Mathematics

Russells Paradox Explained



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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 the set of all things that are not dolphins. Certainly, it does not take a lot of reflection to see that the set of all objects that are not dolphins is itself not a dolphin and, therefore, seems to belong to itself.

Russell, being a genius, was then able to take his novel thinking about sets to a slightly more complex level. Specifically, he began considering sets of sets and, in particular, the set of all sets that are not members of themselves. Ask yourself, as Russell did, if the set of all sets that are not members of themselves is itself a member of itself. Paradoxically, this set is a member of itself only if it is not a member of itself. And if it is not a member of itself then it is a member of itself. The analysis will forever jump back and forth concerning the status of this set, leaving any proposition concerning its status in this regard as undecidable. And this is Russell's famous paradox.

What is of philosophical interest about Russell's Paradox is that it seems to have discovered that there are alternative kinds of meaningful propositions than had been originally supposed. Because of Russell (and others since) mankind is now confronted with propositions which are unanswerable not because of inadequate information but because of the very nature of the propositions themselves. Many philosophers have wrestled with these kinds of paradoxes (including Russell himself) but no fully satisfactory way of ridding the world of them has yet been brought forth.

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