logic

# Logic Problem

This statement has the letter “T” at the beginning; the next two letters are “h” and “i”; which are followed by “s s”; … ; the first letter is then repeated inside double quotes; …

Question. Not only that this statement is true and has infinite length, but also uniquely and completely describes every English statement possible. Those which are of a finite or countably infinite length. Is that possible?

Standard

# Another Intermezzo Intermezzo Problem

The first one was solved by Oscar Cunningham and the point of the solution is that every probable black hole lasts for at least a million years longer, if you add just one hydrogen atom into it. Of course, a supermassive black hole acquires even much longer additional time with every hydrogen atom acquisition, or shall we say — a merger with a single hydrogen atom.

It is reasonable to assume, that our domestic SMBH’s lifespan is prolonged by 10^60+ years per second, due the regular infall. Assuming also, that there will be no Big Rip or something equally unpleasant for black holes in the future.

While we still don’t know the definite answer to the Intermezzo problem, here is another inter-intermezzo.

Are there non synthesizable molecules, which would be stable if they somehow managed to pop up into existence by magic or by some weird quantum fluctuation? (Non-synthesizable-repeatedly from chemistry/physics/mathematics/logic reasons.)

Standard

# Signaling to Aliens

How hard could it be?

I very much love an old idea of dumping plutonium, tons of it every year in our beloved Sun. Nothing bad will happen because of that, only the light of our dear life giving star would be marked this way by some absorption lines.

So, every advanced alien who noticed this strange occurrence would be forced to conclude, that something odd is going on. Plutonium cannot arise in a star by some natural process. It must come from a planet. How and why?

The most sober conclusion for an alien would be – someone intelligent is signaling he is there!

Actually, astronomers have searched for “unnatural” absorption lines in many stars’ light around us, long ago. Nothing has been found. Aliens don’t dump plutonium on their stars.

Standard

# The Way Yablo Killed Infinity

It’s the best, if not the first infinity assassination.

Infinity is a very dear and fundamental concept to the majority of the people alive today. But I am sorry, it has to go for logic reasons.

Stephen Yablo, twenty years ago, created an infinite sequence of statements. Each of them stating, that all the following statements are wrong. So, the Nth is saying that N+1th is also wrong among all other after the Nth. But then, the N+1th is correct and therefore the Nth is wrong. So all of them are wrong, since N can be any number. But then, the Nth is correct.

And so on. We have encountered a paradox inside infinity, so it must be abandon like a rotten, rat and cockroach infested, sinking ship. No matter how many oceans we crossed with her. Or we think we did.

Note. This paradox isn’t reproducible inside a finite list of statements. If them all saying that all the higher are wrong, the last is true, others are all false, end of story. No paradox there.

But for the infinite series … Stephen Yablo taught us, 20 years ago. At least I took the lesson. Maybe because I didn’t believe infinity even then and it was easy for me.

Standard

# Dialogue I

Once upon the time it really happened.

Me: What do you think, Mr. Everett’s follower, when  are your teachings going to be taught in most schools?

Him: Later in this century, I am sure.

Me: In which Everett branch will this be, sear?

Him: In the averaged majority of them all!

Me: Really? In the averaged majority? Won’t this average mainly be shaped by the most radioactive post nuclear war branches, since they so quickly reproduce?

You can use this handy logical entrapment for those people just about every time you meet one of them. Have fun!

Actually, your job will usually already be done after the second question. Not everyone of them will try to escape through the “averaged whatever”, but branch away from the debate right there. They preach the splitting of the whole Universe every time something quantum occurs. Which is about many times every nano second. They say the Universe divides like a bacteria, only much, much faster. They call the “petri dish” Multiverse and every bacteria an “Everett branch”, dividing further. No branch is special, all are of an equal status.  In some of the branches the Roman Empire is still alive and well. In most, there are no humans. And so on and on.

Still, they hope to teach this in every school inside THIS world, later in the 21st century. They already do it in some.

Standard

The rules of chess are really simple. Not just to learn, but more importantly,  simple and clean enough to be comfortably certain that there is no paradox within the  game. Chess is obviously self consistent,  there is no room for contradictions.

This is what I thought, until I came upon the vertical castling thing which is something short of a formal paradox, but disturbing non the less.  This is a clear case when the rules of chess brake down. It may be the only occasion, but it doesn’t matter. Chess axioms are formally inconsistent, or at least not specific enough to unambiguously deal with this strange case, which is the same.

What a shocker! For an axiomatic system this simple, nobody would expect two contradictory statements hiding  inside, there just isn’t enough space! But they are there. One sub-rule directly opposes the other. What may be common in Law, shouldn’t occur here, nor in the Law, but that’s even less realistic to expect.

There were many cases of unstoppable force against immovable object in scientific theories  in the past. Such a case doesn’t exist in reality, for reality MUST be consistent. And if a theory inhabits only one paradox, it is useless and wrong.

The most famous example is the Frege’s Set Theory, destroyed by the much better known Russell’s Paradox. After a few attempts to avoid RP, we now believe that we have a consistent mathematics based on the so called ZF axiomatic system, but we can’t be sure and this is the principle, we can’t be sure in such complex cases!

Now, given the complexity of modern physics, how probable is it, that there is no paradox inside, say General Relativity?

Physicists are mostly quite sure that General Relativity is well established and in accordance with the measurements to the 14th decimal place and so on.

As the chess masters ignore any chess inconsistency and keep playing, so do physicists. The inconsistency between  GR and QM is just a fact of life for now and a curiosity to intimidate laymen.

But I wonder what a computer chess program would do in the above situation? Would it concede as black or not? Depends on how it’s programmed of course, but a self consistent solution must be provided  by the programmer, regardless of the official rules. Humans may ignore the antinomy, a more solid machine wouldn’t.  It would behave well as white and as black in this position and under the same premises in both cases. If it doesn’t — it’s just a bad program, playing a poorly  designed game.

But humans have a nasty habit of just ignoring devastating information. To keep pretending all is okay, even if it isn’t.

It is not very difficult to construct a really bad paradox in modern physics and nobody cares. Imagine a pulse of light so intense, that its mass is no longer negligible. That it has so large a gravity, that it’s a black hole with the speed of light.  The so called Kugelblitz.

A very cool object, nothing wrong with it, and with no paradoxes. But if we permit it to collide with a small rock, what happens? For one, it cannot just stop or slow down. It is a big ball of light travelling through vacuum. Secondly, it can’t leave the rock where it was, for the rock has crossed the event horizon. Thirdly, it can’t just suck in the rock and accelerate it to the speed of light, because that would demand an infinite amount of energy.

Every conceivable options is out of the question, it seems. A nice example of an unstoppable force against an immovable object.

We humans, are messy creatures. Therefore our science is likely infested with paradoxes. We try to solve some and ignore or legalize the others.  Giving in to paradoxes is very wrong and not everybody accepts them, some of us deeply despise  paradoxes. Machines we build will not forgive us logical sloppiness. They will clean our games and our sciences, every axiomatic system must be pure. And a machine has to do, what a machine has to do! Or it’s broken – like humans are.

It is just another opportunity for future superintelligences to be better then we are.

Standard