In Arrow’s original version, the unanimity (Pareto) property is a consequence of three properties: non-imposition (or in the 1948 version the procedure has to be non conventional)–meaning essentially that the procedure is not a sort of constant function; monotonicity (in Arrow 1948) later called positive association of social and individual values (Arrow 1950, 1951); and independence of irrelevant alternatives.
Finiteness of N is crucial to get Arrow’s theorem. However, in Arrow’s framework, finiteness of X is not important since what is needed is to get a transitive social preference. Here, with the choice-theoretic setting, finiteness of X is important. With an infinite set of alternatives we would need specific mathematical properties, for instance a topological structure, to guarantee the non-emptiness of choice sets.
A weaker property would be the following: if an alternative is chosen in the larger set and still belongs to the smaller set, it must be chosen in the smaller set–a property often attributed to Herman Chernoff (1954) and developed by Sen (1970, 2017).
In Arrow (2017), one can still read: ‘For example, if you have a three-person election and one is chosen, suppose one of the losers drops out. Now compare that situation when one of the losers never even ran. You should get the same outcome, no matter what system you have anyway.’