June 06, 2017 Tuesday

Bedtime Story 

Understanding Gödel Numbering System With a Theorem

  

Now let us see how Gödel played it out by considering one example and analyzing it.

Let us take one simple formula from Principia Mathematica and analyze it through Gödel numbering system.

(∃x) (x = sy)

This is quite an elementary theorem.

In English, it translates as ‘there exists x such that x is the immediate successor of y’.

In other words, what it is saying that whatever number y may stand for, there is an immediate successor number to it.

Please keep this theorem of Principia in your memory.

We will return to it at a later stage when we go deeper into the proof of the incompleteness theorems.

Now if you examine this theorem of Principia Mathematica, you will notice that it consists of 10 elementary signs that are a mixture of both constants and variables.

So let us see what Gödel numbers are associated with each of them.

(         ∃         x           )       (      x       =      s       y       )

↓         ↓          ↓          ↓        ↓      ↓       ↓       ↓       ↓      ↓                  

8         4         13        9        8     13     5      7       17     9

These numbers are one too many to handle.

Gödel realized that with so many numbers it would become confounding.

So it was imperative that he found a way out to keep these large numbers manageable.  

How would he do that and still keep it unique?

How would YOU go about this task if you were given this problem?

The task at hand is to find a way to reduce these long strings of unique numbers into something simpler such that the number that would encode the formula would be unique and still contain all the information that had been previously assigned to it by Gödel numbering.

You can either give it a thought and spend some time on it or continue reading further and seen the trick that Gödel played.

He once again invoked those mystical prime numbers which have always fascinated mathematicians.

The unique number that such formulas would we given was to be the product of first prime numbers as many as needed raised to the power equal to the assigned Gödel number of the corresponding sign.

If that is confusing, you will get rid of the confusion by seeing the number I am now assigning the above formula.

2⁸ X 3⁴ X 5¹³ X 7⁹ X 11⁸ X 13¹³ X 17⁵ X 19⁷ X 23¹⁷ X 29⁹

Do you get the pattern?

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