June 05, 2017 Monday

Bedtime Story 

Gödel Numbering for the Three Types of Variables

Having done away with Gödel numbering of the 12 constants and having discussed the Correspondence Lemma, let us now move on to the three types of variables and their Gödel numbering.

The numerical variables go like this:

Numerical                     Gödel                           Possible

Variable                        number                      substitution

x                                  13                               0

y                                  17                               s0

z                                  19                                x

Sentential variables would be represented by the alphabets such as p, q, r… and so on and they would represent the formulas or in other words, they can be substituted by formulas.

The Gödel number that is associated with sentential variables is a unique square of prime number greater than 10.

In tabloid form, sentential variables go as follows:

Sentential                      Gödel                        Possible

Variable                         number                    substitution

p                                   13²                             0 = 0

q                                   17²                       (∃ x) (x = sy)

r                                    19²                             p ⊃ q  

Predicate variables would be represented by alphabets P, Q, R,…and so on.

Predicate variables would represent the predicate which in logic essentially means Boolean-valued function.

So the predicate values would represent something like “is prime” or “is greater than” or in other words, they can be substituted by such functions.

The Gödel number that is associated with predicate variables is a unique cube of prime number greater than 10.

In tabloid form, the predicate variables would look something like this:

Predicate                          Gödel                         Possible

Variable                            number                  Substitution

P                                    13³                                x = sy

Q                                    17³                   ~ (x = ss0 X y)

R                                     19³                (∃ z) (x = y + sz)

This is the basics of Gödel numbering.

A very unique and specific Gödel numbers have been assigned to the elementary signs or the primitive symbols which fall in the two categories of either beings constants or variables.

The variables in turn are of three types whose Gödel numbers are essentially based on prime numbers.

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