June 30, 2017 Friday

Bedtime Story 

Gödel’s Proof, ω-inconsistency and Rosser's Trick

We saw last night that Gödel also showed that if the formal negation of the formula G was demonstrable, then G too would be demonstrable.

Thereby we land up with this formal antinomy or contradiction: G is demonstrable, if and only if, ~G is demonstrable.

I will have to pause and add an important caveat here.

This is not exactly what Gödel had proved in his original paper.

There was an American logician by the name of John Barkley Rosser senior (1907 – 1989) who had done his doctoral research under the great Alonzo Church (most famous for inventing the lambda-calculus).

In 1936, Barkley Rosser proved something that goes by the name of Rosser’s trick.

Now what exactly is this Rosser’s trick?

Well…in a simple way one can say that it is a proof of a stronger version of Gödel’s incompleteness theorem.

Let me explain how.

What Gödel actually proved was that if G is demonstrable, then ~G is demonstrable (rendering the Principia inconsistent) and if ~G is demonstrable, then PM is ω-inconsistent.

So Rosser’s trick is a method of proving Gödel’s theorem without the assumption that the theory being considered is ω-inconsistent.

The difference between Gödel and Rosser can essentially and informally be reduced to these two statements:

While Gödel used the formula that implies the liar paradox and says “This sentence is not provable” in his proof, Rosser used the formula that says “If this sentence is provable, there is a shorter proof of its negation.”

Since Gödel used the term ω-inconsistency, we will need to understand what it means, at least in simple terms.

Rather than defining omega-inconsistency, let us understand ω-consistency.

It was a theory that was introduced by Kurt Gödel during his work on the incompleteness theorems.

It refers to a collection of sentences that are not only syntactically consistent (lacking any contradiction), but also avoids proving certain infinite combination of sentences that are intuitively contradictory.

Let us tackle it slightly more formally now.

This formal consideration of ω-consistency will be carried forward to the nights to come in order to maintain the flow of thought.

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

June 29, 2017 Thursday

Bedtime Story 

How Gödel Proved That G is Demonstrable Only if its Formal Negation ~G is Demonstrable

(2) In the second point, we will try to make sense on how Gödel proved that G is demonstrable only if its formal negation ~G is demonstrable.

This argument bears extremely close semblance to the concept of Richard’s paradox and Richardian number though enshrined with subtle but critical differences.

At the very onset, let me point out the similarities and differences between Gödel’s argument and Richard’s Paradox.

The crucial point that has to be specified is that G is not exactly same as the meta-mathematical statement that it mirrors, but it merely represents it within the Principia.

In the Richard paradox of you recall, the number n is associated with a certain meta-mathematical statement.

In the case of Gödel, the number n is associated with a certain formula that belongs to the Principia, and this formula more by coincidence than intent happens to represent a meta-mathematical statement.

In the concept of Richard’s Paradox, the question that was eventually raised at the end was whether the number n has in it the meta-mathematical property of being Richardian.

In Gödel’s case, the question that was raised is whether the number g = sub (n, 17, n) carries a specific arithmetical property, that property being the assertion ‘dem (x, g)’ holds for no cardinal number x, whatever the number x may be.    

What I am trying to say was although Gödel constructed his argument based on Richard’s paradox, he kept the distinction very clear between the statements within the Principia and the statements about Principia.

Hence it is totally free from any sort of fallacy that many consider exists in Richard Paradox.

Gödel started this part of his proof by showing that if the formula G were to be demonstrable in the formal system, then its formal negation would also be demonstrable.

You would recall what the formula G is.

~(∃x) Dem (x, Sub (n, 17, n)) 

 It stands for the meta-mathematical statement:

‘The formula with the Gödel number sub (n, 17, n) is not demonstrable.’ 

So then, the formal negation of the formula G would be:

‘(∃x) Dem (x, Sub (n, 17, n))’ 

A negation of negation is obtained by simply taking off the tilde sign (negation) from the formula G.

The meta-mathematical interpretation of this formula would be:

‘There exists a demonstration of the formula G within the Principia.’ 

Further on, Gödel also showed that if the formal negation of the formula G was demonstrable, then G too would be demonstrable.

