Infinite Populations, Choice and Determinacy

inﬁnite populations, choice and determinacy

tadeusz litak

informatik 8 FAU erlangen-nurnberg¨

1/54 By all means mathematicians should learn all the pure mathematics they want. But they must also learn applications. Get to be an expert in either the social sciences or in computer science.

John Kemeny

why am I giving this talk?

2/54 why am I giving this talk?

By all means mathematicians should learn all the pure mathematics they want. But they must also learn applications. Get to be an expert in either the social sciences or in computer science.

John Kemeny

2/54 3/54 Logics for Social Behaviour Workshop: 10 - 14 November 2014, Leiden, the Netherlands

Scientific • Frederik Herzberg, Bielefeld U Organizers • Alexander Kurz, U Leicester • Alessandra Palmigiano, TU Delft Invited • Samson Abramsky, U Oxford Speakers • Bob Coecke, U Oxford The Lorentz Center is an international • Martin Hyland, U Cambridge center in the sciences. Its aim is to • Philippe Mongin, CNRS Paris organize workshops for scientists in an atmosphere that fosters collaborative • Ariel Rubinstein, U Tel Aviv work, discussions and interactions. • Carl Wagner, U Tennessee For registration see: www.lorentzcenter.nl

• Viktor Winschel, U Mannheim Academic fencing in Göttingen, Germany (1847). Poster design: SuperNova Studios . NL

a few years ago I was invited to some meetings on Logics for Social Behaviour ...

but seriously, why?

4/54 but seriously, why?

Logics for Social Behaviour Workshop: 10 - 14 November 2014, Leiden, the Netherlands

• Viktor Winschel, U Mannheim Academic fencing in Göttingen, Germany (1847). Poster design: SuperNova Studios . NL

a few years ago I was invited to some meetings on Logics for Social Behaviour ...

4/54 I I didn’t know much about social choice at that time or set theory, for that matter. . . and it’s not that I know that much more now btw

I but I got really bothered about importance of a surprisingly large industry, whose existence I discovered at those meetings

I the original idea, I guess, was to have a look at coalgebraic/algebraic logic for preference aggregation more about this business later

5/54 I but I got really bothered about importance of a surprisingly large industry, whose existence I discovered at those meetings

I the original idea, I guess, was to have a look at coalgebraic/algebraic logic for preference aggregation more about this business later

I I didn’t know much about social choice at that time or set theory, for that matter. . . and it’s not that I know that much more now btw

5/54 I the original idea, I guess, was to have a look at coalgebraic/algebraic logic for preference aggregation more about this business later

I I didn’t know much about social choice at that time or set theory, for that matter. . . and it’s not that I know that much more now btw

I but I got really bothered about importance of a surprisingly large industry, whose existence I discovered at those meetings

5/54 6/54 I and now I got invited to give a talk in Freiburg about that piece . . . I . . . so, without further ado, what is this industry about? I well, formal economists have surprisingly often some use for infinite collections of agents or infinite populations I that, in itself, may be defensible in certain contexts we will briefly overview some of them in the next slide I the problem lies in how they handle them and what they prove

I and then I got invited to submit a paper to a special issue of Studia Logica ...

7/54 I . . . so, without further ado, what is this industry about? I well, formal economists have surprisingly often some use for infinite collections of agents or infinite populations I that, in itself, may be defensible in certain contexts we will briefly overview some of them in the next slide I the problem lies in how they handle them and what they prove

I and then I got invited to submit a paper to a special issue of Studia Logica ... I and now I got invited to give a talk in Freiburg about that piece . . .

7/54 I well, formal economists have surprisingly often some use for infinite collections of agents or infinite populations I that, in itself, may be defensible in certain contexts we will briefly overview some of them in the next slide I the problem lies in how they handle them and what they prove

7/54 I that, in itself, may be defensible in certain contexts we will brieﬂy overview some of them in the next slide I the problem lies in how they handle them and what they prove

I and then I got invited to submit a paper to a special issue of Studia Logica ... I and now I got invited to give a talk in Freiburg about that piece . . . I . . . so, without further ado, what is this industry about? I well, formal economists have surprisingly often some use for inﬁnite collections of agents or inﬁnite populations

7/54 I the problem lies in how they handle them and what they prove

7/54 I and then I got invited to submit a paper to a special issue of Studia Logica ... I and now I got invited to give a talk in Freiburg about that piece . . . I . . . so, without further ado, what is this industry about? I well, formal economists have surprisingly often some use for infinite collections of agents or infinite populations I that, in itself, may be defensible in certain contexts we will briefly overview some of them in the next slide I the problem lies in how they handle them and what they prove

7/54 intergenerational social choice the assumption of an infinite time horizon, e.g., infinitely repeating patterns of behaviour in games or utility streams social choice under uncertainty an infinite set of possible states of the world, each with its own (conceptual) inhabitants see Pivato 2014 for a more detailed analysis voting theory... ??? ...... analogous to that of idealized markets ?? perhaps not coincidentally, most authors do not seem overly explicit about their motivations

whence inﬁnite populations?

8/54 social choice under uncertainty an inﬁnite set of possible states of the world, each with its own (conceptual) inhabitants see Pivato 2014 for a more detailed analysis voting theory... ??? ...... analogous to that of idealized markets ?? perhaps not coincidentally, most authors do not seem overly explicit about their motivations

whence inﬁnite populations?

8/54 voting theory... ??? ...... analogous to that of idealized markets ?? perhaps not coincidentally, most authors do not seem overly explicit about their motivations

whence inﬁnite populations?

idealized markets uncountably many agents when studying an idealized notion of perfect competition (Aumann 1964) the essential idea . . . is that the economy under consideration has a “very large” number of participants, and that the influence of each individual participant is “negligible” . . . one can integrate over a continuum, and changing the integrand at a single point does not affect the value of the integral intergenerational social choice the assumption of an infinite time horizon, e.g., infinitely repeating patterns of behaviour in games or utility streams social choice under uncertainty an infinite set of possible states of the world, each with its own (conceptual) inhabitants see Pivato 2014 for a more detailed analysis

8/54 whence inﬁnite populations?

8/54 Hildenbrand 1970 . . . as an economist, our interest in these “ideal economies” is proportional to how much new information can be derived for large but ﬁnite economies “On economies with many agents”, Journal of Economic Theory Zame 2007 the full power of the Axiom of Choice is almost never used in formal economics or in classical analysis for that matter. What is used is a much weaker axiom, the Axiom of Dependent Choice “Can intergenerational equity be operationalized?”, Theoretical Economics

three quotes to guide us

Aumann 1964 nonmeasurable sets are extremely pathological; it is unlikely that they would occur in the context of an economic model “Markets with a continuum of traders”, Econometrica

9/54 Zame 2007 the full power of the Axiom of Choice is almost never used in formal economics or in classical analysis for that matter. What is used is a much weaker axiom, the Axiom of Dependent Choice “Can intergenerational equity be operationalized?”, Theoretical Economics

three quotes to guide us

9/54 three quotes to guide us

Aumann 1964 nonmeasurable sets are extremely pathological; it is unlikely that they would occur in the context of an economic model “Markets with a continuum of traders”, Econometrica Hildenbrand 1970 . . . as an economist, our interest in these “ideal economies” is proportional to how much new information can be derived for large but ﬁnite economies “On economies with many agents”, Journal of Economic Theory Zame 2007 the full power of the Axiom of Choice is almost never used in formal economics or in classical analysis for that matter. What is used is a much weaker axiom, the Axiom of Dependent Choice “Can intergenerational equity be operationalized?”, Theoretical Economics

9/54 I decades later, papers are still published debating if something useful has been thus achieved I some instances, ﬁrst without being speciﬁc on details (these will come later)

and yet, we have a pattern

I somebody assumes a ZFC-based metatheory and proves a cute mathematical result in formal economics voting theory, intergenerational social choice . . .

10/54 I some instances, ﬁrst without being speciﬁc on details (these will come later)

and yet, we have a pattern

I somebody assumes a ZFC-based metatheory and proves a cute mathematical result in formal economics voting theory, intergenerational social choice . . .

I decades later, papers are still published debating if something useful has been thus achieved

10/54 and yet, we have a pattern

I somebody assumes a ZFC-based metatheory and proves a cute mathematical result in formal economics voting theory, intergenerational social choice . . .

I decades later, papers are still published debating if something useful has been thus achieved I some instances, ﬁrst without being speciﬁc on details (these will come later)

10/54 ﬁghting the same monster: on the seas, in the air, in the hills

Many are the tales that are told . . . I think that no idea is so outlandish that it should not be considered and viewed with a searching, but at the same time, I hope, with a steady eye.

