June 12, 2017 Monday

Bedtime Story 

Testing if True Meta-mathematical Statement Faithfully Maps Itself into Number Theoretical Assertion of the formal Logic

The Gödel number of the formula ‘~ (0=0)’ goes like this:

2¹ x 3⁸ x 5⁶ x 7⁵ x 11⁶ x 13⁹ 

Let us label this Gödel number as ‘a’.

Now let us examine the meta-mathematical statement.

‘The initial symbol of the formula ‘~ (0=0) is a tilde’

This meta-mathematical statement is merely commenting a factual point about ‘a’.

  

Hence, this meta-mathematical statement about ‘a’ can be represented in terms of assertion about the exponent of the smallest prime of this large number as it is this that is representing the tilde sign.

Our true meta-mathematical statement simply asserts that the exponent of the smallest prime (namely 2) factorization of ‘a’ is 1.

Another way of asserting our meta-mathematical statement would be to say that is this:

“2¹ is a factor of ‘a’ but not 2²” 

Thus we have a number theoretical way of expressing our true meta-mathematical assertion ‘The initial symbol of the formula ‘~ (0=0) is a tilde’.

Now is it possible to convert the statement “2 is a factor of ‘a’” into formal string in the language of Principia Mathematica?

Let us try.

It is similar to seeking the formal string of the statement ‘x is a factor of y’.

This statement is equivalent to the statement:

‘There is a number z such that y equals z times x’.

In formal calculus, this turns out to be:

(∃z) (y = z X x) 

(It translated literally as ‘there exists a number z such that y equals z times x).

In formal language, the numerical variables can be replaced by successor function.

Successor function is a truly and perhaps the only formal way of representing natural numbers.

So the natural numbers of the decimal system would be defined as follows:

1 would be represented by s0 (immediate successor of 0)

2 would be represented by ss0

3 would be represented by sss0 and so on

So a formula such as ‘~ (3 = 4)’ which literally means that the statement three is equal to four is not true would in Principia Mathematica be represented as:

‘~ (sss0 = ssss0)’

Then of course, you would need to substitute each symbol with its corresponding designated Gödel number to make it admissible into the Principia.

I urge you to keep in mind this chain of reasoning.

It will demanded of you not only in for the next night’s bedtime story but even the subsequent ones.

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.

                           

  

                

             

Advertisements

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