# Gödel Theorems of Incompleteness

Syed Fawad Bokhari

Particularly on the “Fermaish” of Idrees Azad sb, which was made on the first post of Khawar Asad, i.e. to discuss Gödel Theorems in a separate post. Some part of the this post contains the comments that I made during different discussions on “Mukalama” group.

## Gödel theorems of incompleteness:

The theorems (Theorem I and II) are a hallmark in mathematical logic and in the philosophy of mathematics.

Gödel’s first incompleteness theorem shows that any consistent effective formal system that includes enough of the theory is incomplete: the true mathemtical statements expressible in its (mathematical) language that are unprovable within the system.

Simply speaking: Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete,

and

the direct implication is: For any mathematical system, the consistent mathematical statements are unprovable within the system.

Godel’s 2nd theorem of incompleteness suggest:

————————————————————-

For any formal effectively generated theory, which includes basic arithmetical truths and also certain truths about formal provability, if the theory includes a statement of its own consistency then the theory is inconsistent.

A small explanation:

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One can “ONLY” prove unless and until one is in the domain of “contradictions”. However, the moment one come to the domain of “consistency” then one can have a ONLY a “relative truth” , which is kind of the “data fitting model”. So as long as one is in the domain of consistency, one CANNOT prove anything with “Absolute truthfulness” .

A simple example:

———————–

If one has a calculator which can take two inputs “a” and “b” and then it produced a results “c” which a simple addition of “a+b”. Then one can prove from calculator (containing the algorithm of addition as a “first principle”) all other mathematical theorems which originate from the principle of “addition”, but one would be left with a major problem:

One would never be able to prove the principle of addition, because it has been taken as a “first principle” and the journey started from there on wards. Now, if one has to prove the “first-principle” of addition, then one has to go beyond the boundaries of the “first principles”, which (in the present case) is the principle of “addition”.

In that domain (beyond the domain of addition), there may also exists a calculator which can take the same inputs “a” and “b” but this time it would not add but subtract. Which is in a sense, a contradiction to what has been initially thought. So with in the domain of “addition” calculator, one would find an outstanding consistency with regard to the “addition” but “no absolute prove” and hence no absolute truth.

One implication:

——————–

Even if, one has all the resources, it has been mathematically proven in the field of mathematical logic that one will not never be able to prove (find) the “first principles” that are playing role in the existence of this very world (perceived as a reality). However, in order to so so one has to go into a domain, where exists the contradiction to those laws that are the very foundation of ones own existence. 🙂

……………………………………….

Syed Fawad Bokhari The actual proof of this theorem can be found on this translated version:

http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf

July 17 at 12:05am · Like · 1

Idrees Azad کافی مشکل ہے۔ دوبارہ پڑھتا ہوں۔

July 17 at 12:07am · Like · 2

Syed Fawad Bokhari jee zaroor,

July 17 at 12:07am · Like · 1

Syed Fawad Bokhari Jee Idrees Azad sb, kyaa banaa? Mujhay maloom hai, mairee “report writing” bohat kheraab hai,

July 17 at 12:25am · Like · 2

Idrees Azad سچی بات ہے سمجھ نہیں آیا۔ دو نہیں۔ تین بار پڑھا ہے۔ آپ کے بیان میں کمزوری نہیں۔ میرا ہاتھ ریاضی میں تنگ ہی رہا ہے۔ لیکن ایک تأثر سا ملا ہے۔

if the theory includes a statement of its own consistency then the theory is inconsistent.

July 17 at 1:11am · Like · 2

Idrees Azad آپ کچھ مزید روشنی ڈالیے نا۔ مجھے دلچسپی پیدا ہورہی ہے اس تھیورم کی رُوح کو سمجھنے میں۔

July 17 at 1:12am · Edited · Like · 1

Syed Fawad Bokhari aapko woh “calculator” kay istaaray waali misaal samajh main aayee?

July 17 at 1:30am · Like · 1

Syed Fawad Bokhari aapkaa “like” khooch bataa rahaa hai. dobaaraa koshish kertaa hoon,

July 17 at 1:52am · Like · 2

Idrees Azad میرے نیٹ کی سپیڈ خراب ہے ۔

July 17 at 2:03am · Edited · Like · 1

Syed Fawad Bokhari ?

July 17 at 2:03am · Like · 2

Idrees Azad جی ہاں کچھ کچھ سمجھ آئی۔ اصل میں فرسٹ پرسنپل کیا ہے؟ یہ ٹھیک سے سمجھ آجائے تو بات سمجھ آجائے گی۔

July 17 at 2:03am · Like · 2

Syed Fawad Bokhari First principle is a fundamental proposition or postulate that serves as the starting points of any other proposition and it cannot be deduced from any other proposition or postulate.

In mathematics the postulate is a starting point of reasoning.

July 17 at 2:30am · Like · 2

Idrees Azad دوبارہ حاضری پر بات ہوتی ہے۔ ایک درخواست پیش کرونگا۔

July 17 at 5:24am · Like · 2

حمید نیازی میں بھی کچھ سیکھنا چاہتا ہوں اس موضوع پر- کچھ ایل علم دوستوں کو یہاں ٹیگ کیا جائے-

July 17 at 9:21am · Like · 5

Syed Fawad Bokhari I am posting three links (each containing one part) to one lecture about Gödel and the “incomplete theorem”. Please let me know if they are any help. If, not, then just let me know and I will try to arrange other resources. YouTube is ban is Pakistan,…See More

July 17 at 12:05pm · Edited · Like · 2

Syed Fawad Bokhari Ager ijazat ho tou aik sawal puchoon…

Kyaa kisee nay yea lecture sunay bhee hain kay nahee?

Because, if not then I may devise another way to explain

Thursday at 11:25pm · Like · 1

حمید نیازی Weekend par sunte hain aor baat hoti hai. More interested in conceptualizing the theorem for my understanding rather than the actual technical proof which in some case spans many branches of mathematics even for a one liner elegant problem like Fermat’s Last Theorem.

Friday at 8:44am · Like · 1

Syed Fawad Bokhari Stephen Hawking speech (Godel and the End of the Universe) on the Gödel Theorem

http://www.hawking.org.uk/godel-and-the-end-of-physics.html

Godel and the End of Physics – Stephen Hawking

www.hawking.org.uk

Friday at 10:59pm · Like

Idrees Azad “Some people will be very disappointed if there is not an ultimate theory that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind. I’m now glad that our search for understanding will never come…See More

Friday at 11:15pm · Like

Syed Fawad Bokhari You are welcome

Friday at 11:16pm · Like · 1

Syed Fawad Bokhari So, Idrees Azad sb, kyaa khooch clear hoyaa?

Friday at 11:29pm · Like · 1

Idrees Azad اب ہوگا نا۔ انشااللہ