11/54 Kirman and Sonderman 1972 argue via measure theory and Stone-Cechˇ compactiﬁcations that such social welfare functions are actually invisible dictators “Arrow’s theorem, many agents, and invisible dictators”, Journal of Economic Theory Lauwers and van Liedekerke 1995 claim aggregation procedures corresponding to nonprincipal ultraﬁlters are not appropriate in the context of social choice and exhibit an insuperable arbitrariness in construction “Ultraproducts and aggregation”, Journal of Mathematical Economics Mihara 1997 adds that such non-dictatorial social welfare functions are not computable “Arrow’s theorem, countably many agents, and more visible invisible dictators”, Journal of Mathematical Economics

inﬁnite“solutions” of Arrow’s impossibility

Fishburn 1970 claims solutions exist relative to ZFC “Arrow’s impossibility theorem: Concise proof and inﬁnite voters”, Journal of Economic Theory

12/54 Lauwers and van Liedekerke 1995 claim aggregation procedures corresponding to nonprincipal ultraﬁlters are not appropriate in the context of social choice and exhibit an insuperable arbitrariness in construction “Ultraproducts and aggregation”, Journal of Mathematical Economics Mihara 1997 adds that such non-dictatorial social welfare functions are not computable “Arrow’s theorem, countably many agents, and more visible invisible dictators”, Journal of Mathematical Economics

inﬁnite“solutions” of Arrow’s impossibility

12/54 Mihara 1997 adds that such non-dictatorial social welfare functions are not computable “Arrow’s theorem, countably many agents, and more visible invisible dictators”, Journal of Mathematical Economics

inﬁnite“solutions” of Arrow’s impossibility

Fishburn 1970 claims solutions exist relative to ZFC “Arrow’s impossibility theorem: Concise proof and infinite voters”, Journal of Economic Theory Kirman and Sonderman 1972 argue via measure theory and Stone-Cechˇ compactifications that such social welfare functions are actually invisible dictators “Arrow’s theorem, many agents, and invisible dictators”, Journal of Economic Theory Lauwers and van Liedekerke 1995 claim aggregation procedures corresponding to nonprincipal ultrafilters are not appropriate in the context of social choice and exhibit an insuperable arbitrariness in construction “Ultraproducts and aggregation”, Journal of Mathematical Economics

12/54 inﬁnite“solutions” of Arrow’s impossibility

12/54 Zame 2007 not only cannot be shown to exist on the basis of ZF + DC but is nonmeasurable and undefinable even relatively to full ZFC “Can intergenerational equity be operationalized?”, Theoretical Eco- nomics Chichilnisky 1996 problems in carbon emissions, resource allocation, biodiversity and sustainable development all solved by using finitely additive measures on integers “An axiomatic approach to sustainable development”, Social Choice and Welfare Lauwers 2012 The Chichilnisky criterion is based upon . . . a measure which is a non-constructible object and has therefore no explicit description “Intergenerational equity, efficiency, and constructibility”, Journal of Economic Theory

another example: intergenerational social choice Svensson 1980 an intergenerational ethical preference relation exists relative to ZFC “Equity among generations”, Econometrica

13/54 Chichilnisky 1996 problems in carbon emissions, resource allocation, biodiversity and sustainable development all solved by using ﬁnitely additive measures on integers “An axiomatic approach to sustainable development”, Social Choice and Welfare Lauwers 2012 The Chichilnisky criterion is based upon . . . a measure which is a non-constructible object and has therefore no explicit description “Intergenerational equity, efﬁciency, and constructibility”, Journal of Economic Theory

13/54 Lauwers 2012 The Chichilnisky criterion is based upon . . . a measure which is a non-constructible object and has therefore no explicit description “Intergenerational equity, efﬁciency, and constructibility”, Journal of Economic Theory

another example: intergenerational social choice Svensson 1980 an intergenerational ethical preference relation exists relative to ZFC “Equity among generations”, Econometrica Zame 2007 not only cannot be shown to exist on the basis of ZF + DC but is nonmeasurable and undeﬁnable even relatively to full ZFC “Can intergenerational equity be operationalized?”, Theoretical Eco- nomics Chichilnisky 1996 problems in carbon emissions, resource allocation, biodiversity and sustainable development all solved by using ﬁnitely additive measures on integers “An axiomatic approach to sustainable development”, Social Choice and Welfare

13/54 another example: intergenerational social choice Svensson 1980 an intergenerational ethical preference relation exists relative to ZFC “Equity among generations”, Econometrica Zame 2007 not only cannot be shown to exist on the basis of ZF + DC but is nonmeasurable and undefinable even relatively to full ZFC “Can intergenerational equity be operationalized?”, Theoretical Eco- nomics Chichilnisky 1996 problems in carbon emissions, resource allocation, biodiversity and sustainable development all solved by using finitely additive measures on integers “An axiomatic approach to sustainable development”, Social Choice and Welfare Lauwers 2012 The Chichilnisky criterion is based upon . . . a measure which is a non-constructible object and has therefore no explicit description “Intergenerational equity, efficiency, and constructibility”, Journal of Economic Theory 13/54 recap of ZF(C)

14/54 Zermelo-Fraenkel set theory ZF is formulated in the ﬁrst-order language LZF with one binary primitive ∈ and numerous standard abbreviations like x = {y, z}, ∅, x ∩ y = ∅, ∃x ∈ y.φ(x), ∃!x ∈ y.φ(x), x ⊆ y, etc.

Axioms

Axiom of Extensionality ∀xy.(∀z.z ∈ x ↔ z ∈ y) → x = y. Axiom of Regularity ∀x.x 6= ∅ → ∃y ∈ x.x ∩ y = ∅. Axiom of Union ∀x∃y∀z.z ∈ y ↔ ∃v ∈ x.z ∈ v. Axiom of Powerset ∀x∃y∀z.z ∈ y ↔ ∀v ∈ z.v ∈ x. Axiom of Inﬁnity ∃x.∅ ∈ x ∧ ∀y.y ∈ x → y ∪ {y} ∈ x. Axiom scheme of Restricted Comprehension (also known as Speciﬁcation or Separation) For any LZF formula φ(x, z) with no occurrences of y,

∀x∃y∀z.z ∈ y ↔ (z ∈ x ∧ φ(x, z)).

Axiom scheme of Replacement (also known as Collection) For any LZF formula φ(x, y, z, w) with no occurrences of v,

∀wz.(∀x ∈ w∃!y.φ(x, y, z, w)) → ∃v∀x ∈ w∃y ∈ v.φ(x, y, z, w).

The Zermelo–Fraenkel set theory with Choice( ZFC) extends the above set of axioms with Axiom of Choice (AC)

∀w.(∀x ∈ w.x 6= ∅ ∧ ∀y ∈ w.x 6= y → x ∩ y = ∅) → ∃c∀x ∈ w.∃!y ∈ x.y ∈ c).

15/54 Axiom of Countable Choice (ACω) Every countable family of non-empty sets has a choice function. A somewhat stronger principle proposed by Bernays 1942: Axiom of Dependent Choice (DC) For any X 6= ∅ and any R ⊆ X × X such that ∀x∃y.xRy, there exists f : ω → X s.t. ∀n ∈ ω.f (xn)Rf (xn+1).

Lemma ZF + DC ` ACω. Proof. Let W := {Wn}n∈ω be a family of non-empty sets. Consider X := {(w1,... wn) | n ∈ ω, ∀i ≤ n, wi ∈ Wi}. Deﬁne S (w1,... wm)R(w1,... wm, w) for any w1,..., wm, w ∈ {Wn}n∈ω. A direct application of DC yields a choice function for W.

16/54 Lemma The following facts are provable in ZF + ACω:

I Every inﬁnite set has a countable subset.

I The union of countably many countable sets is countable.

I Cauchy-style (e − δ) and Heine-style (limits of sequences) deﬁnitions of continuity, closedness and compactness are equivalent.

I Every subspace of a separable metric space is separable.

I Lebesgue measure and meager sets have the property of countable additivity.

Lemma The following facts are provable in ZF + DC:

I A linearly ordered set is well-founded iff it contains no inﬁnite descending sequence.

I Urysohn’s Lemma: if X, Y are disjoint closed sets in a T4-space S, then there is a continuous function from S to [0, 1] which takes the value 1 everywhere in X and 0 everywhere in Y. 17/54 some very important consequences of AC...

Strong BPI Every proper filter in a boolean algebra can be extended to an ultrafilter. Weak BPI Every boolean algebra contains an ultrafilter.

Theorem I Over ZF, the weak and the strong variant of BPI are equivalent. I ZFC ` BPI. I ZF + DC 0 BPI. I ZF + BPI 0 AC.

Theorem ZF + BPI implies the existence of a non-Lebesgue measurable set. Proof. Follows from a 1938 result by Sierpinski´ that a free ultraﬁlter over ω understood as a set of reals is not Lebesgue measurable. Vitali 1905: the very ﬁrst proof that ZFC implies the existence of a nonmeasurable set

18/54 I critical or distrustful: Brouwer, Borel, Baire, Lebesgue, Pasch, Lusin, Peano . . . I even in Kuratowski & Mostowski 1968 book, theorems relying AC marked with the ◦ sign (a custom borrowed from Sierpinski)´ I note: permutation models used to show independence of AC from ZFA can be of interest from the point of view of Pareto-compatible permutations and anonymity conditions I iterative concept of set (the conceptual basis of ZFC) conceptually neutral wrt AC (Boolos 1971) I more about more recent constructive criticism at the end

. . . it may surprise scholars working in the ﬁeld of abstract or applied set theory that even after more than half a century of utilizing the axiom of choice and the well-ordering theorem, a number of ﬁrst-rate mathematicians (especially French) have not essentially changed their distrustful attitude; not even such as have been working most successfully in the domain of point sets and of real functions . . .

Fraenkel et al. Foundations of Set Theory

. . . at the moment (1904) when the axiom, explicitly formulated, was used by Zermelo to prove and conﬁrm one of the earliest assertions of Cantor, viz. the well-ordering theorem, mathematical journals were ﬂooded with critical notes rejecting the proof, mostly arguing that our axiom was either illegitimate or meaningless . . .

