous. Further, there are plausible reasons why the postulated choice axioms should not hold. Last, (...) these tests make implicit assumptions about beliefs that further blur the interpretations of the results. The tests therefore say little if anything about the validity of certain preference axioms or the maximization hypothesis. (shrink)

Since Varian (Econometrica 50:945–973, 1982; Review of Economic Studies 50:90–110, 1983) made checking for consistency with revealed preference conditions more accessible to empirical researchers; researchers have often used revealed preference procedures to test their maintained hypotheses and narrow the scope of their demand studies. The tests developed by Varian are for the direct utility function, while researchers estimating demand systems often find it convenient to model consumer behavior with an indirect utility function. Unfortunately structure revealed (...) in the direct utility function does not necessarily follow for the indirect utility function. To remedy this problem, we set forth conditions for checking consumer data for consistency with revealed preference conditions for minimization and weak separability of the indirect utility function. (shrink)

A much discussed topic in the theory of choice is how a preference order among options can be derived from the assumption that the notion of ' choice' is primitive. Assuming a choice function that selects elements from each finite set of options, Arrow (Económica 26: 121-127,1959) already showed how we can generate a weak ordering by putting constraints on the behavior of such a function such that it reflects utility maximization. Arrow proposed that rational agents can be (...) modeled by such choice functions. Arrow's standard model of rationality has been criticized in economics and gave rise to approaches of bounded rationality. Two standard assumptions of rationality will be given up in this paper. First, the idea that agents are utility optimizers (Simon). Second, the idea that the relation of ' indifference' gives rise to an equivalence relation. To account for the latter, Luce (Econometrica 24: 178-191, 1956) introduced semi-orders. Extending some ideas of Van Benthem (Pac Philos Q 63: 193-203, 1982), we will show how to derive semi-orders (and so-called interval orders) based on the idea that agents are utility satisficers rather than utility optimizers. (shrink)

We introduce an Absolute (Relative) Time-consistent Axiom of Revealed Preference which characterizes the consistency of a choice function with the property of absolute (relative) time-consistency and impatience. The axiom requires that the absolute (relative) time-consistent and impatient closure of the revealed preference relation does not conflict with the strict revealed preference relation.

Conventional economic theory assumes that people care only about ultimate outcomes and are indifferent to the decision and allocation processes by which outcomes are brought about. Building on Sen (1997), I relax this assumption, and investigate the formal and philosophical issues that arise. I extend the formal apparatus of preference theory to analyse how processes may enter preferences, and investigate whether traditional invariance requirements like the Weak Axiom of Revealed Preference are still satisfied in this (...) new setting. I show that it is, provided certain conditions of separability hold, and I discuss the plausibility of these conditions. Further, I argue that processes are often valued in a mode that diverges from the conventional modes of instrumental and intrinsic/independent valuation. I introduce the notion of dependent non-instrumental valuation, and show how processes could depend on their instrumental function for their value – making their value dependent – and yet derive their value from something else – making it non-instrumental. Dependent non-instrumental value, I argue, can be explained by symbolic and evidential relations between processes and outcomes. (Published Online July 31 2007) Footnotes1 This article is based on the third chapter of my Ph.D. dissertation (Sandbu 2003). I would like to thank Richard Tuck for many discussions over several years, which helped me develop and elaborate the ideas presented here. I am also very grateful to Amartya Sen, Nien-hê Hsieh, Luc Bovens, and Xaq Pitkow for their close readings of various versions of the paper and their incisive comments, questions, and suggestions. Further thanks go to Christopher Avery, Matthias Benz, Jerry Green, Waheed Hussain, David Laibson, Robert Sugden, Alan Strudler, Justin Wolfers, and seminar participants at Harvard University and the Wharton School of Business. Akshay Jashnani provided helpful research assistance. Most of the ideas in the present article were developed while I was the recipient of a doctoral grant from the Research Council of Norway, which I gratefully acknowledge. (shrink)

This note comments on Professor Majumdar’s concept of an individual who seeks to change his own preferences over time. It is argued that while one can formulate “revealed preference axioms“ which will rationalize the choices of Professor Majumdar’s individual, it is unlikely that the choice of such an individual will reveal his preferences.

