What is Bertrand Russell's Paradox?
The Theory of Types and the Unrestricted Comprehension Axiom
Oct 2, 2009
One of the most notable accomplishments in the life of Bertrand Russell was the development of Russell's Paradox as a direct result of the unrestricted comprehension axiom. The axiom stated that any predicate expression, P(x), which contained x as a free variable, would then determine a set in which the members are the objects which satisfy P(x). It was originally introduced by Georg Cantor, and gave form to the idea that any coherent condition could be used to determine a class (a set).
There have been a lot of attempts at resolving Russell's Paradox. While all have failed, the largest majority of them have made their focus the various ways in which the axiom could be abandoned or restricted.
Bertrand Russell's Paradox and the Theory of Types
Bertrand Russell's response to the paradox came along with an introduction to the Theory of Types. The basic idea of this was that reference to troublesome sets could be avoided by more carefully arranging all sentences in question into a hierarchy. This would begin with sentences regarding individuals at the lowest level, then move to sentences about individuals at the next lowest level, then move to sentences about sets of individuals at the next lowest level, etc. These individuals would always be in sets.
Russell was also able to explain why the unrestricted comprehension axiom fails, using what's called the "vicious circle" principle. It was proposed by Henri Poincaré, and Bertrand Russell used it together with the "no class" theory of classes.
According to Russell, the comprehension axiom fails because "propositional functions, such as the function 'x is a set,' should not be applied to themselves since the self-application would involve a vicious circle." By taking this view, it follows that it's possible to reference a collection of different objects for which any given condition holds, but only if those objects are all on the same level. In other words, they all have to be of the same "type."
Acceptability of Bertrand Russell's Theories and Russell's Paradox
The Theory of Types was first introduced by Russell in 1903 in the Principles, but it didn't find it's real, mature expression until the 1908 article Mathematical Logic as Based on the Theory of Types. It was also much-referenced in a monumental work that Russell co-authored with Alfred North Whitehead, Principia Mathematica (1910, 1912, 1913).
When the details of the theory are examined, it can be seen that there are really two versions of it: "simple theory" and "ramified theory." Both of these have come under attack by various individuals. Some people saw these theories as being too weak because they failed in their task to resolve every known paradox that exists.
For other people, the theories were too strong. They disallowed much that was already defined in mathematics. The consistencies of these mathematical definitions violated the vicious circle principle. In response to this concern, Russell introduced the "axiom of reducibility" within the ramified theory.
Although this axiom moved to successfully lessen the vicious circle principle's scope of application, a lot of people still claimed that it was simply too ad hoc, and therefore it wasn't philosophically justified. Like other philosophers, Russell had those who accepted his work and those who shunned it, for various reasons and beliefs.
Russell's Paradox Resources
Readers may also enjoy reading further information on Russell's Paradox, along with information on the Russell Family.
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