Search results
Results from the WOW.Com Content Network
In class theories such as Von Neumann–Bernays–Gödel set theory and Morse–Kelley set theory, there is an axiom called the axiom of global choice that is stronger than the axiom of choice for sets because it also applies to proper classes.
Social choice theory is a branch of welfare economics that extends the theory of rational choice to collective decision-making. [1] Social choice studies the behavior of different mathematical procedures ( social welfare functions ) used to combine individual preferences into a coherent whole.
The strength of a belief can vary from person to person. Furthermore, a social axiom is different from a normative belief. Normative beliefs tell us what we ought to do, e.g., be polite to everyone. Social axioms are a guide as to what it is "possible" to do. [4] Leung and Bond (2008) provide a formal definition of social axioms:
Introducing one additional axiom—the nonexistence of Dutch Books, or equivalently that social choice behaves according to the axioms of rational choice—implies that the social choice function must be the utilitarian rule, i.e. the weighting function () must be equal to the utility functions of each individual.
Together with the axiom of choice (see below), these are the de facto standard axioms for contemporary mathematics or set theory. They can be easily adapted to analogous theories, such as mereology. Axiom of extensionality; Axiom of empty set; Axiom of pairing; Axiom of union; Axiom of infinity; Axiom schema of replacement; Axiom of power set ...
A variation on the method of forcing can also be used to demonstrate the consistency and unprovability of the axiom of choice, i.e., that the axiom of choice is independent of ZF. The consistency of choice can be (relatively) easily verified by proving that the inner model L satisfies choice.
The precise definition varies across fields of study. In classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. [3] In modern logic, an axiom is a premise or starting point for reasoning. [4] In mathematics, an axiom may be a "logical axiom" or a "non-logical axiom".
Consider a hypothetical “run-off vote” between say only 2 available social states. The social choice is associated with the sets of rankings for that subset, not with rankings of unavailable social states beyond the subset. Thus, that social choice for the subset is unaffected by say a change in orderings only beyond the subset.