19/54 I even in Kuratowski & Mostowski 1968 book, theorems relying AC marked with the ◦ sign (a custom borrowed from Sierpinski)´ I note: permutation models used to show independence of AC from ZFA can be of interest from the point of view of Pareto-compatible permutations and anonymity conditions I iterative concept of set (the conceptual basis of ZFC) conceptually neutral wrt AC (Boolos 1971) I more about more recent constructive criticism at the end

Fraenkel et al. Foundations of Set Theory

I critical or distrustful: Brouwer, Borel, Baire, Lebesgue, Pasch, Lusin, Peano . . .

19/54 I note: permutation models used to show independence of AC from ZFA can be of interest from the point of view of Pareto-compatible permutations and anonymity conditions I iterative concept of set (the conceptual basis of ZFC) conceptually neutral wrt AC (Boolos 1971) I more about more recent constructive criticism at the end

Fraenkel et al. Foundations of Set Theory

19/54 I iterative concept of set (the conceptual basis of ZFC) conceptually neutral wrt AC (Boolos 1971) I more about more recent constructive criticism at the end

Fraenkel et al. Foundations of Set Theory

19/54 I more about more recent constructive criticism at the end

Fraenkel et al. Foundations of Set Theory

19/54 . . . at the moment (1904) when the axiom, explicitly formulated, was used by Zermelo to prove and conﬁrm one of the earliest assertions of Cantor, viz. the well-ordering theorem, mathematical journals were ﬂooded with critical notes rejecting the proof, mostly arguing that our axiom was either illegitimate or meaningless . . .

Fraenkel et al. Foundations of Set Theory

I critical or distrustful: Brouwer, Borel, Baire, Lebesgue, Pasch, Lusin, Peano . . . I even in Kuratowski & Mostowski 1968 book, theorems relying AC marked with the ◦ sign (a custom borrowed from Sierpinski)´ I note: permutation models used to show independence of AC from ZFA can be of interest from the point of view of Pareto-compatible permutations and anonymity conditions I iterative concept of set (the conceptual basis of ZFC) conceptually neutral wrt AC (Boolos 1971) I more about more recent constructive criticism at the end 19/54 . . . there are diverse distinct concepts of set, each instantiated in a corresponding set-theoretic universe, which exhibit diverse set-theoretic truths. Each such universe exists independently in the same Platonic sense that proponents of the universe view regard their universe to exist. Many of these universes have been already named and intensely studied in set theory

. . . there seems to be no reason to restrict inclusion only to ZFC models, as we can include models of weaker theories ZF, ZF−, KP and so on, perhaps even down to second order number theory, as this is set-theoretic in a sense . . .

Hamkins 2012

20/54 the Hildenbrand(–Aumann) criterion

21/54 I how are we going to separate legitimate and illegitimate uses of set theory in formal economics? . . . and perhaps elsewhere . . .

I on a broader conceptual level, we have seen that people expressed the right sentiments quite often I from all the quoted classical references, I think Werner Hildenbrand came closest

I back to our main problem

22/54 I on a broader conceptual level, we have seen that people expressed the right sentiments quite often I from all the quoted classical references, I think Werner Hildenbrand came closest

I back to our main problem I how are we going to separate legitimate and illegitimate uses of set theory in formal economics? . . . and perhaps elsewhere . . .

22/54 I from all the quoted classical references, I think Werner Hildenbrand came closest

I back to our main problem I how are we going to separate legitimate and illegitimate uses of set theory in formal economics? . . . and perhaps elsewhere . . .

I on a broader conceptual level, we have seen that people expressed the right sentiments quite often

22/54 I back to our main problem I how are we going to separate legitimate and illegitimate uses of set theory in formal economics? . . . and perhaps elsewhere . . .

I on a broader conceptual level, we have seen that people expressed the right sentiments quite often I from all the quoted classical references, I think Werner Hildenbrand came closest

22/54 23/54 . . . But, as an economist, our interest in these “ideal economies” is proportional to how much new information can be derived for large but ﬁnite economies. In other words, the relevance of the ideal case to the ﬁnite case has to be established

Werner Hildenbrand, “On economies with many agents”, J. Economic Theory, 1970

Instead of considering a sequence of economies and looking for an asymptotic identity one may reason “in the limit”, i.e., one considers economic systems with more than ﬁnitely many participants and proves that the identity holds in this case.

24/54 Instead of considering a sequence of economies and looking for an asymptotic identity one may reason “in the limit”, i.e., one considers economic systems with more than finitely many participants and proves that the identity holds in this case. . . . But, as an economist, our interest in these “ideal economies” is proportional to how much new information can be derived for large but finite economies. In other words, the relevance of the ideal case to the finite case has to be established

Werner Hildenbrand, “On economies with many agents”, J. Economic Theory, 1970

24/54 I even when considering infinite populations of agents as limit or ideal generalizations of very large finite ones, the results thus obtained should remain effective. I Moving to the limit should mean precisely this and nothing more: making the roleˆ of any particular individual (or perhaps even any particular generation!) infinitesimally negligible. I Not an excuse for skyhooks or pseudo-solutions, which by nature cannot correspond to any meaningful algorithm or definable strategy!

the Hildenbrand criterion

I . . . or the Hildenbrand–Aumann criterion

25/54 I Moving to the limit should mean precisely this and nothing more: making the roleˆ of any particular individual (or perhaps even any particular generation!) inﬁnitesimally negligible. I Not an excuse for skyhooks or pseudo-solutions, which by nature cannot correspond to any meaningful algorithm or deﬁnable strategy!

the Hildenbrand criterion

25/54 I Not an excuse for skyhooks or pseudo-solutions, which by nature cannot correspond to any meaningful algorithm or deﬁnable strategy!

the Hildenbrand criterion

I . . . or the Hildenbrand–Aumann criterion I even when considering infinite populations of agents as limit or ideal generalizations of very large finite ones, the results thus obtained should remain effective. I Moving to the limit should mean precisely this and nothing more: making the roleˆ of any particular individual (or perhaps even any particular generation!) infinitesimally negligible.

25/54 the Hildenbrand criterion

25/54 > there are many equivalents of AC, not all of them immediately recognized w/o training > few people will listen anyway > not a winning strategy to tell entire ﬁelds not to use certain results w/o offering something to stimulate imagination

how to ensure this though?

I ... independence proofs in ZFC an enigma for most economists, even formal ones . . . we’ve seen them being used, but this will always remain a niche

26/54 > there are many equivalents of AC, not all of them immediately recognized w/o training > few people will listen anyway > not a winning strategy to tell entire ﬁelds not to use certain results w/o offering something to stimulate imagination

I . . . it’s easy to try and tell use only DC or don’t use non-principal ultraﬁlters but

how to ensure this though?

I ... independence proofs in ZFC an enigma for most economists, even formal ones . . . we’ve seen them being used, but this will always remain a niche

I . . . even more so varieties of constructive mathematics... I will say more about this towards the end, but see above

how to ensure this though?

I ... independence proofs in ZFC an enigma for most economists, even formal ones . . . we’ve seen them being used, but this will always remain a niche

26/54 > few people will listen anyway > not a winning strategy to tell entire ﬁelds not to use certain results w/o offering something to stimulate imagination

how to ensure this though?

I ... independence proofs in ZFC an enigma for most economists, even formal ones . . . we’ve seen them being used, but this will always remain a niche

26/54 > not a winning strategy to tell entire ﬁelds not to use certain results w/o offering something to stimulate imagination

how to ensure this though?

I ... independence proofs in ZFC an enigma for most economists, even formal ones . . . we’ve seen them being used, but this will always remain a niche

I . . . even more so varieties of constructive mathematics... I will say more about this towards the end, but see above I . . . it’s easy to try and tell use only DC or don’t use non-principal ultraﬁlters but > there are many equivalents of AC, not all of them immediately recognized w/o training > few people will listen anyway

26/54 how to ensure this though?

I ... independence proofs in ZFC an enigma for most economists, even formal ones . . . we’ve seen them being used, but this will always remain a niche

26/54 I . . . an intuitive alternative axiom directly highlighting problems with the nature of inﬁnity and continuity... I . . . preferably not leading to all sorts of paradoxical constructions AC leads to . . . I . . . and ideally, having a nice game-theoretic ﬂavour as this is something economists understand best!

what we’re looking for

I . . . something which instead of just weakening ZFC, would illustrate that rich alternative universes are possible . . .

27/54 I . . . preferably not leading to all sorts of paradoxical constructions AC leads to . . . I . . . and ideally, having a nice game-theoretic ﬂavour as this is something economists understand best!

what we’re looking for

27/54 I . . . and ideally, having a nice game-theoretic ﬂavour as this is something economists understand best!

what we’re looking for

I . . . something which instead of just weakening ZFC, would illustrate that rich alternative universes are possible . . . I . . . an intuitive alternative axiom directly highlighting problems with the nature of inﬁnity and continuity... I . . . preferably not leading to all sorts of paradoxical constructions AC leads to . . .

27/54 what we’re looking for

27/54 28/54 I building formal economics in an universe contradicting AC would seem an even more extravagant exercise . . . although I can imagine that one can get a neat publication or two this way!

