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Essay on Russells paradox and Incompleteness theorem

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Introduction

Neil Sanghrajka Essay on Russell?s paradox and Incompleteness theorem For me mathematics is like a tall building, and from time to time mathematicians create new floors of formulas and theorems. A new floor cannot be built without a previous floor, similarly in Mathematics new theorems rely on deductive reasoning using previous knowledge. Mathematics is an axiomatic system; these ?truths? build the foundation of the field. However Russell?s paradox and the incompleteness theorem state that the very foundation of mathematics is inconsistent. Russell?s paradox shows an inconsistency in one of the axioms of the set theory. ...read more.

Middle

To prove or disprove a statement, axioms are used in the proof. However if there is an inconsistency in the axiom itself, the proof can never be definite. An example is the set of all even natural numbers is infinite, and the set of all natural numbers is infinite. However the set of all natural numbers must be bigger than the set of all natural even numbers, but strangely both are infinite and hence this statement cannot be proved. The incompleteness theorem and Russell?s paradox conclude that in order to have a complete system, it must not contain axioms. ...read more.

Conclusion

Generally, mathematics obeys the axioms and has no inconsistencies. I fail to understand the practical purpose of these irregularities within the system. On a philosophical level it proves that like other areas of knowledge Mathematics is indefinite and we can never know the completeness of the system. However using simple mathematics, we can still perform our daily tasks, and still perform calculations strong enough to help us send a rover to Mars. I believe that even if the axioms are wrong, the inconsistency is repeated throughout the system and hence there is still validity to Mathematics. Having a system that avoids self referencing and axioms, will make Mathematics an area only for the scholars and not common people, as it will be difficult to understand for the masses. -490 words ...read more.

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