## Friday, August 12, 2005

### Supposed purity, paradox

One of the weirder assertions in modern intellectual thought is of the "logical paradox" which makes no sense. Consider a supposed example where here's what Wikipedia says:

The Barber paradox is a paradox that relates to mathematical logic and set theory. The paradox considers a town with a male barber who shaves every man who does not shave himself, and no one else. Such a town cannot exist:

* 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.

Thus the rule results in an impossible situation.

Whoever wrote that Wikipedia article or edited it to add that such a town cannot exist, deferred to practical reality, which does not allow those conditions, yet supposedly there is a paradox in logic there!!!

What gives?

The conditions as given reduce to the assertion that a set includes and excludes itself, so there is a direct contradiction--not a paradox--but mathematicians and logicians teach it as a paradox, so they are not telling the truth.

To understand the reduction, just read the description that's in the Wikipedia article! The requirement that the barber shaves every man who does not shave himself in the town directly contradicts with the barber shaving himself or not shaving himself, so you get that reduction to a statement that the barber shaves and does not shave himself.

You might think it a trivial issue, but let's resolve it anyway by relying on the real world, where it makes sense if in the town the barber shaves every man who does not shave himself, and no one else, except himself, as that barber can exist!!!

So you have the idea of an exception--the barber.

Seem trivial?

Well now we get to consider THE set that only has sets that do not include themselves and has all such sets--except itself.

Necessarily, the set made up of all and only sets that do not include themselves, except itself, cannot include itself, as then you'd have a contradiction as it would both include and exclude itself.

There must, logically then, be a set of all sets that exclude themselves, except itself.

I call that set a superset, as there is only one set that only has sets that exclude themselves and has them all, except itself.

Let that sink in--there can be only one.

Super sets of that type are unknown in "modern" set theory as it is primitive.

You need the real world to understand logic.

If mathematicians and logicians focused in the real world versus being "pure" they could have resolved the supposed paradox by seeing the contradiction and hypothesized the existence of the superset with the easy proof of its existence.

Concentrate your mind on a set that includes all sets that exclude themselves, except itself.

To me that is an example of super-information as in any reality any sentient beings who came across this issue would, if smart enough, eventually hypothesize the exact same set. There can be only one across all realities, all means of existence.

That's kind of cool.

James Harris

## Tuesday, August 09, 2005

### Logical theory versus illogical logicians

There's this guy named Russell who put forward what he thought was a logical paradox:

See http://www.iep.utm.edu/p/par-russ.htm

or

http://www.cut-the-knot.org/selfreference/russell.shtml

One variant of it is called the "Barber's Paradox" and I use it for laypeople as it might be easier to understand than starting with the full abstraction:

Consider a small village where all men are shaved, the barber is a man, and the barber shaves all and only those men who do not shave themselves.

But who shaves the barber?

The answer is that it's impossible for the barber to shave himself and not shave himself, which is what those conditions would require.

That is, the supposed paradox is with assertions that would require that the barber shave himself and not shave himself, which is a direct contradiction.

More abstractly, consider a set made up only of sets that exclude themselves that includes all such sets, but stop, before you run around with that idea, as it reduces to: consider a set that includes and excludes itself.

So why is this a big deal in logic?

Because, I guess, people a while back didn't deal much with exceptions.

But we're modern, well-educated people so let's toss in exceptions:

Consider a small village where all men are shaved, the barber is a man, and the barber shaves all and only those men who do not shave themselves, except himself, as the barber shaves himself

Easy.

So let's handle the abstraction: consider a set that includes all and only sets that exclude themselves, except itself.

And just like that I've handled a big enough issue in logical theory that you can find quite a bit on it in books and on the web, and look at them to see that they do not provide the correct solution.

What? Yup. Look at them and they do not provide the correct solution.

Why don't they?

Best guess is that we have the benefit of something modern: computers.

So our perspective is broader.

James Harris