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

June 28, 2017 Wednesday

Bedtime Story 

Explanation Why Gödel number g of the Definite Formula G has to be sub (n, 17, n) 

Last night we were discussing the definite formula G of the Principia and that its Gödel number g had to be sub (n, 17, n).

Here I need to bring to your notice an important point.

There is an important difference between the number and its mirror equivalent in the Principia.

The number is sub (n, 17, n) with a small s.

Its equivalent in the Principia is the string of symbols that we had abbreviated to ‘Sub (n, 17, n)’ with a capital S.

In simpler terms, ‘sub (n, 17, n)’ denotes an actual quantity in a manner similar to the way ’10 X 10’ denotes the quantity of 100.

On the other hand ‘Sub (n, 17, n)’ stands for meaningless strings of symbols of Principia that signifies a number, very much like a number-naming string such as:

‘ssssssssss0 X ssssssssss0’.

If you look up, you will see that I had said that the Gödel number small g had to be sub (n, 17, n).

Let us try to see why this is so.

We had admitted earlier that sub (n, 17, n) is the Gödel number of any formula that emerges when the numeral n is substituted for the variable  with Gödel number 17 (which is represented by the variable y) inside the formula whose Gödel number is that very n.

Yet the formula G was also obtained in exactly the same way.

While trying to get the formula G, we had started out with the formula that had the Gödel number n.

Then all the copies of ‘y’ in it were replaced with the numeral n. 

This them makes it obvious that sub (n, 17, n) is the Gödel number of G.

We also know that G is a formula within the Principia that mirrors the meta-mathematical statement:

‘The formula with Gödel number g is not demonstrable.’

That implies that within the Principia, the formula G stands for the meta-mathematical statement:

‘The formula G is not demonstrable’.

So what does this imply?

Simply that the formula G of the Principia is stating that it is not a theorem of the Principia.

This is the end of the discussion of the first point of Gödel’s argument.

Now we will move on to the elaboration of the second point of his argument.

I am sure the first point will leave you sleepless the whole night if you are in the Eastern hemisphere and keep you thinking the entire day if you are in the Western hemisphere.

Of course, if you are in the Middle East somewhere, try to keep yourself safe from the religious madness and mass butchery of human apes.

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

June 27, 2017 Tuesday

Bedtime Story 

Construction of the Formula 'G' with the meta-mathematical meaning ‘The formula with the Gödel number sub (n, 17, n) is not demonstrable.’ 

Last night we were dealing with a formula that we had labeled as Formula 1.

~(∃x) Dem (x, Sub (y, 17, y))  

This formula represents the meta-mathematical statement:

‘The formula with the Gödel number sub (y, 17, y) is not demonstrable.’ 

Though an alluring statement, it cannot a confirmatory statement since it involves a variable ‘y’.

It can only become a definite statement if y is replaced with one specific numeral.

What numeral can it be?

Let us see what Gödel showed.

The Formula 1 ~(∃x) Dem (x, Sub (y, 17, y)), as already stated, is from the Principia.

So it must have a Gödel number associated with it.

The number would be huge and tediously long.

Yet we need not worry and we need not take the pains of calculating it.

Let us simply assign its value to be represented by the letter ‘n’.

Then we will agree that whenever and wherever the variable ‘y’ appears, we will replace it with the number n, or to be more accurate, with the numeral for the number n.

We shall represent this numeral for the number n as ‘n’.

(This is much like writing ‘17’ when we know what we really mean to write is actually sssssssssssssssss0.)

Doing so will result in a new formula that will look like this:

~(∃x) Dem (x, Sub (n, 17, n))  

We shall call this formula ‘G’.

Now finally here we have the formula that is promising.

It is based exactly on the Formula 1 but a very specialized form of it.

Its meta-mathematical meaning is simple:

‘The formula with the Gödel number sub (n, 17, n) is not demonstrable.’ 

Since it has no variable, its meaning is definite.

Now that this formula G is definite and occurs within the Principia, it must have a Gödel number associated with it.

Let us name the associated Gödel number as g.

Now what can this number g be?

If you study the lines carefully, you will come to the conclusion that the number g has to be equal to:

g = sub (n, 17, n)

If you are not convinced about the number g, it is understandable.

