by Andrew Edge

Created on: July 07, 2009

It would not be at all unusual for an individual to believe that there exist only two kinds of meaningful propositions. First, there are those propositions (or assertions) that are demonstrably true or false. An example of one of these is the proposition that it is currently raining in a particular locale. In order to determine whether this is or is not the case one must simply turn to various widely accepted methods of gathering contemporaneous weather data. Alternatively, there are those propositions that are not demonstrably true or false. These would be the propositions that indeed are either true or false but we, as mankind, do not yet have enough information to make the proper determination. A fine example of this, of course, is the proposition that God exists. Logically, whether or not God exists is demonstrable but the human race has not yet developed the means and/or had the opportunity to make a widely accepted determination as to which is the case.

What follows from the above discussion is the idea that all meaningful propositions that assert a truth or falsehood are decidable if the totality of facts concerning the universe were readily available. It will be shown below, however, that strictly speaking this very idea is a false one. Enter: Bertrand Russell.

At the turn of the Twentieth Century Bertrand Russell (1872-1970) was among a small group of brilliant intellectuals who were examining the foundations of mathematics. Russell believed and hoped that the whole of mathematics could be grounded in a framework of formal logic. Russell felt, in fact, that the not fully formal art of mathematics was simply logic disguised in a different language. His work at this time strived to demonstrate that the processes behind formal logic were exactly equivalent to the processes behind mathematics.

During the course of his philosophical endeavor Russell began to think about the concept of sets (or groups). A set is a collection of objects that share a like characteristic or like characteristics. Sets are conceptual in nature. An example would be the set of all dolphinsor, the set of all things that are not dolphins. Russell soon noticed an interesting characterization of sets, namely that they can either be or not be members of themselves. To explain, consider the set of all dolphins. Now ask yourself: is the set of all dolphins itself a dolphin? The answer, of course, is no. Next consider

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