In [10], we have shown that the statement that all ∑ 1 1 partitions are Ramsey is deducible over ATR 0 from the axiom of ∑ 1 1 monotone inductive definition,but the reversal needs П 1 1 - CA 0 rather than ATR 0 . By contrast, we show in this paper that the statement that all ∑ 0 2 games are determinate is also deducible over ATR 0 from the axiom of ∑ 1 1 monotone inductive definition, (...) but the reversal is provable even in ACA 0 . These results illuminate the substantial differences among lightface theorems which can not be observed in boldface. (shrink)

My aim in this paper is to argue that the recent philosophical defenses of revealed preference theory do not withstand scrutiny. Towards this aim, I will first outline revealed preference theory. I will then briefly present the two most common arguments that the received view offers against it. Afterwards, I will outline three argumentative strategies for rehabilitating revealed preference theory, and successively rebut each of them.

Independence is the condition that, if X is preferred to Y, then a lottery between X and Z is preferred to a lottery between Y and Z given the same probability of Z. Is it rationally required that one’s preferences conform to Independence? The main objection to this requirement is that it would rule out the alleged rationality of Allais and Ellsberg Preferences. In this paper, I put forward a sequential dominance argument with fairly weak assumptions for a variant (...) of Independence (called Independence for Constant Prospects), which shows that Allais and Ellsberg Preferences are irrational. Hence this influential objection (that is, the alleged rationality of Allais and Ellsberg Preferences) can be rebutted. I also put forward a number of sequential dominance arguments that various versions of Independence are requirements of rationality. One of these arguments is based on very minimal assumptions, but the arguments for the versions of Independence which are strong enough to serve in the standard axiomatization of Expected Utility Theory need notably stronger assumptions. (shrink)

This paper defends revealed preference theory against a pervasive line of criticism, according to which revealed preference methodology relies on appealing to some mental states, in particular an agent’s beliefs, rendering the project incoherent or unmotivated. I argue that all that is established by these arguments is that revealed preference theorists must accept a limited mentalism in their account of the options an agent should be modelled as choosing between. This is consistent both with (...) an essentially behavioural interpretation of preference and with standard revealed preference methodology. And it does not undermine the core motivations of revealed preference theory. (shrink)

This chapter explores reasons to doubt the defensibility of the “weak theory of sustainability” that informs and justifies the use of cost-benefit analysis by environmental regulators. As the argument reveals, inasmuch as the weak theory equates what is sustainable with what sustains the satisfaction of human preferences, it has the surprising philosophical wherewithal to make climate-changing activities sustainable, at least in principle. This would be so if human ingenuity made possible the replacement of ecosystem services with technological alternatives. (...) And it would be particularly so if the aesthetic goods that derive from nature – goods which are resistant to quantification – are excluded from environmental cost-benefit analyses. But it is also true if those aesthetic goods are reduced to mere human preferences that can be measured through indirect market-based means, for aesthetic preferences are remarkably adaptive.Inasmuch as people can be expected to come to appreciate landscapes degraded by climate change, those who defend a weak theory of sustainability can expect that climate change will not long be at odds with what people find aesthetically pleasing. However, as the growing literature on “nature-deficit disorder” suggests, while aesthetic preferences may be elastic, aesthetic needs are not. As climate change progressively strips us of nature’s goods, we will lose the ability to meet crucial aesthetic needs. The irony is clear: As our aesthetic preferences bend towards the acceptance of ecological loss, we will predictably lose our preferences for meeting our own enduring aesthetic needs. We can thus expect climate change to cause us to prefer what we do not need and to need what we do not prefer. And this implies that climate change is the reductio ad absurdum of the weak theory of sustainability that dominates contemporary environmental regulation. (shrink)

