• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Essay on Russells paradox and Incompleteness theorem

Extracts from this document...

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.

The above preview is unformatted text

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. Extended Essay- Math

    A'�$u�'�l'��#e�(��eh��v��d�%Q��\�� ��`'"�"��1/2-�"\Ô�z9n�M�6��W2�'�?�� sw� ���w�X�!(tm)��nB&qĹ"-�y�\'���(��(a��)[$��" �aҳ�"�"O>Iz "�W0P�isI�m, ��[email protected]��##EN�4)?�r�'�G"�9��'�����p'm.��P�v:e� ,w e��(c)hãµï¿½ ��;-\�'a"-�%�%��% ��#��'���V�T9u�T�f��l ��R�F�S0P� "*-�9�Âl_q*�i'�" 3f���"�aʳ����?���{�Æ)���p(c)rI �E�.��AEw �v�m'��#u���;}G�d5��% &�EXm*�o�AJ14�p�]w�ms�v>}���ρC4���*�"�$ �a!-�]' �D���ŨQ�Æk��[email protected]"�4���*�"�"��$�6U�)�\> ݱ I%yy�+'�mL(tm)8�*_�%É�W$_�7��'�nT�V�d��$ bz"�?�(tm)@�h��TK�K�...��y��$t�-й�G�zh����(tm)p F$�pn'��/T!"���%9.'0I(c)���� R��E�9Ҫ�Y8"S\'�U3/4PÍ�;-%.�n��*���"�S�B�5�)�_� U�~':����"w"ZZE�p" �k��h�' �v�--��o"��-D�7i���?�y'��- Yr�'%:�!}Ò�$ ^u�Ub'�A�%N�v��aM��-�#���3ck����%Q��­(r)�! ��@�F�T�xx�#yv�"J���.� �sI\Û¸q- �[email protected]@Rd�T��B�O'��'�["q.�~�z�m�M��Öl0 ���U�'2!N>�d�(� �^<�3^x��f��|�'w�u)�.R�-#�I\$��Y�f�?Õ� t�t�"�-Ò¸K����M�c��1/4y� '�P2����Hl'$��]S?��ү_�_|�b80!�*�z�+�1/4B �{�1/2×µkWË­O�-Z1/4��[-S?\.���,�E][,Ҫ�a�~�"�" nnh��_��'�tÒ¯~�+$w )�L׸ ��i+6�����n"�'3/4Wer"��R�"�aK��'.l�/-���?��?"��!�2�A"� �F�@��&�%�"$ �x<]g�,�W�^�E�N�rz�'�"��_���/~��+����Zϸ��%�J��=� ^2����h�"o��-L&�� � >-�ĸt�(c)�"Nb�;��#8PP�)l �ܷo��+�1/4�6d"�]B'�$qd;v" ��L�4,R�vB�P�"A�.9��H��I�(�Bo��v�Æko3f� q'�t�(c)<�C�X�n(c)��"�>�o%�q����pQ�uV���`�z�m�é£P�����'N���G-[��ã>�hp�"�/��<�0i�Ͽ���I�NR'@hT�@�`+"��ZÆ�'"a'r�Ämq��-�Q���wB 4p�ҥyq�Ê0�믿3/4�K(r)

  2. Future Career Paths in the Field of Cosmetic Surgery

    The fact is, before the actually surgery, the surgeon has to make a lot of precise evaluations and procedures. During the whole process, mathematic calculation could decide how ideal the computer model will turn out to be. What the surgeon does is scan the patient's face with safe white light

  1. Stellar Numbers. In this task geometric shapes which lead to special numbers ...

    1 As this is a quadratic equation, a graph was plotted to demonstrate how it expanded Stage Number Number of Points Notes and Observations 4S0 1 None 4S1 9 Adding 8 to previous 4S2 25 Adding 8x2 to previous 4S3 49 Adding 8x3 to previous 4S4 81 Adding 8x4 to

  2. Investigating ratio of areas and volumes

    This conjecture can be proven using calculus. The area under a graph is equal to the integral of the graph in between the two arbitrary points: Area B: Area B is the area under the graph of y = xn in between the arbitrary points a and b.

  1. Salida del sol en NY

    m�s temprano esto es en la semana 24 que ser�a el mes de agosto, y para el hemisferio norte le corresponder�a la estaci�n del verano, donde los d�as son m�s largos, en cambo el d�a donde sale m�s tarde el sol, se encuentra en la primera semana de Enero, c�mo

  2. Infinity Essay

    In geometry one might define a point of infinity as the point of and intersection of two parallel lines. Cantor defined two kinds of infinite numbers, the ordinal numbers; is identified in a stopping point when counting; and the cardinal numbers, define how many numbers a set may contain; when that set is too large it is called uncountable.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work