June 07, 2017 Wednesday

Bedtime Story 

How to Gödel number a Theorem and Sequence of Theorems

Look at these numbers again carefully and the pattern will be revealed. 

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

All the bases of the exponentials are the first ten prime numbers and the powers to which they have been raised are the Gödel numbers that was associated with each elementary sign of that formula

Since it is a product, it is now a one number.

The formula – that also happens to be a theorem of the Principia - is now defined by this one number, massive as it may be.

Since it is a large number, it can be represented by an alphabet say m.

This example shows us that any formula of the formal system that is derived using transformation rules and rules of inference can be assigned a unique number.

That number will have as many consecutive primes as there are elementary signs in that formula, each prime being raised to the power that corresponds to the Gödel number of that sign.  

You will see that now we have built up a system of coding wherein a specific Gödel numbers lies within another unique Gödel number.

This can be carried on further.

Let me show you how Gödel did it.

I have already shown you ways of handling Gödel’s number with respect to elementary signs and a formula.

Now let us consider a sequence of formulas that may be encountered such as these ones, again from Principia Mathematica.

(∃x) (x = sy)

(∃x) (x = s0)

The lower formula simply means that zero has an immediate successor.

And you will agree that it can be easily derived from the first formula by substituting 0 for y.

This uses the rule of inference that allows the substitution of any number to a numerical variable (that variable here being ‘y’).

The first formula as we had agreed upon was assigned the Gödel number m.

We can, using the same long process, eventually assign the Gödel number n to the second formula.

Now, like before, we would like to reduce them to one number.

You may ask why?

We shall go over it in the nights to come.

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