Revealed preference approaches to modelling agents’ choices face two seemingly devastating explanatory objections. The no self-explanation objection imputes a problematic explanatory circularity to revealed preference approaches, while the causal explanation objection argues that, all things equal, a scientific theory should provide causal explanations, but revealed preference approaches decidedly do not. Both objections assume a view of explanation, the constraint-based view, that the revealed preference theorist ought to reject. Instead, the revealed (...) class='Hi'>preference theorist should adopt a unificationist account of explanation, allowing her to escape the two explanatory problems discussed in this paper. (shrink)

This paper evaluates how Amartya Sen’s critique of revealed preference theory stands against the latter’s contemporary, ‘neo-Samuelsonian’ version. Neo- Samuelsonians have argued that Sen’s arguments against RPT are innocuous, in particular once it is acknowledged that RPT does not assume away the existence of motivations or other latent psychological or cognitive processes. Sen’s claims that preferences and choices need to be distinguished and that external factors need to be taken into account to analyze the act of choice then (...) appear to be irrelevant. However, while it is true that contemporary revealed preference theory partially evades Sen’s critique, I show that the latter is still relevant outside the restricted areas of consumer choice and market dynamics. In particular, Sen’s views regarding the importance of incomplete preferences and the multiplicity of levels of agency can hardly be integrated into the framework of CRPT. This is a significant limit, given the imperialistic claims of some of the proponents of the latter. (shrink)

A weak form of intuitionistic set theory WST lacking the axiom of extensionality is introduced. While WST is too weak to support the derivation of the law of excluded middle from the axiom of choice, we show that bee.ng up WST with moderate extensionality principles or quotient sets enables the derivation to go through.

In this paper we study some statements similar to the Partition Principle and the Trichotomy. We prove some relationships between these statements, the Axiom of Choice, and the Generalized Continuum Hypothesis. We also prove some independence results. MSC: 03E25, 03E50, 04A25, 04A50.

This paper examines methodological issues raised by revealed preference theory in economics: particularly contemporary revealed preference theory. The paper has three goals. First, to make the case that revealed preference theory is a broad research program in choice theory—not a single theory—and understanding this diversity is essential to any methodological analysis of the program. Second, to explore some of the existing criticisms of revealed preference theory in a way that emphasizes how the (...) effectiveness of the critique depends on the particular version of revealed preference under consideration. Finally, three additional criticisms are presented that are aimed specifically at contemporary revealed preference theory. (shrink)

(2) Mental preferences: These describe the mental attitude of an individual toward the objects. They can be defined in contexts which do not involve actual choice. In particular, preferences can describe tastes (such as a preference for one season over another) or can refer to situations which are only hypothetical (such as the possible courses of action available to an individual were he to become Emperor of Rome) or which the individual does not fully control (such as a game (...) situation in which a player has preferences over the entire set of outcomes). (shrink)

Rationality postulates for preferences are developed from two basic decision theoretic principles, namely: (1) the logic of preference is determined by paradigmatic cases in which preferences are choice-guiding, and (2) excessive comparison costs should be avoided. It is shown how the logical requirements on preferences depend on the structure of comparison costs. The preference postulates necessary for choice guidance in a single decision problem are much weaker than completeness and transitivity. Stronger postulates, such as completeness and transitivity, can (...) be derived under the further assumption that the original preference relation should also be capable of guiding choice after any restriction of the original set of alternatives. (shrink)