We shall discuss it over 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

June 26, 2017 Monday

Bedtime Story 

Kurt Gödel Begins Weaving His Magic with the Formula ‘~(∃x) Dem (x, z)’ 

Last night we were on this formula:

‘(∃x) Dem (x, z)’

Just to bring to your attention that the words affirmation or demonstrable refers to formula being derivable within the Principia.

Adding a tilde sign before this formula gives us its formal negation.

Then we get this:

‘~(∃x) Dem (x, z)’

I am sure you can deduce what this formula states.

This formula is a formal representation of the meta-mathematical statement:

‘The formula with the Gödel number z is not demonstrable.’

Another way of putting it would be:

‘No proof can be shown for the formula with the Gödel number z.’

Gödel in his theorems had shown that a specific case of the formula was not demonstrable.

We will see how he constructed this special case.

Consider the formula below that we will label as Formula 1.

~(∃x) Dem (x, Sub (y, 17, y))   

This formula has been taken straight out of Principia Mathematica.

Now you will understand why so few apes (even mathematical apes) have ever had the courage to read this book.

The book is an unending series of - pages after pages and chapter after chapter - such notations.

Yet, since it is a formula of the Principia it is bound to have a meta-mathematical statement that must correspond with it.

What is that meta-mathematical statement?

You will recall from the past bedtime stories that the expression ‘Sub (y, 17, y)’ represents a number.

It needs to be pointed out immediately that even though the expression ‘Sub (y, 17, y)’ has been taken out from the Principia, it is not a formula but a kind of function for identifying a number.

The fact that ‘y’ is a variable and not a numeral, the expression ‘Sub (y, 17, y)’ cannot represent one specific number.

The number that it represents is the Gödel number of the formula obtained from the formula with Gödel number y by substituting for the variable with Gödel number 17 for the numeral for y.

In that case the Formula 1 represents the meta-mathematical statement:

‘The formula with the Gödel number sub (y, 17, y) is not demonstrable.’    

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

June 25, 2017 Sunday

Bedtime Story 

Gödel was Mindful of the Subversive Implications of his Paper

The last two lines of Gödel’s paper reflect modesty, or perhaps a fear that lurks in the mind of any scientist who is set to publish a path-breaking paper.

Very much like Charles Darwin, Gödel was mindful of the subversive implications of his paper and was apprehensive of ripples of shock waves it was bound to send into the universe of mathematics.

That is the reason why he was gentle in his conclusion that his arguments need to be further proved for other formal systems.

In other words, he was soothing some nerves beforehand in declaring that the proof of the generalization (applicability to other formal systems) of his radical idea would follow in his ensuing papers.

Little did he realize that the arguments that he developed in his paper were so convincing that there was little need of any further elaboration.

Hence it is crucial to stress that Gödel’s results were not an outcome of some flaws or insufficiency of the system of Principia Mathematica.

Gödel’s implication would hold firm and true to any system that is structured upon the arithmetic of integers, something that involves as basic as addition and multiplication.

Of course, only truly brilliant minds were capable of grasping the ramification of Gödel’s proofs.

One such mind was the great John von Neumann.

When Neumann saw this paper, with exasperation he exclaimed, “It’s all over”.

Now my dear fellow cousin apes, it is time to go back and quickly look at the five points under which Gödel’s reasoning is being scrutinized.

We shall tear apart and probe each of these five arguments in depth.

This will probably be the most punishing chapter of Gödel bedtime series, but perhaps also the most rewarding if you are able to comprehend the logic.

So buckle up and get set go.

(1) If you go back and look up (unless you have an incredible memory like mon ami), you will find that the formula ‘Dem (x, z)’ has been defined.

Within the Principia, it mirrors the meta-mathematical statement:

‘The sequence of the formulas with the Gödel number x is a proof for the formula with the Gödel number z.’

Now let us add to this formula the existential quantifier of the predicate logic “there is”.

Then we get:

‘(∃x) Dem (x, z)’

What does this formula say?

It simply says, ‘There exists a sequence of formulas (with Gödel number x) that embodies the affirmation of the formula with Gödel number z.’

Or in more simple terms, ‘The formula with Gödel number z is demonstrable.’

