In the mathematical theory of social welfare, it is possible to create a list of axioms that lead to a contradiction. For example, in voting theory, the following axioms are considered reasonable for a voting system to qualify as being fair:

1. Each voter can have any set of rational preferences. This requirement is called “universal admissibility”.

2. If a voter prefers Candidate A to Candidate B, and Candidate B to Candidate C, then he/she prefers A to C. This requirement is called “transitivity”.

3. If every voter prefers A to B, then the group prefers A to B. This is sometimes called the “unanimity” condition.

4. If every voter prefers A to B, then any change in preferences that does not affect this relationship must not affect the group preference for A over B. For example, if a set of historians unanimously decides that Abraham Lincoln was a better president than Chester A. Arthur, a changing opinion of Bill Clinton should not affect this decision. This more subtle requirement is called “independence from irrelevant alternatives”.

5. There are no dictators. In other words, no individual exists whose preferences determines the preferences of the group.

The mathematical economist Kenneth Arrow showed in a landmark paper that one obtains a contradiction if all five assumptions are assumed to hold. In fact, Assumptions (1) - (4) imply the existence of a dictator. However, these assumptions seem fairly reasonable and consistent, so the fact that they are contradictory is why Arrow named his paper “A Difficulty in the Concept of Social Welfare”. His result is known nowadays as Arrow's Impossibility Theorem.

In the mathematical theory of social welfare, it is possible to create a list of axioms that lead to a contradiction. For example, in voting theory, the following axioms for a voting system are considered reasonable in order for the system to qualify as being fair:

1. Each voter can have any set of rational preferences. This requirement is called “universal admissibility”.

2. If a voter prefers Candidate A to Candidate B, and Candidate B to Candidate C, then he/she prefers A to C. This requirement is called “transitivity”.

3. If every voter prefers A to B, then the group prefers A to B. This is sometimes called the “unanimity” condition.

4. If every voter prefers A to B, then any change in preferences that does not affect this relationship must not affect the group preference for A over B. For example, if a set of historians unanimously decides that Abraham Lincoln was a better president than Chester A. Arthur, a changing opinion of Bill Clinton should not affect this decision. This more subtle requirement is called “independence from irrelevant alternatives”.

5. There are no dictators. In other words, no voter exists whose preferences determine the preferences of the whole group.

The mathematical economist Kenneth Arrow showed in a landmark paper (stemming from his PhD thesis) that one obtains a contradiction if all five assumptions are assumed to hold. In fact, Assumptions (1) - (4) imply the existence of a dictator. However, these assumptions seem fairly reasonable and consistent, so the fact that they are contradictory is why Arrow named his paper “A Difficulty in the Concept of Social Welfare”. His result is known nowadays as Arrow's Impossibility Theorem.