I . . . what we’re looking for is an axiom allowing safe constructions: those consistent with ZF + ACω or perhaps even ZF + DC ... I . . . and at the same provided a more intuitive way of killing “illegal” ones than, say, independence proofs . . . I . . . and, of course, there is a set-theoretic principle which does exactly all we want!

just to clarify . . .

I no, I am not looking for a replacement for AC!

29/54 I . . . what we’re looking for is an axiom allowing safe constructions: those consistent with ZF + ACω or perhaps even ZF + DC ... I . . . and at the same provided a more intuitive way of killing “illegal” ones than, say, independence proofs . . . I . . . and, of course, there is a set-theoretic principle which does exactly all we want!

just to clarify . . .

I no, I am not looking for a replacement for AC! I building formal economics in an universe contradicting AC would seem an even more extravagant exercise . . . although I can imagine that one can get a neat publication or two this way!

29/54 I . . . and at the same provided a more intuitive way of killing “illegal” ones than, say, independence proofs . . . I . . . and, of course, there is a set-theoretic principle which does exactly all we want!

just to clarify . . .

I . . . what we’re looking for is an axiom allowing safe constructions: those consistent with ZF + ACω or perhaps even ZF + DC ...

29/54 I . . . and, of course, there is a set-theoretic principle which does exactly all we want!

just to clarify . . .

29/54 just to clarify . . .

29/54 the axiom of determinacy (AD)

30/54 I meets all the conditions listed in the preceding slides Mycielski and Steinhaus were careful to state that they did not question the validity of AC in the “absolute universum of sets”. Mycielski called the inconsistency of AD and AC a “sad fact”.

I largely a candidate for a proposition valid in an inner model containing R, especially smallest such model, i.e., L(R) I from the point of view of consistency strength, a much more powerful axiom than AC

the axiom of determinacy (AD)

I proposed by Mycielski and Steinhaus in 1962

31/54 I largely a candidate for a proposition valid in an inner model containing R, especially smallest such model, i.e., L(R) I from the point of view of consistency strength, a much more powerful axiom than AC

the axiom of determinacy (AD)

I proposed by Mycielski and Steinhaus in 1962 I meets all the conditions listed in the preceding slides Mycielski and Steinhaus were careful to state that they did not question the validity of AC in the “absolute universum of sets”. Mycielski called the inconsistency of AD and AC a “sad fact”.

31/54 I from the point of view of consistency strength, a much more powerful axiom than AC

the axiom of determinacy (AD)

I largely a candidate for a proposition valid in an inner model containing R, especially smallest such model, i.e., L(R)

31/54 the axiom of determinacy (AD)

31/54 > the ∃-player makes the ﬁrst move > wins if the sequence created this way belongs to A > otherwise the game is won by the ∀-player

A is what is usually called the payoff for GX(A)

I with every such X and A, associate an inﬁnite two-person game GX(A) with perfect information I two players ∀ and ∃ take turns to choose elements of X:

I if either of the two players has a winning strategy, GX(A) is determined.

I AD says simply that all games of the form Gω(A), i.e., whose payoffs are (identiﬁable with) subsets of R, are determined I sounds natural enough, right? well . . .

ω I consider any set X and A ⊆ X if X = ω, Xω identiﬁed with R via a standard argument

32/54 > the ∃-player makes the ﬁrst move > wins if the sequence created this way belongs to A > otherwise the game is won by the ∀-player

A is what is usually called the payoff for GX(A)

I two players ∀ and ∃ take turns to choose elements of X:

I if either of the two players has a winning strategy, GX(A) is determined.

I AD says simply that all games of the form Gω(A), i.e., whose payoffs are (identiﬁable with) subsets of R, are determined I sounds natural enough, right? well . . .

ω I consider any set X and A ⊆ X if X = ω, Xω identiﬁed with R via a standard argument

I with every such X and A, associate an inﬁnite two-person game GX(A) with perfect information

32/54 > the ∃-player makes the ﬁrst move > wins if the sequence created this way belongs to A > otherwise the game is won by the ∀-player

A is what is usually called the payoff for GX(A)

I if either of the two players has a winning strategy, GX(A) is determined.

I AD says simply that all games of the form Gω(A), i.e., whose payoffs are (identiﬁable with) subsets of R, are determined I sounds natural enough, right? well . . .

ω I consider any set X and A ⊆ X if X = ω, Xω identiﬁed with R via a standard argument

I with every such X and A, associate an inﬁnite two-person game GX(A) with perfect information I two players ∀ and ∃ take turns to choose elements of X:

32/54 > wins if the sequence created this way belongs to A > otherwise the game is won by the ∀-player

A is what is usually called the payoff for GX(A)

I if either of the two players has a winning strategy, GX(A) is determined.

I AD says simply that all games of the form Gω(A), i.e., whose payoffs are (identiﬁable with) subsets of R, are determined I sounds natural enough, right? well . . .

ω I consider any set X and A ⊆ X if X = ω, Xω identiﬁed with R via a standard argument

I with every such X and A, associate an inﬁnite two-person game GX(A) with perfect information I two players ∀ and ∃ take turns to choose elements of X: > the ∃-player makes the ﬁrst move

32/54 > otherwise the game is won by the ∀-player

A is what is usually called the payoff for GX(A)

I if either of the two players has a winning strategy, GX(A) is determined.

I AD says simply that all games of the form Gω(A), i.e., whose payoffs are (identiﬁable with) subsets of R, are determined I sounds natural enough, right? well . . .

ω I consider any set X and A ⊆ X if X = ω, Xω identiﬁed with R via a standard argument

I with every such X and A, associate an inﬁnite two-person game GX(A) with perfect information I two players ∀ and ∃ take turns to choose elements of X: > the ∃-player makes the ﬁrst move > wins if the sequence created this way belongs to A

32/54 I if either of the two players has a winning strategy, GX(A) is determined.

I AD says simply that all games of the form Gω(A), i.e., whose payoffs are (identiﬁable with) subsets of R, are determined I sounds natural enough, right? well . . .

ω I consider any set X and A ⊆ X if X = ω, Xω identiﬁed with R via a standard argument

A is what is usually called the payoff for GX(A)

32/54 I AD says simply that all games of the form Gω(A), i.e., whose payoffs are (identiﬁable with) subsets of R, are determined I sounds natural enough, right? well . . .

ω I consider any set X and A ⊆ X if X = ω, Xω identiﬁed with R via a standard argument

A is what is usually called the payoff for GX(A)

I if either of the two players has a winning strategy, GX(A) is determined.

32/54 I sounds natural enough, right? well . . .

ω I consider any set X and A ⊆ X if X = ω, Xω identiﬁed with R via a standard argument

A is what is usually called the payoff for GX(A)

I if either of the two players has a winning strategy, GX(A) is determined.

I AD says simply that all games of the form Gω(A), i.e., whose payoffs are (identiﬁable with) subsets of R, are determined

32/54 ω I consider any set X and A ⊆ X if X = ω, Xω identiﬁed with R via a standard argument

A is what is usually called the payoff for GX(A)

I if either of the two players has a winning strategy, GX(A) is determined.

I AD says simply that all games of the form Gω(A), i.e., whose payoffs are (identiﬁable with) subsets of R, are determined I sounds natural enough, right? well . . .

32/54 Corollary Under ZF + AD, every ultraﬁlter over ω is principal.

Later on, we will see a natural strategy-stealing argument as used, e.g., in Kanamori’s The Higher Inﬁnite: Large Cardinals in Set Theory from Their Beginnings. We’ll apply it directly in the context of preference aggregation

About the only “paradoxical” consequence of AD I’m aware of follows from Sierpinski´ 1947: R/Q may have more elements than R itself; the former set cannot be even linearly ordered

Theorem (Mycielski, Swierczkowski,´ Mazur, Banach, Davis) Under ZF + AD, every set of reals is Lebesgue measurable, has the Baire property and the perfect set property.

33/54 Later on, we will see a natural strategy-stealing argument as used, e.g., in Kanamori’s The Higher Inﬁnite: Large Cardinals in Set Theory from Their Beginnings. We’ll apply it directly in the context of preference aggregation

About the only “paradoxical” consequence of AD I’m aware of follows from Sierpinski´ 1947: R/Q may have more elements than R itself; the former set cannot be even linearly ordered

Theorem (Mycielski, Swierczkowski,´ Mazur, Banach, Davis) Under ZF + AD, every set of reals is Lebesgue measurable, has the Baire property and the perfect set property.

Corollary Under ZF + AD, every ultraﬁlter over ω is principal.

33/54 About the only “paradoxical” consequence of AD I’m aware of follows from Sierpinski´ 1947: R/Q may have more elements than R itself; the former set cannot be even linearly ordered

Theorem (Mycielski, Swierczkowski,´ Mazur, Banach, Davis) Under ZF + AD, every set of reals is Lebesgue measurable, has the Baire property and the perfect set property.

Corollary Under ZF + AD, every ultraﬁlter over ω is principal.

33/54 Theorem (Mycielski, Swierczkowski,´ Mazur, Banach, Davis) Under ZF + AD, every set of reals is Lebesgue measurable, has the Baire property and the perfect set property.

Corollary Under ZF + AD, every ultraﬁlter over ω is principal.

About the only “paradoxical” consequence of AD I’m aware of follows from Sierpinski´ 1947: R/Q may have more elements than R itself; the former set cannot be even linearly ordered

33/54 ω I consider any A ⊆ X I the existence of a winning strategy for the ∃-player can be reformulated as

∃x0.∀x1.∃x2.... (xi)i∈ω ∈ A,

I the existence of a winning strategy for the ∀-player as

∀x0.∃x1.∀x2.... (xi)i∈ω 6∈ A.

