May 04, 2017 Thursday

Bedtime Story 

The Crux of Gödel's Reasoning (Most Important Bedtime Story)

We are however not interested in the idea of complexity at this moment; for now the engaging issue is that of establishing the consistency.

The idea of creating the formal system of Principia Mathematica was to show that such a rigid formal system would not end up in generating contradictory theorems. 

   

That is the sole test of consistency.

To show or prove that it would be impossible from the system that we have set up and using its Transformation rules and Rules of Inference to generate both the formula S and its formal negation ~S.

Now you will have to concentrate and apply your mind here as this is the tough bit but the crucial part and forms the basis of Gödel’s incompleteness theorems.

There is a well established theorem of this formal system of Principia what has been already derived.

It goes as follows:

‘p ⊃ (~p ⊃ q)’

In English, it can be translated as: If p, then if not-p, then q.

Let us then assume that there exists a formula S and its formal negation ~S, and both of them can be derived from this system.

Now we use the Rule of Substitution and replace the variable p with the formula S.

Then we get ‘S ⊃ (~S ⊃ q)’

Now we use the rule of inference modus ponens (Remember, if P implies Q, and if P is true then Q is true?).

That will give us ~S ⊃ q  

But since we are assuming ~S to be also derivable in this system, and then we apply once again modus ponens on this formula, we end up in getting q.

What did this operation tell us?

That we can get any formula q (remember, it is a sentential variable) using both S and its formal negation.

In short, if both S and its formal negation ~S were true, any and every formula would a theorem of this system.

What this says in effect that if this formal system is not consistent (both S and ~S are derivable), then every formula is a theorem.

Or, from a contradictory set of axioms, any formula can be derived!

This I think was not too difficult.

Yet one can go even further if one sees the flip side logic of it.

The counter argument to the above logic is this:

If not every formula is a theorem, or if there is at least one theorem that is not derivable from the axioms, then that formal logic system is consistent.

Hence, the other way a consistency of a system can be proved is by showing that there is at least one formula that is cannot be derived from the axioms.

I hope you got this reasoning right as this would be the crux of Gödel’s argument in his incompleteness theorems.

 

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