Resumen: En el presente artículo se realiza una comparación crítica de dos enfoques sobre los fundamentos de la teoría microeconómica del comportamiento del consumidor, a saber: el enfoque utilitarista y el de la preferencia revelada. Se lleva a cabo un análisis de la preferencia revelada en el contexto de la introducción de restricciones a la función de elección en la teoría de la decisión. Se evalúan sus ventajas y limitaciones en contraste con el problema de la utilidad propio del punto (...) de vista subjetivista de la teoría utilitarista. Y se confronta la capacidad explicativa de los distintos modelos de agencia racional sobre las bases de las curvas de demanda. Asimismo, se desarrolla una crítica formal al método de la preferencia revelada en entornos estratégicos de decisión.: This article makes a critical comparison of two approaches to the foundations of the microeconomic theory of consumer behavior, namely: the utilitarian approach and that of the revealed preference theory. An analysis of revealed preference is carried out in the context of introducing restrictions to the choice function in decision theory. Its advantages and limitations are evaluated in contrast to the utility problem proper to the subjectivist point of view of utilitarian theory. And the explanatory capacity of the different models of rational agency is confronted on the bases of the demand curves. Furthermore, a formal critique of the revealed preference method is developed in strategic decision-making environments. (shrink)

It is shown that AC, the axiom of choice for families of non-empty subsets of the real line ℝ, does not imply the statement PW, the powerset of ℝ can be well ordered. It is also shown that the statement “the set of all denumerable subsets of ℝ has size 2math image” is strictly weaker than AC and each of the statements “if every member of an infinite set of cardinality 2math image has power 2math image, then the union (...) has power 2math image” and “ℵ ≠ ℵω” is Hartogs' aleph, the least ℵ not ≤ 2math image), is strictly weaker than the full axiom of choice AC. (shrink)

The notion of ‘revealed preference’ is unclear and should be abandoned. Defenders of the theory of revealed preference have misinterpreted legitimate concerns about the testability of economics as the demand that economists eschew reference to (unobservable) subjective states. As attempts to apply revealed-preference theory to game theory illustrate with particular vividness, this demand is mistaken.

The question of how to frame agential preferences in economics finds one caught between Scylla and Charybdis. If preferences are framed in as minimal and deflationary a manner as revealed preference theory recommends, the theory falls prey to objections about its predictiveness and explanatory power. Alternatively, if too many cognitive and causal intricacies are incorporated into the preference concept, revealed preference models will violate pragmatic norms of model construction, surrendering model simplicity and generality. This paper (...) charts a middle course, arguing that the path to salvation lies through an understanding of revealed preference models as program explanations. (shrink)

We introduce the anti-rectangle refining property for forcing notions and investigate fragments of Martin’s axiom for ℵ1 dense sets related to the anti-rectangle refining property, which is close to some fragment of Martin’s axiom for ℵ1 dense sets related to the rectangle refining property, and prove that they are really weaker fragments.

We investigate, within the framework of Zermelo-Fraenkel set theory ZF, the interrelations between weak forms of the Axiom of Choice AC restricted to sets of reals.

This paper examines the incompleteness of collective preference. We provide a series of Arrovian impossibility theorems without completeness. First, we consider the notion of regularity introduced by Eliaz and Ok (2006, Games and Economic Behavior 56, 61–86); it is an appropriate richness property for strict preference when preference is allowed to be incomplete. We examine the implication of imposing regularity on collective preference. Second, we propose responsiveness, a variation of positive responsiveness. This axiom requires that (...) some changes in individual preferences make an alternative weakly better than another. Third, we consider coherency conditions for collective preferences; this conditionally requires the existence of comparable pairs in a certain manner. We prove an impossibility result for each condition using Arrovian axioms. (shrink)

The “Asian disease” problem (Tversky & Kahneman, 1981) demonstrated behaviour in contradiction to the invariance axiom of EU theory. However, the risky choice behaviour was simply seen by the equate-to-differentiate model as a choice between the best possible outcomes or a choice between the worst possible outcomes. It was then argued that a way in which frame influences choice is through the perceived difference between possible outcomes. A “judgement” task was designed to examine whether the knowledge of “the value (...) difference between each possible outcome and the certain outcome” will permit prediction of preference in the choice pattern related to the Asian disease problem. Participants were exposed to an anthrax disease problem (the original or probabilistic version of the Asian disease problem) and a SARS problem (the fuzzy version of the Asian disease problem). It was shown that the empirical evidence in relation to the Asian disease problem could be satisfactorily accounted for by the generalised weak dominance strategy revealed by the judgement results. (shrink)

