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Let us take the set T, the set of all truths.

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But there can be no set of all truths.

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Originally posted by TGB

Sahid: When did you prove that T cannot exist?

Tom: T is, presumably, the set of ALL truths. But T's power set, P(T) contains truths that are not members of T.

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Tom: T is, presumably, the set of ALL truths. But T's power set, P(T) contains truths that are not members of T.

Scott: It seems to me that the demonstration fails at this point. In what sense is some subset of all truths itself a truth?

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T is, presumably, the set of ALL truths. But T's power set, P(T) contains truths that are not members of T. Thus, there are truths that exist outside of T, in which case T is not really the set of ALL truths. If you enlarge T to contain the truths that correspond to each set of P(T), to form T1 (the new and improved set of all truths), you face the same problem again with the power set of T1, or P(T1), which then contains truths not members of T1.

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The set of all truths T is (S, G, W)
The power set of T is P(T) and is made up of ({S}{G}{W}{S, G}{S, W}{G, W})

Now, here are the truths generated by each subset that are not members of T:

1. S is a member of T
2. G is a member of T
3. W is a member of T
4. S and G are two members of T
5. S and W are two members of T
6. G and W are two members of T
7. S, G, and W are members of T

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Plantinga to Grim: As you point out, it's plausible to think there is no such set [of all truths]. Still, there are truths of the sort: every proposition is true or false (or if you don't think that's a truth, every proposition is either true or not-true). This doesn't require that there be a set of all truths: why buy the dogma that quantification essentially involves sets?

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Another supporting argument for constructing formal systems is that uninhibited use of intuition leads to paradoxes, or self-referencing statements, like:

This statement is false.

This self-referencing statement is false if true, and true if false. It is inherently contradictory, or so the argument goes.

This argument, though, is weakened by the work of G. Spencer Brown, who developed a calculus of forms in which self-referencing statements can have a truth value different from true or false (namely a third truth value, called imaginary). This proposed solution is the analog to the solution developed in algebra to allow imaginary numbers to deal with self-referencing equations like.

x2 + 1 = 0.

This equation has solutions only if imaginary numbers are allowed. The solutions are:

+√ -1 and -√ -1

Similarly, paradoxes would have solutions if imaginary statements were allowed.

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SELF-REFERENTIAL SYSTEMS

Why does such a paradox arise? The problem is because with both the "Barber Paradox" and in the paradox of the "set of all sets which do not contain themselves", we are dealing with self-referential systems. Self-referential systems are systems which refer to themselves. For example: consider the following sentence: "This sentence has five words." The sentence does, in fact, have five words, so the statement, though self-referential, is true. There is also no problem with false self-referential sentences, such as "This sentence has four words," which is false because it actually has five words. However, some self-referential statements present a more paradoxical situation: "This sentence is false." It's clearly self-referential, but is it true or false? When we try to decide, we find ourselves stuck in a vicious cycle. The truth or falsity of the statement is indeterminable.

As this simple example demonstrates, self-referential systems quickly become too complex for logic. The self-referential system par excellence is the mind. Since the mind is the only tool that we have for thinking about anything, that means that the mind is also the only tool which we can use to think about the nature of the mind. Jung was fond of emphasizing "the fundamental fact that in psychology the object of knowledge is at the same time the organ of knowledge, which is true of no other science." (32)

However, Jung was wrong in assuming that the problem is confined to psychology; if we think deeply enough about anything, we are led ineluctably to think about the process of thought. As an example which we have already discussed at some length, Kant ran up against the same issue in philosophy. Once the Renaissance ideal led men and women to separate themselves from nature in order to observe nature objectively, it was inevitable that they would eventually turn that observation upon themselves. Berkeley and Hume were merely the first in the Western World to demonstrate the nasty little paradoxes that appear when the mind thinks about itself.

Anthropologist Gregory Bateson stressed that all living creatures are complex feed-back systems. A feed-back system provides information to itself about its performance, which in turn is used to change that performance. That change leads to further feed-back, thus forming a continuous loop. The information, which the system feeds back to itself so that it can better adapt to reality, is of a higher order of reality than the behavior that it comments on. Hence living creatures are complex self-referential systems. (33)

As physicist Werner Heisenberg [1901-1976] discovered, even confining our observation to non-living systems is not sufficient to protect us from the problem. At the sub-atomic level, our observation changes the object under observation, which is another way of saying that the subject, the object, and the act of observation are actually a single system--the split between subject and object is artificial. Thus, even in physical experiments, the subject is once again involved in a self-referential system. (34)

There is no necessary paradox in self-referential systems as long as the systems are finite. Mathematicians have been able to successfully prove the consistency and completeness of many finite mathematical systems. If the system is small enough and simple enough, they simply develop every implication of the system for every member of the system. Thus, if human beings can be viewed independently from the world around them, and if they are nothing more than the finite amount of "things" (such as proteins and nucleic acid, etc.) that make up their physical composition, there is a way out of the problem of self-referencing. The mind is then reducible to the brain, and the problem of the brain thinking about itself eventually winds down since it is at most a finite process.

That is still the most popular scientific position. The point of this book is that such a position seems increasingly untenable as we dig deeper into any of the systems that explore the universe, from philosophy to mathematics to physics to psychology; in fact, to any field at all. All fields of thought eventually involve themselves in infinite self-referential systems. (35) And the only way out of that dilemma lies in an archetypal hypothesis, such as developed by Jung and Gödel in their separate fields. But we are getting ahead of ourselves and need to get back to Russell's project.

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