June 03, 2017 Saturday

Bedtime Story 

Proposition V of Gödel’s 1931 paper

If you care to go back, you will notice that some of the signs or symbols are tagged with their usual meanings, whereas the others like the tilde are not.

Now why was this done?

There was a reason for it.

This was done to see how these symbols behaved; more precisely, to keep an eye if the behavior of the symbols in the derived theorems were consistent to the assigned meanings.

So for example, if in the derived theorems it was found that the ‘+’ symbol was behaving exactly like an addition symbol, the ‘=’ symbol was behaving like an equal to sign ought to behave and if the ‘0’ was behaving like a zero, then that would reassure us that the tilde sign ‘~’ is behaving like a ‘not’ symbol.

So for example, if Gödel was able to derive the following three theorems from his formal calculus:

‘0 + 0 = 0’

‘0 + s0 = s0’

‘s0 + s0 = ss0’

Then he would be able to rest assured that since the three symbols ‘0’, ‘+’ and ‘=’ were behaving in the manner as their interpretation demanded, the other symbols should also be behaving as they have randomly assigned.

Of course, you may well argue that just having these three theorems cannot be seen as enough evidence that the interpretations of the assigned symbols are reliable and dependable.

Only if the theorems are able to capture large family of truths can we have the satisfaction of their reliability.

This is exactly what Gödel went on to do.

To clear all doubts regarding the consistency of interpretations of the assigned symbols, one only has to look at the Proposition V of Gödel’s 1931 paper.

Let me state it the way it is in the paper.

For every recursive relation R(x₁,…, x_n), there is an n-ary ‘predicate r’ (with ‘free variables’ u₁,…,u_n) such that, for all…

You see how technical it is.

People like me who am virtually untrained in higher mathematics and formal logic will simply get lost in the technical jargon.

So let me tell you the simple English translation of this Proposition V.

Gödel showed that Principia Mathematica contains within it an infinite class of theorems which if interpreted in the sense of meaning as has been enlisted in the table, will express arithmetical truths.

On the flip side, there exists an infinite class of primitive recursive truths (or arithmetical truths) which if converted to the formal language using the above table will end up in giving the theorems of Principia Mathematica.

These are profound statements mon ami which the visual system of an average, untrained ape will simply gloss over without getting its significance.

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