I thus, determinacy of GX(A) can be rewritten as

¬∃x0.∀x1.∃x2.... A(x0, x1, x2 ... ) ↔

∀x0.∃x1.∀x2.... ¬A(x0, x1, x2 ... ).

I one can see AD as an inﬁnitary generalization of the De Morgan law for quantiﬁers

34/54 I the existence of a winning strategy for the ∃-player can be reformulated as

∃x0.∀x1.∃x2.... (xi)i∈ω ∈ A,

I the existence of a winning strategy for the ∀-player as

∀x0.∃x1.∀x2.... (xi)i∈ω 6∈ A.

I thus, determinacy of GX(A) can be rewritten as

¬∃x0.∀x1.∃x2.... A(x0, x1, x2 ... ) ↔

∀x0.∃x1.∀x2.... ¬A(x0, x1, x2 ... ).

I one can see AD as an inﬁnitary generalization of the De Morgan law for quantiﬁers ω I consider any A ⊆ X

34/54 I the existence of a winning strategy for the ∀-player as

∀x0.∃x1.∀x2.... (xi)i∈ω 6∈ A.

I thus, determinacy of GX(A) can be rewritten as

¬∃x0.∀x1.∃x2.... A(x0, x1, x2 ... ) ↔

∀x0.∃x1.∀x2.... ¬A(x0, x1, x2 ... ).

I one can see AD as an inﬁnitary generalization of the De Morgan law for quantiﬁers ω I consider any A ⊆ X I the existence of a winning strategy for the ∃-player can be reformulated as

∃x0.∀x1.∃x2.... (xi)i∈ω ∈ A,

34/54 I thus, determinacy of GX(A) can be rewritten as

¬∃x0.∀x1.∃x2.... A(x0, x1, x2 ... ) ↔

∀x0.∃x1.∀x2.... ¬A(x0, x1, x2 ... ).

I one can see AD as an inﬁnitary generalization of the De Morgan law for quantiﬁers ω I consider any A ⊆ X I the existence of a winning strategy for the ∃-player can be reformulated as

∃x0.∀x1.∃x2.... (xi)i∈ω ∈ A,

I the existence of a winning strategy for the ∀-player as

∀x0.∃x1.∀x2.... (xi)i∈ω 6∈ A.

34/54 I one can see AD as an inﬁnitary generalization of the De Morgan law for quantiﬁers ω I consider any A ⊆ X I the existence of a winning strategy for the ∃-player can be reformulated as

∃x0.∀x1.∃x2.... (xi)i∈ω ∈ A,

I the existence of a winning strategy for the ∀-player as

∀x0.∃x1.∀x2.... (xi)i∈ω 6∈ A.

I thus, determinacy of GX(A) can be rewritten as

¬∃x0.∀x1.∃x2.... A(x0, x1, x2 ... ) ↔

∀x0.∃x1.∀x2.... ¬A(x0, x1, x2 ... ).

34/54 I . . . namely, the law of excluded middle (Diaconescu, Goodman–Myhill) I “natural generalizations” of ﬁnitary laws can conﬂict with each other

I contrast this with the fact that AC can also be seen as an inﬁnitary generalization of a ﬁnitary law . . .

35/54 I “natural generalizations” of ﬁnitary laws can conﬂict with each other

I contrast this with the fact that AC can also be seen as an inﬁnitary generalization of a ﬁnitary law . . . I . . . namely, the law of excluded middle (Diaconescu, Goodman–Myhill)

35/54 I contrast this with the fact that AC can also be seen as an infinitary generalization of a finitary law . . . I . . . namely, the law of excluded middle (Diaconescu, Goodman–Myhill) I “natural generalizations” of finitary laws can conflict with each other

35/54 Kechris If the model of ZF based on L(R) is assumed to satisfy AD, then DC holds therein Woodin In ZFC enriched with axioms ensuring the existence of inﬁnitely many Woodin cardinals with a measurable cardinal above them, L(R) is a model of ZF + AD Woodin ZF + AD is equiconsistent with ZFC enriched with axioms ensuring the existence of inﬁnitely many Woodin cardinals

the ﬁrst two clauses do not say that AD implies ACω! A construction by Woodin, which assumes that L(R) taken as a model of ZF satisﬁes AD, constructs its generic

extension where ACω fails. By the way: a model example of the process described by Hamkins in the multiverse paper

Consistency strength Theorem Mycielski AD implies that every countable family of non-empty subsets of R has a choice function

36/54 Woodin In ZFC enriched with axioms ensuring the existence of inﬁnitely many Woodin cardinals with a measurable cardinal above them, L(R) is a model of ZF + AD Woodin ZF + AD is equiconsistent with ZFC enriched with axioms ensuring the existence of inﬁnitely many Woodin cardinals

the ﬁrst two clauses do not say that AD implies ACω! A construction by Woodin, which assumes that L(R) taken as a model of ZF satisﬁes AD, constructs its generic

extension where ACω fails. By the way: a model example of the process described by Hamkins in the multiverse paper

36/54 Woodin ZF + AD is equiconsistent with ZFC enriched with axioms ensuring the existence of inﬁnitely many Woodin cardinals

the ﬁrst two clauses do not say that AD implies ACω! A construction by Woodin, which assumes that L(R) taken as a model of ZF satisﬁes AD, constructs its generic

extension where ACω fails. By the way: a model example of the process described by Hamkins in the multiverse paper

Consistency strength Theorem Mycielski AD implies that every countable family of non-empty subsets of R has a choice function Kechris If the model of ZF based on L(R) is assumed to satisfy AD, then DC holds therein Woodin In ZFC enriched with axioms ensuring the existence of inﬁnitely many Woodin cardinals with a measurable cardinal above them, L(R) is a model of ZF + AD

36/54 the ﬁrst two clauses do not say that AD implies ACω! A construction by Woodin, which assumes that L(R) taken as a model of ZF satisﬁes AD, constructs its generic

extension where ACω fails. By the way: a model example of the process described by Hamkins in the multiverse paper

36/54 the ﬁrst two clauses do not say that AD implies ACω! A construction by Woodin, which assumes that L(R) taken as a model of ZF satisﬁes AD, constructs its generic

extension where ACω fails. By the way: a model example of the process described by Hamkins in the multiverse paper

36/54 Consistency strength Theorem Mycielski AD implies that every countable family of non-empty subsets of R has a choice function Kechris If the model of ZF based on L(R) is assumed to satisfy AD, then DC holds therein Woodin In ZFC enriched with axioms ensuring the existence of inﬁnitely many Woodin cardinals with a measurable cardinal above them, L(R) is a model of ZF + AD Woodin ZF + AD is equiconsistent with ZFC enriched with axioms ensuring the existence of inﬁnitely many Woodin cardinals

the ﬁrst two clauses do not say that AD implies ACω! A construction by Woodin, which assumes that L(R) taken as a model of ZF satisﬁes AD, constructs its generic

extension where ACω fails. By the way: a model example of the process described by

Hamkins in the multiverse paper 36/54 37/54 38/54 I we are going to follow on its formalization as proposed by Fishburn and Kirman & Sondermann I essentially following his 1963 book earlier version of the result used stronger assumptions

I as the wikipedia entry makes clear, a result which caused decades of discussion

39/54 I essentially following his 1963 book earlier version of the result used stronger assumptions

I as the wikipedia entry makes clear, a result which caused decades of discussion I we are going to follow on its formalization as proposed by Fishburn and Kirman & Sondermann

39/54 I as the wikipedia entry makes clear, a result which caused decades of discussion I we are going to follow on its formalization as proposed by Fishburn and Kirman & Sondermann I essentially following his 1963 book earlier version of the result used stronger assumptions

39/54 asymmetry aRb implies not bRa, negative transitivity for any a, b, c ∈ Options, aRb implies aRc or cRb

I PO(Options) be the set of preference orders on Options, i.e., those R ⊆ Options × Options satisfying

I these conditions imply transitivity I furthermore, denote Situations := Voters → PO(Options) and SWFs := Situations → PO(Options) SWF stands for “Social Welfare Function” a.k.a. preference aggregation rule or ranked voting electoral system Situations a.k.a. proﬁles of preferences

I let Voters, Options be arbitrary sets

40/54 asymmetry aRb implies not bRa, negative transitivity for any a, b, c ∈ Options, aRb implies aRc or cRb I these conditions imply transitivity I furthermore, denote Situations := Voters → PO(Options) and SWFs := Situations → PO(Options) SWF stands for “Social Welfare Function” a.k.a. preference aggregation rule or ranked voting electoral system Situations a.k.a. proﬁles of preferences

I let Voters, Options be arbitrary sets I PO(Options) be the set of preference orders on Options, i.e., those R ⊆ Options × Options satisfying

40/54 negative transitivity for any a, b, c ∈ Options, aRb implies aRc or cRb I these conditions imply transitivity I furthermore, denote Situations := Voters → PO(Options) and SWFs := Situations → PO(Options) SWF stands for “Social Welfare Function” a.k.a. preference aggregation rule or ranked voting electoral system Situations a.k.a. proﬁles of preferences

I let Voters, Options be arbitrary sets I PO(Options) be the set of preference orders on Options, i.e., those R ⊆ Options × Options satisfying asymmetry aRb implies not bRa,

