Tagged: Continuum Hypothesis

Mathematical Infinity

“There are two approaches to mathematical infinity. It can be seen as defining limiting cases that can never be realized or as existing in some philosophical sense. These mathematical approaches parallel approaches to meaning and value that I call absolutist and evolutionary. The absolutist sees ultimate meaning as something that exists most commonly in the form of an all powerful infinite God. The evolutionary sees life and all of a creation as an ever expanding journey with no ultimate or final goal. There is only the journey. There is no destination. This video argues for an evolutionary view in our sense of meaning and values and in our mathematical understanding. There is a deep connection between the two with profound implications for the evolution of consciousness and human destiny.” (Paul Budnik)

Continuum Hypothesis

Georg Cantor originally proposed that there is no infinite set with a cardinal number between that of the “small” infinite set of natural numbers (N) and the “large” infinite set of real numbers C (the “continuum”).  “There is no set whose cardinality is strictly between that of the integers and that of the real numbers.”  This means that N=Aleph0, C=Aleph1. Gödel showed that no contradiction arises if the continuum hypothesis is added to conventional Zermelo-Fraenkel set theory. However, Paul Cohen proved in 1963 that no contradiction arises if the negation of the continuum hypothesis is added to set theory. Together,...