June 13, 2017 Tuesday

Bedtime Story

(∃z) ) (sss…ssss0 = z x ss0) . ~ (∃z) (sss…ssss0 = z x (ss0 x ss)) is the formal way of saying ‘The first symbol of the formula "~ (0=0)" is a tilde’

Last night we discussed the successor function and saw how it is the only formal way of representing the natural numbers.

So similarly, in formal logic to state (∃z) (y = z X x), the proper way would be:

(∃z) ) (sss…ssss0 = z x ss0) . ~ (∃z) (sss…ssss0 = z x (ss0 x ss))

This is heavily coded so let me try to deconstruct it for you.

Recall that we had designated the Gödel number of the blatantly wrong formula ‘~ (0=0)’ to be ‘a’.

Since our idea of y = z X x would involve factorization of a, then y would be represented by sss…sss0 where s would be repeated a number of times (Yes, it’s a huge many times!).

The dot that is seen between the two formulas represents ‘and’.

The second part I am sure now is not very difficult to unravel.

The tilde represents ‘not’.

So the second part says that there is no number z such that a equals z times the quantity 2 x 2.

This is so because if you care to look back and see that we had agreed that “2¹ is a factor of ‘a’ but not 2²” 

In totality, the whole message of this elaborate formula is this:

‘There is a number z such that ‘a’ equals z times 2 and that there is no number z such that ‘a’ equals z times the quantity 2 x 2.’

I know you will complain that if all the ‘s’s were to be listed, it would become a preposterously long formula.

That is totally a valid point, yet conceptually it is a simple idea.

Yet, this is the formal logic at its stupefying glory.

It is a totally formal way in pure symbols of stating the mathematical truth that ‘The first symbol of the formula that was the subject of discussion is a tilde’.

It is expressing a meta-mathematical assertion about the identity of the first symbol of another one of its formulas.

There is yet another thing interesting about the idea that we have been discussing.

‘x is a factor of y’ is a arithmetical predicate.

You would recall the word predicate variables that we assigned Gödel numbers to.

The predicates include terms like “is prime”, “is greater than” and so on.     

As I had told you earlier, a predicate is a statement that can be true or false depending upon the value of its variables.

So ‘is prime’ predicate will be true or false depending on what number it is being linked to.

Such is the case with ‘x is a factor of y’ too.

I am afraid I will have to break for the night and continue the chain of thought 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