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Question. Let $\sf{AI}_{\kappa}$ (Arrow's Impossibility for $\kappa$ voters) denote the assertation that the axioms (A1)-(A4) are inconsistent under the condition $|\sf{Voters}$$|=\kappa$. As a summary of the mentioned results by Arrow, Fishburn, Litak and Woodin, we know:

 

$\sf{ZF}$$~\vdash \forall n\in [1,\aleph_0) ~~~$$\sf{AI}_{n}$

 

$\sf{ZF+AC}$$~\vdash \neg $$\sf{AI}_{\aleph_0}$

 

$\sf{ZF+AD}$$~\vdash$$~\sf{AI}_{\aleph_0}$

 

$\sf{Con(ZFC+\text{$\omega$-many Woodin cardinals and a measurable above them})}$$\Rightarrow$$\sf{Con(ZF+\sf{AI}_{\aleph_0})}$

 

The questions simply are:

 

$\sf{Con(ZFC+\text{?})}$$\Rightarrow$$\sf{Con(ZF+\forall \kappa\in Card^{>0}~~~ \sf{AI}_{\kappa})}$

 

$\sf{Con(ZFC+\text{?})}$$\Rightarrow$$\sf{Con(ZF+\exists \kappa\geq \aleph_{1}~~~ \sf{AI}_{\kappa})}$

 

In other words:

 

1. What is the consistency strength of having Arrow's Impossibility Theorem in societies of arbitrarily large size in the absence of the Axiom of Choice?

2. Define an Arrow cardinal to be an uncountable cardinal $\kappa$ which $\sf{AI}_{\kappa}$ holds. What is the consistency strength of having at least one such cardinal in the absence of $\sf{AC}$?

Question. Let $\sf{AI}_{\kappa}$ (Arrow's Impossibility for $\kappa$ voters) denote the assertation that the axioms (A1)-(A4) are inconsistent under the condition $|\sf{Voters}$$|=\kappa$. As a summary of the mentioned results by Arrow, Fishburn, Litak and Woodin, we know:

$\sf{ZF}$$~\vdash \forall n\in [1,\aleph_0) ~~~$$\sf{AI}_{n}$

$\sf{ZF+AC}$$~\vdash \neg $$\sf{AI}_{\aleph_0}$

$\sf{ZF+AD}$$~\vdash$$~\sf{AI}_{\aleph_0}$

$\sf{Con(ZFC+\text{$\omega$-many Woodin cardinals and a measurable above them})}$$\Rightarrow$$\sf{Con(ZF+\sf{AI}_{\aleph_0})}$

The questions simply are:

$\sf{Con(ZFC+\text{?})}$$\Rightarrow$$\sf{Con(ZF+\forall \kappa\in Card^{>0}~~~ \sf{AI}_{\kappa})}$

$\sf{Con(ZFC+\text{?})}$$\Rightarrow$$\sf{Con(ZF+\exists \kappa\geq \aleph_{1}~~~ \sf{AI}_{\kappa})}$

In other words:

1. What is the consistency strength of having Arrow's Impossibility Theorem in societies of arbitrarily large size in the absence of the Axiom of Choice?

2. Define an Arrow cardinal to be an uncountable cardinal $\kappa$ which $\sf{AI}_{\kappa}$ holds. What is the consistency strength of having at least one such cardinal in the absence of $\sf{AC}$?

Arrow's Impossibility Theorem (published in 1951) is one of the most central (and controversial) theorems in the social choice theory with various applications in politics and economics. Roughly speaking, it asserts that any democratic society in which finite number of voters participate in a fair election gives rise to a dictator, an individual whose personal preference always determines the fate of the whole society! In other words, a true democracy is impossible! In this post, I am going to deal with some of the set-theoretic aspects of this theorem.

Before getting into technicalities let me mention that you are not alone if you think that Arrow's Impossibility Theorem is weird! In fact, the notion of election as an interpretation process which transfers individual preferences of a group of people to a conclusion (i.e. the so-called community's preference) may affect the original information included in the individual votes drastically. So the result of an election highly depends on its interpretation method. Consequently, it is not immediately clear what the community's opinion is even if every single person's opinion is known! A curious reader may take a look at Condorcet paradox and this and this episodes of PBS Infinite series show for a quick introduction to the subject. (For more advanced stuff see this paper of Shelah on Arrow property in which he answered a question of Gil Kalai affirmatively).

This question is loosely inspired by the following paper of Shelah on Arrow property in which he answered a question of Gil Kalai affirmatively.

• Shelah, Saharon, On the Arrow property. Adv. in Appl. Math. 34 (2005), no. 2, 217–251. (MR2110551)(Arxiv link)

Arrow's Impossibility Theorem (published in 1951) is one of the most central (and controversial) theorems in the social choice theory with various applications in politics and economics. Roughly speaking, it asserts that any democratic society in which finite number of voters participate in a fair election gives rise to a dictator, an individual whose personal preference always determines the fate of the whole society! In other words, a true democracy is impossible! The following post deals with some of the set-theoretic aspects of this theorem. The terminology and notation that is being used are standard.

Before getting into technicalities let me mention that you are not alone if you think that Arrow's Impossibility Theorem is weird! In fact, the notion of election as an interpretation process which transfers individual preferences of a group of people to a conclusion (i.e. the so-called community's preference) may affect the original information included in the individual votes drastically. So the result of an election highly depends on its interpretation method. Consequently, it is not immediately clear what the community's opinion is even if every single person's opinion is known! A curious reader may take a look at Condorcet paradox and this and this episodes of PBS Infinite series show for a quick introduction to the subject. See also this related MathOverflow question: Is there a truly general voting impossibility theorem that applies to real elections?

Question. Let $\sf{AI}_{\kappa}$ (Arrow's Impossibility for $\kappa$ voters) denote the assertation that the axioms (A1)-(A4) are consistent under the condition $|\sf{Voters}$$|=\kappa$. As a summary of the mentioned results, we know:

Question. Let $\sf{AI}_{\kappa}$ (Arrow's Impossibility for $\kappa$ voters) denote the assertation that the axioms (A1)-(A4) are inconsistent under the condition $|\sf{Voters}$$|=\kappa$. As a summary of the mentioned results by Arrow, Fishburn, Litak and Woodin, we know: