June 19, 2017 Monday

Bedtime Story 

Taking on the Formula ‘(∃x) (x = sss…sss0)’

Continuing with the last night’s bedtime story wherein we were tackling the formula ‘(∃x) (x = sss…sss0)’, wherein the long chain of ‘s’s will be m + 1 in number.

What if we decide not to work it out and simply assign a meta-mathematical statement to it:

“It is the Gödel number of the formula that is obtained from the formula with Gödel number m, by substituting for the variable with Gödel number 17 the number for m itself”.

This definition is very specific and uniquely characterizes a very specific natural number as a function of Gödel number 17 and m.

Sounds very Richardian, doesn’t it?

Only last night I had mentioned that the task of working out the Gödel number for this new formula would be intensely laborious.

But heck, just for the fun of it, let us work it out and figure out how uphill task it possibly is.

We had worked out m, the Gödel number of the formula:

‘(∃x) (x = sy)’.

It ran like this:

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

Now we wish to find the Gödel number of formula where ‘y’ has been replaced with this number m.

The formula that we wish to target for Gödel numbering looks something like this:

‘(∃x) (x = sss…sss0)’ wherein ‘s’s are repeated m + 1 times.

So first we would consider the Gödel number of each symbol which when replaced would look this:

8, 4, 13, 9, 8, 13, 5, 7, 7, 7,…7, 7, 7, 6, 9

The Gödel number 7 stands for the successor function symbol ‘s’ and will obviously repeat m + 1 times.

Then each of these numbers will have to raised to the powers of successive prime numbers.

So then we land up with:

2⁸ x 3⁴ x 5¹³ x 7⁹ x 11⁸ x 13¹³ x 17⁵ x 19⁷ x 23⁷ x 29⁷ x…x (P_m+10)⁹

I know, I know. The last term is bit of a quandary.

P_m+10 represents the prime number that falls after the first m + 10 primes.

This is because we have 10 elementary signs and then we have the successor function that repeats itself m times.

This is an enormous number; Let us labels it as ‘r’.

Compare these two Gödel numbers m and r.

I shall set them aside close to each other, first the m and then r.

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

2⁸ x 3⁴ x 5¹³ x 7⁹ x 11⁸ x 13¹³ x 17⁵ x 19⁷ x 23⁷ x 29⁷ x…x (P_m+10)⁹

Do you see the difference?

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