Wow ! Just now understood something about the incompleteness theorem. I am really amazed by the keen insight of Kurt Godel. While Principia Mathematica (PM) by Alfred North Whitehead and Bertrand Russell needed the necessity of having self-contradicting statements for the existence of a mathematical system, Godel proved the unnecessity of having such statements. Let me try to explain. Consider the following statement, S.
S : This statement is false.
If asked whether 'S' is true or false, we'll get into a paradox. These types of paradox is fundamental in any logical system we build. Although, we feel 'S' is a genuine statement we are not able to ascertain any truth to it. So Whitehead and Russel admitted these statements as an integral part of a logical system. Here is where the genius of Godel lies. He made a 'coding machine' which would transform 'S' (or any other statement) to a statement in Number Theory. He then showed the impossibility of existence of a proof for the transformed statement within the framework of PM. Thus, 'S' is 'true' but we cannot prove or disprove it within the system of PM. Hence, the present mathematical system is 'incomplete' as it is not possible to verify the truth. Moreover, the greatness about Godel is that he showed that this 'incompleteness' is just not restricted to the system of PM but to any system that tries to achieve the aims of PM.
The genius Godel differentiated the genuine 'feeling' about 'S' from examining its truth. He showed the triumph of Truth over Provability. He showed the triumph of Self-awareness over Self-contradiction. He showed the triumph of God over Man !
S : This statement is false.
If asked whether 'S' is true or false, we'll get into a paradox. These types of paradox is fundamental in any logical system we build. Although, we feel 'S' is a genuine statement we are not able to ascertain any truth to it. So Whitehead and Russel admitted these statements as an integral part of a logical system. Here is where the genius of Godel lies. He made a 'coding machine' which would transform 'S' (or any other statement) to a statement in Number Theory. He then showed the impossibility of existence of a proof for the transformed statement within the framework of PM. Thus, 'S' is 'true' but we cannot prove or disprove it within the system of PM. Hence, the present mathematical system is 'incomplete' as it is not possible to verify the truth. Moreover, the greatness about Godel is that he showed that this 'incompleteness' is just not restricted to the system of PM but to any system that tries to achieve the aims of PM.
The genius Godel differentiated the genuine 'feeling' about 'S' from examining its truth. He showed the triumph of Truth over Provability. He showed the triumph of Self-awareness over Self-contradiction. He showed the triumph of God over Man !
7 comments:
How do you say "S is true"???
S is a geniune statement without gramatical errors and so it is true.
The key thing to note here is when I say "S is true", I think about 'S' from a more general system of languages than confining to the world of mathematics. And this is where Godel made a difference. The 'actual' truth may be different from 'relative' truth.
Another way of looking at this is that, when I say "S is true", I mean the "existence of the statement S" is true. This is much more philosophical and fundamental than looking 'S' from the system of languages.
I think you mean to say that "S is a valid proposition (since it is grammatically correct) but not a theorem"
Hi Swami,
In the system of formal languages the notion of a 'theorem' is much more generalized. There they precisely define what is meant by a theorem, proof etc. Of course, these notions coincide with what we mean by theorem in mathematics. There a 'valid proposition' becomes a theorem and Godel showed that there is no proof for the theorem within the scope of the generalized system. So, 'S' is a theorem but it cannot have any proof. In other words, S is a true statement(or theorem) without any proof. I would suggest you to read GEB (I have a post abt it). You'll definitly like it.
From what I understood, S is not a theorem unless it can be formally deduced from axioms.
However, there can be propositions (which are just valid constructions based on the axioms and rules of the language) that are not theorems. S is one such.
And we know S is one such - because if we assume S to be a theorem, we can deduce that S is not a theorem.
Godel's paper starts "On formally undecidable propositions... " not theorems...
Am I making sense??
I think you are right here. Not all valid propositions can be theorems and S is one such. Thanks da !
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