Essay on Russells paradox and Incompleteness theorem

by neilsanghrajka (student)

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. This example shows how mathematics fails the coherent truth test. The coherent truth test states that the premise for deductive reasoning must be logically consistent. Russell’s paradox gives an example of the incompleteness of mathematics.

Gödel’s incompleteness theorem says that any axiomatic system cannot be complete and consistent at the same time. Using deductive reasoning, we may reach a point where a mathematical statement can neither be proved nor disproved. This is a simpler interpretation of the incompleteness theorem. 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 ...

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The incompleteness theorem and Russell’s paradox conclude that in order to have a complete system, it must not contain axioms. Axioms are assumptions whose validity is generally accepted. However a system will remain incomplete until all its axioms are proved correctly. For instance Euclid’s fifth postulate regarding parallel lines cannot be proved because in order to show that these two lines never meet, someone needs to travel till infinity to show that these lines do not intersect which is not possible.

Though both these flaws in mathematic make logical sense and I understand the incompleteness of Mathematics, I believe that these pertain to very special cases. 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.

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