Vicious circle principle

Mathematical-logical tenet purposed to disallow circular reasoning The vicious circle principle is a principle that was endorsed by many predicativist mathematicians in the early 20th century to prevent contradictions. The principle states that no object or property may be introduced by a definition that depends on that object or property itself. In addition to ruling out definitions that are explicitly circular (like "an object has property P iff it is not next to anything that has property P"), this principle rules out definitions that quantify over domains which include the entity being defined. Thus, it blocks Russell's paradox, which defines a set R that contains all sets which do not contain themselves. This definition is blocked because it defines a new set in terms of the totality of all sets, of which this new set would itself be a member. However, it also blocks one standard definition of the natural numbers. First, we define a property as being "hereditary" if, whenever a number n has the property, so does n +1. Then we say that x has the property of being a natural number if and only if it has every hereditary property that 0 has. This definition is blocked, because it