40/54 I these conditions imply transitivity I furthermore, denote Situations := Voters → PO(Options) and SWFs := Situations → PO(Options) SWF stands for “Social Welfare Function” a.k.a. preference aggregation rule or ranked voting electoral system Situations a.k.a. proﬁles of preferences

I let Voters, Options be arbitrary sets I PO(Options) be the set of preference orders on Options, i.e., those R ⊆ Options × Options satisfying asymmetry aRb implies not bRa, negative transitivity for any a, b, c ∈ Options, aRb implies aRc or cRb

40/54 I furthermore, denote Situations := Voters → PO(Options) and SWFs := Situations → PO(Options) SWF stands for “Social Welfare Function” a.k.a. preference aggregation rule or ranked voting electoral system Situations a.k.a. proﬁles of preferences

40/54 I let Voters, Options be arbitrary sets I PO(Options) be the set of preference orders on Options, i.e., those R ⊆ Options × Options satisfying asymmetry aRb implies not bRa, negative transitivity for any a, b, c ∈ Options, aRb implies aRc or cRb I these conditions imply transitivity I furthermore, denote Situations := Voters → PO(Options) and SWFs := Situations → PO(Options) SWF stands for “Social Welfare Function” a.k.a. preference aggregation rule or ranked voting electoral system Situations a.k.a. proﬁles of preferences

40/54 Options (A1) |Options| ≥ 3.

Unanimity/Pareto efﬁciency (A3)

af [Voters]b implies aσ(f )b,

Independence of Irrelevant Alternatives (A4)

f =a,b g implies σ(f ) =a,b σ(g).

I one rather trivial axiom being imposed on Options:

I further axioms imposed on SWFs for all a, b ∈ Options, f , g ∈ Situations and σ ∈ SWFs:

I some additional conventions for a, b ∈ Options, v ∈ Voters, U ⊆ Voters and f , g ∈ Situations: af [U]b if for all v ∈ U, af (v)b,

f (v) =a,b g(v) if (af (v)b iff ag(v)b) and( bf (v)a iff bg(v)a),

f =a,b g if for all v ∈ Voters, f (v) =a,b g(v).

41/54 Unanimity/Pareto efﬁciency (A3)

af [Voters]b implies aσ(f )b,

Independence of Irrelevant Alternatives (A4)

f =a,b g implies σ(f ) =a,b σ(g).

Options (A1) |Options| ≥ 3.

I further axioms imposed on SWFs for all a, b ∈ Options, f , g ∈ Situations and σ ∈ SWFs:

I some additional conventions for a, b ∈ Options, v ∈ Voters, U ⊆ Voters and f , g ∈ Situations: af [U]b if for all v ∈ U, af (v)b,

f (v) =a,b g(v) if (af (v)b iff ag(v)b) and( bf (v)a iff bg(v)a),

f =a,b g if for all v ∈ Voters, f (v) =a,b g(v). I one rather trivial axiom being imposed on Options:

41/54 Unanimity/Pareto efﬁciency (A3)

af [Voters]b implies aσ(f )b,

Independence of Irrelevant Alternatives (A4)

f =a,b g implies σ(f ) =a,b σ(g).

I further axioms imposed on SWFs for all a, b ∈ Options, f , g ∈ Situations and σ ∈ SWFs:

I some additional conventions for a, b ∈ Options, v ∈ Voters, U ⊆ Voters and f , g ∈ Situations: af [U]b if for all v ∈ U, af (v)b,

f (v) =a,b g(v) if (af (v)b iff ag(v)b) and( bf (v)a iff bg(v)a),

f =a,b g if for all v ∈ Voters, f (v) =a,b g(v). I one rather trivial axiom being imposed on Options: Options (A1) |Options| ≥ 3.

41/54 Unanimity/Pareto efﬁciency (A3)

af [Voters]b implies aσ(f )b,

Independence of Irrelevant Alternatives (A4)

f =a,b g implies σ(f ) =a,b σ(g).

I some additional conventions for a, b ∈ Options, v ∈ Voters, U ⊆ Voters and f , g ∈ Situations: af [U]b if for all v ∈ U, af (v)b,

f (v) =a,b g(v) if (af (v)b iff ag(v)b) and( bf (v)a iff bg(v)a),

f =a,b g if for all v ∈ Voters, f (v) =a,b g(v). I one rather trivial axiom being imposed on Options: Options (A1) |Options| ≥ 3.

I further axioms imposed on SWFs for all a, b ∈ Options, f , g ∈ Situations and σ ∈ SWFs:

41/54 Independence of Irrelevant Alternatives (A4)

f =a,b g implies σ(f ) =a,b σ(g).

I some additional conventions for a, b ∈ Options, v ∈ Voters, U ⊆ Voters and f , g ∈ Situations: af [U]b if for all v ∈ U, af (v)b,

f (v) =a,b g(v) if (af (v)b iff ag(v)b) and( bf (v)a iff bg(v)a),

f =a,b g if for all v ∈ Voters, f (v) =a,b g(v). I one rather trivial axiom being imposed on Options: Options (A1) |Options| ≥ 3.

I further axioms imposed on SWFs for all a, b ∈ Options, f , g ∈ Situations and σ ∈ SWFs: Unanimity/Pareto efﬁciency (A3)

af [Voters]b implies aσ(f )b,

41/54 I some additional conventions for a, b ∈ Options, v ∈ Voters, U ⊆ Voters and f , g ∈ Situations: af [U]b if for all v ∈ U, af (v)b,

f (v) =a,b g(v) if (af (v)b iff ag(v)b) and( bf (v)a iff bg(v)a),

f =a,b g if for all v ∈ Voters, f (v) =a,b g(v). I one rather trivial axiom being imposed on Options: Options (A1) |Options| ≥ 3.

I further axioms imposed on SWFs for all a, b ∈ Options, f , g ∈ Situations and σ ∈ SWFs: Unanimity/Pareto efﬁciency (A3)

af [Voters]b implies aσ(f )b,

Independence of Irrelevant Alternatives (A4)

f =a,b g implies σ(f ) =a,b σ(g).

41/54 I Given any σ ∈ SWFs, deﬁne

Uσ := {U ⊆ Voters | ∃a, b ∈ Options ∃f ∈ Situations.af [U]b & bf [Voters − U]a & aσ(f )b}, 0 Uσ := {U ⊆ Voters | ∃a, b ∈ Options ∀f ∈ Situations.(af [U]b & bf [Voters − U]a ⇒ aσ(f )b)}, 00 Uσ := {U ⊆ Voters | ∀a, b ∈ Options ∀f ∈ Situations.(af [U]b & bf [Voters − U]a ⇒ aσ(f )b)}.

Lemma (Kirman and Sondermann) For any σ ∈ SWFs satisfying unanimity (A3) and independence (A4) above: 0 00 > Uσ = Uσ = Uσ ; 0 00 > whenever Options satisfies (A1), Uσ (= Uσ = Uσ ) is an ultrafilter on Voters and furthermore: > Uσ is the unique ultrafilter U with the property

∀U ∈ U, a, b ∈ Options, f ∈ Situations.(af [U]b ⇒ aσ(f )b).

> Uσ is principal iff σ satisﬁes

Dictatorship (non-A5) there is v0 ∈ Voters such that for any f ∈ Situations, a, b ∈ Options, af (v0)b implies aσ(f )b. 42/54 I for an infinite countable set of Voters, assume that Dictatorship does not hold I define a game where the players pick finite, mutually disjoint subsets of Voters.

I The ∃-player wins if the sum E of all Voters chosen by her belongs to Uσ, i.e., if it holds that

∀a, b ∈ Options, f ∈ Situations.(af [E]b & bf [Voters − E]a ⇒ aσ(f )b).

I A play of this game is of the form e0, a0, e1, a1 . . . where ei and ai are ﬁnite, mutually disjoints subsets of Voters S S I The sums of these choices denoted as E = ei and A = ai. i∈ω i∈ω

Theorem For any countable set of Voters, any Options satisfying (A1) and any σ satisfying (A3) and (A4), ZF + AD implies the Dictatorship (non-A5) condition.

I if Voters is ﬁnite, Uσ must be a principal ultraﬁlter over Voters

43/54 I deﬁne a game where the players pick ﬁnite, mutually disjoint subsets of Voters.

I The ∃-player wins if the sum E of all Voters chosen by her belongs to Uσ, i.e., if it holds that

∀a, b ∈ Options, f ∈ Situations.(af [E]b & bf [Voters − E]a ⇒ aσ(f )b).

I A play of this game is of the form e0, a0, e1, a1 . . . where ei and ai are ﬁnite, mutually disjoints subsets of Voters S S I The sums of these choices denoted as E = ei and A = ai. i∈ω i∈ω

Theorem For any countable set of Voters, any Options satisfying (A1) and any σ satisfying (A3) and (A4), ZF + AD implies the Dictatorship (non-A5) condition.

I if Voters is finite, Uσ must be a principal ultrafilter over Voters I for an infinite countable set of Voters, assume that Dictatorship does not hold

43/54 I The ∃-player wins if the sum E of all Voters chosen by her belongs to Uσ, i.e., if it holds that

∀a, b ∈ Options, f ∈ Situations.(af [E]b & bf [Voters − E]a ⇒ aσ(f )b).

