June 10, 2017 Saturday

Bedtime Story 

Decoding a Godel Number and Beginning of Arithmetization of Meta-Mathematics 

So how would you go about decoding a Gödel number 243,000,000 given that it is first of all a Gödel number and second that it stores information about a very specific formula? 

What would be your first move?

Yes, the very step would be the prime factorization.

243,000,000 = 2⁶ x 3⁵ x 5⁶

Now we need to go back and see what do the 6 and the 5 stand for.

Zero was assigned number 6 and “equals’ was assigned number 5.

So then we get 0 = 0

So in short, 243 million as a Gödel number represents the equation 0 = 0.

This of course was an extremely simple specimen of decoding.

Yet is serves as a powerful illustration of the point that given any number, it is possible to dismantle it apart, work out how it was constructed and then derive the information or the expression that is stored with it.

With this, we have completed the first and a very crucial part of understanding Gödel’s theorems, the Gödel numbering.

This I think is the easiest part, ingenious as it may have been for the inventor.

Now we will go to the second part which is the arithmetization of meta-mathematics and is an even more creative step.

You see mon ami, Gödel had by now brilliantly converted his system of formal logic – which was essentially Principia Mathematica – into a precise and perfectly coded numbering system.

The next thing that his brilliant mind was considering was how to devise a method of converting the meta-mathematical statements into numbers.

  

How could one conceivably convert the statements of ordinary language – which in Gödel’s case was German – which describe mathematics into mathematical symbols?

He found a way out.

Let me explain his idea with a hospital analogy.

If you have visited any major hospital, especially something organized and systemic like a one under the National Health Service (NHS) of Great Britain, you will get it rather quickly.

Any patient that comes into the clinic is given a token that assigns a number to him or her.

So say we have Mrs. Ape with a token number 15 and Mrs. Chimp with a token number 25.

So instead of advising Mrs. Chimp that her turn will come after Mrs. Ape, all we need to tell Mrs. Chimp is that 25 is great than 15.

This can be done for all the patients and then we have only to deal with a set of natural numbers.

We, in a manner of speaking, have arithmetized the consulting order of the patients.

Stay tuned to the voice of an average story storytelling chimpanzee or login at http://panarrans.blogspot.in/

                              

Good night mon ami and my fellow cousin ape.

                           

  

                

             

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Another great educator and a teacher that I am aware of is Professor Subhashish Chattopadhyay in Bangalore, India.

While I narrate stories, Professor Subhashish an electronic engineer and a former professor at BARC, does and teaches real mathematics and physics.

He started the participation of Indian students at the International Physics Olympiad.

Do visit him here:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg