June 20, 2017 Tuesday

Bedtime Story 

r is Derivable From m

Please have a deep look at these two numbers m and r once again and spot the difference.

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

The number m contains a prime number raised to the factor 17 because it contained the numerical variable y.

The number r contains all the factors that the prime numbers have been raised to in the number m and many more, but none of the prime numbers contained in it would be raised to the power 17.

But can number r be obtained from m?

Gödel proved in his paper that is possible.

The number r can be obtained from the number m by replacing the prime factor in m that is raised to the 17^th power by other primes raised to various other powers different from 17.

The mathematics that Gödel used to show this is highly notational and intensely formal and so would be extremely tedious to follow.

Most of us are not only untrained but lack both the insight and mathematical maturity to comprehend that kind of formal logic.

The whole point of this little bit of digression was to convince you that the number r is well-defined number function of m and 17.

This whole concept of taking a formula’s or string’s Gödel number and substituting it into the string itself and then deriving its Gödel number is a demonstration of this man’s remarkable perspicacity.

Though the entire operation seems to be extremely anfractuous, Gödel convincingly showed that this function is rather elementary in principle; so much so that it could rightly be labeled as primitive recursive and thereby have Correspondence Lemma applicable to it.

Having accepted this, we can tag the new Gödel number as a function of the old Gödel number x as:

‘sub (x, 17, x)’

This will surely confuse you but let me explain.

All this new number asks of you is to take the formula of the old Gödel number x and wherever in it you come across the presence of the numerical variable ‘y’, substitute it by the numeral for x.

Note that I have used the word numeral and not number.

There is a difference between the two that need to be brought out.

The difference between ‘the numeral for x’ and ‘the number x’ is something similar to our discussion that involved the distinction between mathematics and meta-mathematics.

I shall highlight the disparity between the two 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