This paper examines the aggregation of preferences with a finitely additive measure space of agents. We consider three types of non-dictatorship axioms: non-dictatorship, coalitional non-dictatorship, and atomic non-dictatorship. First, we show that the existence of an atom is a necessary and sufficient condition for the existence of a social welfare function that satisfies weak Pareto, independence of irrelevant alternatives, and coalitional non-dictatorship. Second, we simultaneously impose non-dictatorship and coalitional non-dictatorship, and specify a necessary and sufficient condition for the finitely (...) additive measure that guarantees the compatibility among the axioms. Third, we impose all non-dictatorship axioms and show that the corresponding measure is extremely restricted. (shrink)

We characterize general constraints under which rational choices are characterized by asymmetric revealed preferences. A key feature of our main characterization result is expressed by the leveling axiom. We also consider the special case of a law-abiding decision maker who chooses optimally among legal options. We show that the law does not necessarily satisfy the leveling axiom and, therefore, transitivity adds empirical content to law-abiding choices.

(1) This paper uses the following binary relations: > (is preferred to); ⩽ (is not preferred to); < (is less preferred than); ~ (is indifferent to). (2) Savage used primitive ⩾, postulated to be connected and transitive onA (the set of acts), to define the others: [x ~ y ⇔ (x ⩽ y and y ⩽ x)]; [y < x ⇔ notx ⩽ y]; [x > y ⇔ y < x]. Independently of the axioms, this definition implies that ⩽ and (...) > are complementary relations onA: [x < y ⇔ notx > y]. (3) Pratt, Raiffa and Schlaifer used primitive ⩽, postulated to be transitive onL (the set of lotteries), to define the others with a different expression for <: [x < y ⇔ (x ⩽ y and noty ⩽ x)]. Thus, ⩽ and > are not necessarily complementary onL; since ⩽ is not postulated to be connected onL, but connected ⩽ is necessary and sufficient for such complementarity. Since the restriction of ⩽ to the subsetA ofL is connected, ⩽ and > are complementary onA. (4) Fishburn used primitive < onA to define the others with different expression for ~ and ⩽: [x ~ y ⇔ (notx < y and noty < x)]; [x ⩽ y ⇔ (x < y orx ~ y)]. His version of Savage's theory then assumed that < is asymmetric and negatively transitive onA. Thus, ⩽ and > are complementary, since asymmetric < is necessary and sufficient for such complementarity. (5) This analysis provides a new proof that the same list of elementary properties of binary relations onA applies to all three theories: ⩽ is connected, transitive, weakly connected, reflexive, and negatively transitive; while both < and > are asymmetric, negatively transitive, antisymmetric, irreflexive, and transitive; but only ~ is symmetric. (shrink)

This essay gives necessary and sufficient conditions for recovering expected utility from choice behavior in several popular models of uncertainty. In particular, these techniques handle a finite state model; a model for which the choice space consists of probability densities and the expected utility representation requires bounded, measurable utility; and a model for which the choice space consists of Borel probability measures and the expected utility representation requires bounded, continuous utility. The key result is the identification of the continuity condition (...) necessary for the revelation of linear utility. (shrink)

We work in set-theory without choice ZF. A set is Countable if it is finite or equipotent with ${\Bbb N}$ . Given a closed subset F of [0, 1] I which is a bounded subset of $\ell ^{1}(I)$ (resp. such that $F\subseteq c_{0}(I)$ ), we show that the countable axiom of choice for finite sets, (resp. the countable axiom of choice AC N ) implies that F is compact. This enhances previous results where AC N (resp. the (...) class='Hi'>axiom of Dependent Choices) was required. If I is linearly orderable (for example $I={\Bbb R}$ ), then, in ZF, the closed unit ball of the Hilbert space $\ell ^{2}(I)$ is (Loeb-)compact in the weak topology. However, the weak compactness of the closed unit ball of $\ell ^{2}(\scr{P}({\Bbb R}))$ is not provable in ZF. (shrink)