I A play of this game is of the form e0, a0, e1, a1 . . . where ei and ai are ﬁnite, mutually disjoints subsets of Voters S S I The sums of these choices denoted as E = ei and A = ai. i∈ω i∈ω

Theorem For any countable set of Voters, any Options satisfying (A1) and any σ satisfying (A3) and (A4), ZF + AD implies the Dictatorship (non-A5) condition.

I if Voters is finite, Uσ must be a principal ultrafilter over Voters I for an infinite countable set of Voters, assume that Dictatorship does not hold I define a game where the players pick finite, mutually disjoint subsets of Voters.

43/54 I A play of this game is of the form e0, a0, e1, a1 . . . where ei and ai are ﬁnite, mutually disjoints subsets of Voters S S I The sums of these choices denoted as E = ei and A = ai. i∈ω i∈ω

Theorem For any countable set of Voters, any Options satisfying (A1) and any σ satisfying (A3) and (A4), ZF + AD implies the Dictatorship (non-A5) condition.

I The ∃-player wins if the sum E of all Voters chosen by her belongs to Uσ, i.e., if it holds that

∀a, b ∈ Options, f ∈ Situations.(af [E]b & bf [Voters − E]a ⇒ aσ(f )b).

43/54 S S I The sums of these choices denoted as E = ei and A = ai. i∈ω i∈ω

Theorem For any countable set of Voters, any Options satisfying (A1) and any σ satisfying (A3) and (A4), ZF + AD implies the Dictatorship (non-A5) condition.

I The ∃-player wins if the sum E of all Voters chosen by her belongs to Uσ, i.e., if it holds that

∀a, b ∈ Options, f ∈ Situations.(af [E]b & bf [Voters − E]a ⇒ aσ(f )b).

I A play of this game is of the form e0, a0, e1, a1 . . . where ei and ai are ﬁnite, mutually disjoints subsets of Voters

43/54 Theorem For any countable set of Voters, any Options satisfying (A1) and any σ satisfying (A3) and (A4), ZF + AD implies the Dictatorship (non-A5) condition.

I The ∃-player wins if the sum E of all Voters chosen by her belongs to Uσ, i.e., if it holds that

∀a, b ∈ Options, f ∈ Situations.(af [E]b & bf [Voters − E]a ⇒ aσ(f )b).

I A play of this game is of the form e0, a0, e1, a1 . . . where ei and ai are ﬁnite, mutually disjoints subsets of Voters S S I The sums of these choices denoted as E = ei and A = ai. i∈ω i∈ω

43/54 I Analogously, a strategy for the ∀-player is thus a function [ 2n+1 τ∀ : (Pﬁn(Voters)) → Pﬁn(Voters), n∈ω

and the ∀-player plays according to τ∀ if the play of the game is of the form

e0, τ∀he0i, e1, τ∀he0, τ∀he0i, e1i ...

I We will show that both the assumption of the existence of a winning τ∃ and the assumption of the existence of a winning τ∀ lead to a contradiction.

proof ctd. I A strategy for the ∃-player is thus a function [ 2n τ∃ : (Pﬁn(Voters)) → Pﬁn(Voters), n∈ω

and the ∃-player plays according to τ∃ if the play of the game is of the form

τ∃∅, a0, τ∃h∅, a0i, a1, τ∃hτ∃∅, a0, τ∃h∅, a0i, a1i ...

44/54 I We will show that both the assumption of the existence of a winning τ∃ and the assumption of the existence of a winning τ∀ lead to a contradiction.

proof ctd. I A strategy for the ∃-player is thus a function [ 2n τ∃ : (Pﬁn(Voters)) → Pﬁn(Voters), n∈ω

and the ∃-player plays according to τ∃ if the play of the game is of the form

τ∃∅, a0, τ∃h∅, a0i, a1, τ∃hτ∃∅, a0, τ∃h∅, a0i, a1i ...

I Analogously, a strategy for the ∀-player is thus a function [ 2n+1 τ∀ : (Pﬁn(Voters)) → Pﬁn(Voters), n∈ω

and the ∀-player plays according to τ∀ if the play of the game is of the form

e0, τ∀he0i, e1, τ∀he0, τ∀he0i, e1i ...

44/54 proof ctd. I A strategy for the ∃-player is thus a function [ 2n τ∃ : (Pﬁn(Voters)) → Pﬁn(Voters), n∈ω

and the ∃-player plays according to τ∃ if the play of the game is of the form

τ∃∅, a0, τ∃h∅, a0i, a1, τ∃hτ∃∅, a0, τ∃h∅, a0i, a1i ...

I Analogously, a strategy for the ∀-player is thus a function [ 2n+1 τ∀ : (Pﬁn(Voters)) → Pﬁn(Voters), n∈ω

and the ∀-player plays according to τ∀ if the play of the game is of the form

e0, τ∀he0i, e1, τ∀he0, τ∀he0i, e1i ...

I We will show that both the assumption of the existence of a winning τ∃ and the assumption of the existence of a

winning τ∀ lead to a contradiction. 44/54 0 I Let now e0, a0, e1, a1 . . . be a play according to τ∀.

I by assumption on τ∃, we have that e0 ∪ A ∈ Uσ.

I As Uσ is nonprincipal, we have A ∈ Uσ.

I Thence, as A ∩ E = ∅, it cannot be the case that E ∈ Uσ and 0 thus τ∀ is a winning strategy; a contradiction.

proof ctd.

I assume ﬁrst a winning τ∃ exists. Deﬁne

0 τ∀hs0, s1 ..., s2ni := τ∃hs1 ..., s2ni − s0.

45/54 I by assumption on τ∃, we have that e0 ∪ A ∈ Uσ.

I As Uσ is nonprincipal, we have A ∈ Uσ.

I Thence, as A ∩ E = ∅, it cannot be the case that E ∈ Uσ and 0 thus τ∀ is a winning strategy; a contradiction.

proof ctd.

I assume ﬁrst a winning τ∃ exists. Deﬁne

0 τ∀hs0, s1 ..., s2ni := τ∃hs1 ..., s2ni − s0.

0 I Let now e0, a0, e1, a1 . . . be a play according to τ∀.

45/54 I As Uσ is nonprincipal, we have A ∈ Uσ.

I Thence, as A ∩ E = ∅, it cannot be the case that E ∈ Uσ and 0 thus τ∀ is a winning strategy; a contradiction.

proof ctd.

I assume ﬁrst a winning τ∃ exists. Deﬁne

0 τ∀hs0, s1 ..., s2ni := τ∃hs1 ..., s2ni − s0.

0 I Let now e0, a0, e1, a1 . . . be a play according to τ∀.

I by assumption on τ∃, we have that e0 ∪ A ∈ Uσ.

45/54 I Thence, as A ∩ E = ∅, it cannot be the case that E ∈ Uσ and 0 thus τ∀ is a winning strategy; a contradiction.

proof ctd.

I assume ﬁrst a winning τ∃ exists. Deﬁne

0 τ∀hs0, s1 ..., s2ni := τ∃hs1 ..., s2ni − s0.

0 I Let now e0, a0, e1, a1 . . . be a play according to τ∀.

I by assumption on τ∃, we have that e0 ∪ A ∈ Uσ.

I As Uσ is nonprincipal, we have A ∈ Uσ.

45/54 proof ctd.

I assume ﬁrst a winning τ∃ exists. Deﬁne

0 τ∀hs0, s1 ..., s2ni := τ∃hs1 ..., s2ni − s0.

0 I Let now e0, a0, e1, a1 . . . be a play according to τ∀.

I by assumption on τ∃, we have that e0 ∪ A ∈ Uσ.

I As Uσ is nonprincipal, we have A ∈ Uσ.

I Thence, as A ∩ E = ∅, it cannot be the case that E ∈ Uσ and 0 thus τ∀ is a winning strategy; a contradiction.

45/54 I this new strategy satisﬁes furthermore the condition that in any play e0, a0, e1, a1 . . . according to it, we have that A ∪ E = Voters.

I as Uσ is an ultraﬁlter, τ∀ is winning and A ∩ E = ∅, we have that A ∈ Uσ. + I But now one can turn τ∀ into a winning strategy for the ∃-player simply by augmenting the input with ∅, a contradiction.

proof ctd.

46/54 I as Uσ is an ultraﬁlter, τ∀ is winning and A ∩ E = ∅, we have that A ∈ Uσ. + I But now one can turn τ∀ into a winning strategy for the ∃-player simply by augmenting the input with ∅, a contradiction.

proof ctd.

I assume now a winning τ∀ exists. Note ﬁrst that a slightly tweaked version of it is a winning strategy too: ( + τ∀hs0, s1,..., s2ni ∪ {n} if n 6∈ s0 ∪ s1 · · · ∪ s2n τ∀ hs0, s1 ..., s2ni := τ∀hs0, s1,..., s2ni otherwise. I this new strategy satisﬁes furthermore the condition that in any play e0, a0, e1, a1 . . . according to it, we have that A ∪ E = Voters.

46/54 + I But now one can turn τ∀ into a winning strategy for the ∃-player simply by augmenting the input with ∅, a contradiction.

proof ctd.

I as Uσ is an ultraﬁlter, τ∀ is winning and A ∩ E = ∅, we have that A ∈ Uσ.

46/54 proof ctd.

