A paradox is a paradox?

September 14, 2007

As a computer scientist i cannot delete the previous ‘Hello World!’ post! But actually this is my first post.

I am starting this blog by explaining its title. I can remember when i attended high school that our philosophy teacher proposed us different sort of paradoxes and that no one convinced me. For example one was about Achille and the turtle , it says that if the turtle have some metres of advantage Achilles (also know as “swift-footed“) cannot catch it, the reason is that before reaching it he has to cover a half of the distance, but then another half and so on, while the turtle is still moving.

At the time we were proposed this solution:

\sum_{i=1}^{\infty} \frac{1}{2^{i}} =1,

but i was not convinced because it looked like a trick (and does not work if we say that Achilles should cover a third of the distance and so on). Right now i think at it in a very different way, no tricks, no analysis, it is simply a bug in our way of thinking; unfortunately this notion of bug is an act of faith, cannot believe it?

Let’s look to another famous paradox, a slightly modified version of the Liar paradox:

“this sentence is false”.

If it is true it is false, but if it is false it is true, ok, we cannot say it is true or it is false. Let’s say it is a paradox, that is, it has no meaning. This bring us to the very unlucky sentence ” ‘ “this sentence is false” has no meaning’ is true”, but then it is also true that ” ‘ “this sentence is false” has meaning’ is false”. If something is false it has a meaning and so the sentence can be rewritten ” ‘ “this sentence is false” has meaning’ has meaning” and so we get a contradiction with the fact it had no meaning (proof of Karl Popper).

We could solve the problem by saying ” ‘ “this sentence is false” has no meaning’ has no meaning” is true… ops I did it again! It looks like this is an unsolvable problem, but, oh my god! Can i say that it is?.

Actually someone, Kurt Godel, saved us. He said that it is NORMAL. He proved that every logical formal system has a statements that are not provable inside the system itself. In particular he introduces two statements: “this statement is not provable” and “this formal system is consistent”. The proof is technical indeed, but logic is technique!

Godel and other logicians in Europe are not the only ones who faced the matter, this stuff has been studied also on the opposite side of the world in Zen Buddhism, they are called zen koans, and their science is explained here.

[EDIT 28/09]: I am realizing that this post is a bit mystical, while the interested reader would like a technical detailed reason for which we can not say that ‘this sentence is false’ is a paradox. Saying that is a paradox is not a solution at all because introduces a new dichotomy. When we deal with this sentence we are in this situation:

true or false?

We define trueness as being where we pretend to be, and as falseness as not being as we pretend to be. For example the sentence ‘The word ‘short’ is short’ pretends to speak about reality, and it is real, while the sentence ‘The word ‘long’ is long’ pretends to speak about reality, but effectively it is not! So we say that the first is true and the second is false. In the case of ‘This sentence is false’ we have a boiling potato, or something more similar to the china syndrom, that says ‘I can’t stay here’ by the definition of falseness, everywhere we try to put it. Even if we define a new set:

true_false_no_meaningg.jpg

The problem is that the sentence is always in a wrong place and dividing the world, by default, is buggy.

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