Russel's Paradox Again: Singularity Paradox

 There is a very nice Blog titled Thinking on Thinking at http://pmulder.blogspot.com/ It has a posting on the Russel Paradox. (By the by, we are talking Bertrand Russell here, not Jack Russell.) Tuesday, February 20, 2007  
Russell's Paradox "Nothing contains everything", applied: A man of Seville is shaved by the Barber of Seville if and only if the man does not shave himself. Does the barber shave himself ? * If the barber does not shave himself, he must abide by the rule and shave himself. * If he does shave himself, according to the rule he will not shave himself.
And, another one:"One of themselves, even a prophet of their own, said, the Cretians (sic) are alway liars, evil beasts, slow bellies. This testimony is true."Titus 1:12-14 (King James Version)

We may describe the Cretan Liar as follows:
Epimenides says "All Cretans are liars." Epimenides is a Cretan. If Epimenides' statement is true, then this implies that Epimenides always tells falsehoods. Hence, Epimenides' statement is false. Or, the truth of Epimenides statement implies the falsity of Epimenides' statement. Both paradoxes have been seen as problems in statements that are self-referential.


I do not consider these two paradoxes to be about the same logical thing. The Cretan Liar says that Cretans are liars, evil beasts, and slow bellies; only one of these is a problem: liars. We would not even blink an eye if all Cretans were slow bellies. We would not be talking about their bowels at all.
"All Cretans are Liars" is a statement that is an appraisal and, as such, should not be assigned a truth value. If I were to say "The Cure ruled!" someone else would say, "Only the early, real Cure ruled!"
Difference of opinion.

Now, the Barber.  
Observe that the Barber paradox disappears if we allow that there are 2 barbers. Then barber 2 may shave barber 1, and barber 1 may shave barber 2. It disappears if there are 3 barbers. Barber 1 shaves barber 2, barber 2 shaves barber 3, barber 3 shaves barber 1. { Of course, we have to transform the original statement of the paradox to: A man of Seville is shaved by a Barber of Seville. ( not "the" barber, not just one and no more.) }
If all men in Seville are barbers, they all shave each other in some sort of ghastly daisy-chain rig-a-marole. In fact, they form sort of a circular group of order N, where N equals the number of men in Seville.

From this I conclude that the Barber Paradox is a Singularity Paradox.
It is not dependent upon Set Theory and whether there is a Set that contains all items of a population. It is dependent on the fact that Singularities are inherently paradoxical. If there is only one and exactly one Barber of Seville... and no one else can fill in for him on a sick-day... then we have a paradox.

For example, look at the work done by Linde on Inflation Theory cosmology. This was inspired, in part, by the cosmological problems created by the singularity of the original Big Bang theory. In essence, where did the singularity come from? Similarly, we see the problem in arguments for the existence of God from a chain of causes, wherein God becomes the First Cause. As observed by Schopenhauer, the Principle of Causality is not a hansom cab that we take only so far, and then send away when we judged that we have arrived at our destination. Thus, the Principle of Causality cannot be used to establish a Terminus at the beginning of a chain of caused events.

Similarly, we find paradox in religious notions of singularity. More power to 'em, I say. Paradox is the heavenly mother of earthly Irony. Many Biblical Parables are paradoxical like Zen Ko'an to get people out of mental ruts.

I have often wondered "I wonder what it was like for the first men or women who uttered speech? Was there anyone else around to understand them?" Scenarios like this, dealing with the first occurence, are very, very foggy. Was there a "first" person to speak? We often talk as if there were. So, prove it. Singularities are funny creatures. Ask Professor Hawking if you do not believe me. Two paradoxes down.