46/54 Corollary In ZFC enriched with axioms ensuring the existence of inﬁnitely many Woodin cardinals with a measurable cardinal above them, L(R) is a model of ZF + DC s.t. for any countable set of Voters, any Options satisfying (A1) and any σ satisfying (A3) and (A4), the Dictatorship (non-A5) condition holds.

Related results obtained by Brunner and Mihara, “Arrow’s theorem, Weglorz’ models and the Axiom of Choice”, Mathematical Logic Quarterly 2000

47/54 I AD does not kill all nonprincipal ultrafilters I there are uncountable sets for which AD allows the existence of free ultrafilters. I AD is a good tool to verify conformity with the Hildenbrand criterion, but it is not infallible. I Alternatively, you need to provide an argument why your uncountable set of voters with its nonprincipal ultrafilter meets the Hildenbrand criterion

I we restricted attention to countable sets of voters

48/54 I there are uncountable sets for which AD allows the existence of free ultraﬁlters. I AD is a good tool to verify conformity with the Hildenbrand criterion, but it is not infallible. I Alternatively, you need to provide an argument why your uncountable set of voters with its nonprincipal ultraﬁlter meets the Hildenbrand criterion

I we restricted attention to countable sets of voters I AD does not kill all nonprincipal ultraﬁlters

48/54 I AD is a good tool to verify conformity with the Hildenbrand criterion, but it is not infallible. I Alternatively, you need to provide an argument why your uncountable set of voters with its nonprincipal ultraﬁlter meets the Hildenbrand criterion

I we restricted attention to countable sets of voters I AD does not kill all nonprincipal ultraﬁlters I there are uncountable sets for which AD allows the existence of free ultraﬁlters.

48/54 I Alternatively, you need to provide an argument why your uncountable set of voters with its nonprincipal ultraﬁlter meets the Hildenbrand criterion

I we restricted attention to countable sets of voters I AD does not kill all nonprincipal ultraﬁlters I there are uncountable sets for which AD allows the existence of free ultraﬁlters. I AD is a good tool to verify conformity with the Hildenbrand criterion, but it is not infallible.

48/54 I we restricted attention to countable sets of voters I AD does not kill all nonprincipal ultrafilters I there are uncountable sets for which AD allows the existence of free ultrafilters. I AD is a good tool to verify conformity with the Hildenbrand criterion, but it is not infallible. I Alternatively, you need to provide an argument why your uncountable set of voters with its nonprincipal ultrafilter meets the Hildenbrand criterion

48/54 AD and Intergenerational Equity

49/54 > displays intergenerational equity, i.e., is invariant under ﬁnite permutations of ω, > respects the weak Pareto ordering, i.e., if xn > yn for all n, then x y, > is linear or total, i.e., makes any two elements comparable; the authors in this line of work often choose to use the word complete instead. Theorem The existence of an ethical preference relation on the space of utility streams X = [0, 1]ω is incompatible with ZF + AD. Proof. Zame 2007 shows that a graph of an ethical preference relation would be a nonmeasurable subset of X, hence yielding a nonmeasurable subset of R.

I one seeks a strict ordering on X which is an ethical preference relation, that is:

I this line of work deals with the space of utility streams X = [0, 1]ω.

50/54 > displays intergenerational equity, i.e., is invariant under ﬁnite permutations of ω, > respects the weak Pareto ordering, i.e., if xn > yn for all n, then x y, > is linear or total, i.e., makes any two elements comparable; the authors in this line of work often choose to use the word complete instead. Theorem The existence of an ethical preference relation on the space of utility streams X = [0, 1]ω is incompatible with ZF + AD. Proof. Zame 2007 shows that a graph of an ethical preference relation would be a nonmeasurable subset of X, hence yielding a nonmeasurable subset of R.

I this line of work deals with the space of utility streams X = [0, 1]ω. I one seeks a strict ordering on X which is an ethical preference relation, that is:

50/54 > respects the weak Pareto ordering, i.e., if xn > yn for all n, then x y, > is linear or total, i.e., makes any two elements comparable; the authors in this line of work often choose to use the word complete instead. Theorem The existence of an ethical preference relation on the space of utility streams X = [0, 1]ω is incompatible with ZF + AD. Proof. Zame 2007 shows that a graph of an ethical preference relation would be a nonmeasurable subset of X, hence yielding a nonmeasurable subset of R.

50/54 > is linear or total, i.e., makes any two elements comparable; the authors in this line of work often choose to use the word complete instead. Theorem The existence of an ethical preference relation on the space of utility streams X = [0, 1]ω is incompatible with ZF + AD. Proof. Zame 2007 shows that a graph of an ethical preference relation would be a nonmeasurable subset of X, hence yielding a nonmeasurable subset of R.

50/54 Theorem The existence of an ethical preference relation on the space of utility streams X = [0, 1]ω is incompatible with ZF + AD. Proof. Zame 2007 shows that a graph of an ethical preference relation would be a nonmeasurable subset of X, hence yielding a nonmeasurable subset of R.

I this line of work deals with the space of utility streams X = [0, 1]ω. I one seeks a strict ordering on X which is an ethical preference relation, that is: > displays intergenerational equity, i.e., is invariant under ﬁnite permutations of ω, > respects the weak Pareto ordering, i.e., if xn > yn for all n, then x y, > is linear or total, i.e., makes any two elements comparable; the authors in this line of work often choose to use the word complete instead.

50/54 Proof. Zame 2007 shows that a graph of an ethical preference relation would be a nonmeasurable subset of X, hence yielding a nonmeasurable subset of R.

50/54 I this line of work deals with the space of utility streams X = [0, 1]ω. I one seeks a strict ordering on X which is an ethical preference relation, that is: > displays intergenerational equity, i.e., is invariant under ﬁnite permutations of ω, > respects the weak Pareto ordering, i.e., if xn > yn for all n, then x y, > is linear or total, i.e., makes any two elements comparable; the authors in this line of work often choose to use the word complete instead. Theorem The existence of an ethical preference relation on the space of utility streams X = [0, 1]ω is incompatible with ZF + AD. Proof. Zame 2007 shows that a graph of an ethical preference relation would be a nonmeasurable subset of X, hence yielding a nonmeasurable subset of R. 50/54 coda: broader picture

51/54 I furthermore, in the long run, I think exploiting the connection with ﬁnite model theory is much more promising even at that Logic for Social Behaviour meeting, an economist gave a talk about a problem suspiciously similar to provenance problems in database theory

I incidentally, yet another area where games plays a central roleˆ

I let me clarify again: the roleˆ proposed for AD here is entirely negative

52/54 I incidentally, yet another area where games plays a central roleˆ

I let me clarify again: the roleˆ proposed for AD here is entirely negative I furthermore, in the long run, I think exploiting the connection with ﬁnite model theory is much more promising even at that Logic for Social Behaviour meeting, an economist gave a talk about a problem suspiciously similar to provenance problems in database theory

52/54 I let me clarify again: the roleˆ proposed for AD here is entirely negative I furthermore, in the long run, I think exploiting the connection with ﬁnite model theory is much more promising even at that Logic for Social Behaviour meeting, an economist gave a talk about a problem suspiciously similar to provenance problems in database theory

I incidentally, yet another area where games plays a central roleˆ

52/54 I Bell’s topos-oriented local set theories view and Hellman’s modal structuralism seem to share many common features with Hamkin’s multiverse in fact, the topos-based view can be seen as an extension of multiverse perspective: one can cast forcing arguments in this context

I more broadly, intuitionistic type theories: Martin–Lof¨ distinguishing between intensional and extensional version of AC the extensional version is the problematic one. The intensional one semantically amounts to a harmless statement about dependent products

I still another take provide by Homotopy Type Theory, where the powerful/problematic form of the axiom is the one for propositional truncations I doing social choice in all these settings . . . ? (still, I think the ﬁnite model theory route is the promising one)

I we have already touched on the issue of constructive mathematics: another line of attack on classical reasoning

53/54 I more broadly, intuitionistic type theories: Martin–Lof¨ distinguishing between intensional and extensional version of AC the extensional version is the problematic one. The intensional one semantically amounts to a harmless statement about dependent products

I we have already touched on the issue of constructive mathematics: another line of attack on classical reasoning I Bell’s topos-oriented local set theories view and Hellman’s modal structuralism seem to share many common features with Hamkin’s multiverse in fact, the topos-based view can be seen as an extension of multiverse perspective: one can cast forcing arguments in this context

53/54 I still another take provide by Homotopy Type Theory, where the powerful/problematic form of the axiom is the one for propositional truncations I doing social choice in all these settings . . . ? (still, I think the ﬁnite model theory route is the promising one)

53/54 I doing social choice in all these settings . . . ? (still, I think the ﬁnite model theory route is the promising one)

I still another take provide by Homotopy Type Theory, where the powerful/problematic form of the axiom is the one for propositional truncations

53/54 I we have already touched on the issue of constructive mathematics: another line of attack on classical reasoning I Bell’s topos-oriented local set theories view and Hellman’s modal structuralism seem to share many common features with Hamkin’s multiverse in fact, the topos-based view can be seen as an extension of multiverse perspective: one can cast forcing arguments in this context

53/54 postscriptum

I two more references I did not mention during the lecture: I Shelah, On the Arrow property http://www.sciencedirect.com/science/article/pii/ S0196885804000338

I Rubinstein, Fishburn, Algebraic aggregation theory https://www.sciencedirect.com/science/article/pii/ 0022053186900888

54/54