As usual, there is no need to hurry and we will carry it on 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

June 24, 2017 Saturday

Bedtime Story 

The Fifth and the Final Point of Gödel’s Argument 

Today we will briefly discuss the last and the fifth point of Gödel’s argument before going into elaborate discussion of all the five.

(5) Gödel finally constructed a formula A of the Principia that mirrored the meta-mathematical statement:

‘Principia Mathematica is consistent’.

We will study this formula A and its construction in greater detail later on.

Then he showed that the formula A  G can be derived within the Principia.

It essentially means that if A is true then G is also true.

Remember that formula G of Principia represents the meta-mathematical statement:

“The formula G is not demonstrable using the rules of Principia”.

Gödel demonstrated that the formula A cannot be derived within the axioms of the Principia.

Since the formula A could not be derived within the formal system, it logically follows the Principia as a formal system is not consistent.

In other words, not only the consistency of Principia cannot be established within the system but furthermore the consistency is not reflected within the system. 

These are the five main points around which the argument of Gödel was based and I will devote my next few bedtime stories in developing them in greater detail.

One thing I wish to clarify before that.

It is of great relevance since it mattered a great deal to Gödel.

You would have noticed all the while what I have been using the phrase “in the Principia” all the time when describing the formal system.

This perhaps might lead you to conclude, quite understandably, that these theorems merely reflect the incompleteness and inconsistency of the Principia Mathematica that was devised by Alfred North Whitehead and Bertrand Russell; One specific formal system.

Gödel suspected that the readers of his proofs who did manage to comprehend them would fall for this trap and he was particularly concerned about this matter.

Hence he made it a point to specifically point out that his results do not pertain to merely one formal system but is a generality that concerns and pertains even “to related systems”.

Let me quote the two sentences that Gödel wrote down towards the end of his paper:

“Throughout this work we have virtually confined ourselves to the system Principia Mathematica, and have merely indicated the applications to other systems.

The results will be stated and proved in fuller generality in a forthcoming sequel.”

We shall discuss in the nights to come about what he meant by these two slightly enigmatic statements.  

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

June 23, 2017 Friday

Bedtime Story 

The Second, Third and the Fourth Point of Gödel’s argument

We will continue where we left last night with the first point of Gödel’s argument that pertained to the construction of the formula G.

Gödel associated the formula G with its Gödel number g, and the construction is such that it says:

‘The formula that has a Gödel number g is not demonstrable.’

(2) Further, as a part of his second argument, Gödel went on to show that the formula G is demonstrable if and only if its formal negation ~G is demonstrable.

If you remember, this is once again very similar to Richard paradox where we came to the conclusion that ‘n is Richardian if, and only if, n is non Richardian’.

Yet we have agreed that in any formal system if both its formula and its negation are demonstrable, then that system is inconsistent.

A direct corollary of the above statement is that if the Principia is consistent, then neither G nor ~G ought to be derivable from the axioms.

This means to say that if Principia is consistent, then G is a formally undecidable formula.

(3) Gödel in his third argument showed that regardless of the fact that G is not demonstrable, it still is a true arithmetical formula.

G makes a statement that no integer can have a certain arithmetical property that Gödel defined.

Since Gödel proved that, it implies that G has to be true.

(4) Once it is accepted that G is both true and formally undecidable within the Principia, then it behooves us to accept the only logical conclusion; which is that the Principia is incomplete.

Once you accept that the Principia is incomplete, it stands to reason that all the arithmetical truths cannot be derived from its axioms and using its rules.

Gödel did not stop at that.

He destroyed Hilbert’s dream by stating that not only was the Principia incomplete, but it was essentially incomplete.

It is reasonable if you are wondering about the difference between being incomplete and being essentially incomplete.

By being essentially incomplete, Gödel was emphasizing on the attribute that even one were to bring into play ancillary axioms and rules into the Principia such that formula G would then be derivable, then another true formula G’ could then be constructed in the same manner which would remain undecidable and underivable within that new reinforced system.

And as you will understand, making further additions into the Principia to make the new formula G’ derivable would similarly lead to yet another true formula G’’ that would be underivable within this twice reinvigorated system – and so on and so forth.

This is what is meant by the system being “essentially incomplete”.

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