Ever since Sen criticized the notion of internal consistency of choice, there exists a widespread perception that the standard rationalizability approach to the theory of choice has difficulties in coping with the existence of external norms. We introduce a concept of norm-conditional rationalizability and show that external norms can be made compatible with the methods underlying the traditional rationalizability approach. To do so, we characterize norm-conditional rationalizability by means of suitable modifications of revealed preference axioms that are well (...) established in the theory of rational choice on general domains as analysed in contributions by Richter and Hansson, for example. We compare our approach to alternative suggestions that have appeared in response to Sen's criticisms, and we discuss its links to Sen's notion of self-imposed choice constraints. (shrink)

Abstract: Don Ross’ Economic Theory and Cognitive Science (2005) provides an elaborate philosophical defense of neoclassical economics. He argues that the central features of neoclassical theory are associated with what he calls the Robbins-Samuelson argument pattern and that it can be reconciled with recent developments in experimental and behavioral economics, as well as contemporary cognitive science. This paper argues that Ross’ Robbins-Samuelson argument pattern is not in the work of either Robbins or Samuelson and in many ways is in conflict (...) with their own versions, and defenses, of neoclassical theory. (shrink)

It is shown that for compact metric spaces the following statements are pairwise equivalent: “X is Loeb”, “X is separable”, “X has a we ordered dense subset”, “X is second countable”, and “X has a dense set G = ∪{Gn : n ∈ ω}, ∣Gn∣ < ω, with limn→∞ diam = 0”. Further, it is shown that the statement: “Compact metric spaces are weakly Loeb” is not provable in ZF0 , the Zermelo-Fraenkel set theory without the axiom of regularity, (...) and that the countable axiom of choice for families of finite sets CACfin does not imply the statement “Compact metric spaces are separable”. (shrink)

We study theorems from Functional Analysis with regard to their relationship with various weak choice principles and prove several results about them: “Every infinite‐dimensional Banach space has a well‐orderable Hamel basis” is equivalent to ; “ can be well‐ordered” implies “no infinite‐dimensional Banach space has a Hamel basis of cardinality ”, thus the latter statement is true in every Fraenkel‐Mostowski model of ; “No infinite‐dimensional Banach space has a Hamel basis of cardinality ” is not provable in ; “No (...) infinite‐dimensional Banach space has a well‐orderable Hamel basis of cardinality ” is provable in ; (the Axiom of Choice for denumerable families of non‐empty finite sets) is equivalent to “no infinite‐dimensional Banach space has a Hamel basis which can be written as a denumerable union of finite sets”; Mazur's Lemma (“If X is an infinite‐dimensional Banach space, Y is a finite‐dimensional vector subspace of X, and, then there is a unit vector such that for all and all scalars α”) is provable in ; “A real normed vector space X is finite‐dimensional if and only if its closed unit ball is compact” is provable in ; (Principle of Dependent Choices) + “ can be well‐ordered” does not imply the Hahn‐Banach Theorem () in ; and “no infinite‐dimensional Banach space has a Hamel basis of cardinality ” are independent from each other in ; “No infinite‐dimensional Banach space can be written as a denumerable union of finite‐dimensional subspaces” lies in strength between (the Axiom of Countable Choice) and ; implies “No infinite‐dimensional Banach space can be written as a denumerable union of closed proper subspaces” which in turn implies ; “Every infinite‐dimensional Banach space has a denumerable linearly independent subset” is a theorem of, but not a theorem of ; and “Every infinite‐dimensional Banach space has a linearly independent subset of cardinality ” implies “every Dedekind‐finite set is finite”. (shrink)

We show that the countable axiom of choice CAC strictly implies the statements “Lindelöf metric spaces are second countable” “Lindelöf metric spaces are separable”. We also show that CAC is equivalent to the statement: “If is a Lindelöf topological space with respect to the base ℬ, then is Lindelöf”.