A complete analysis of Shannon's telegraph channel is given, making use of matrices with elements from a division ring. A notation is developed for representing the set of signals of a communication channel

The concept of functional dependencies in databases is generalized and called generic dependencies. Just as Karnaugh Map exhibits all the functional dependencies in a relation, Entropy Map represents all the generic dependencies. A generalized normal form useful in database design is defined

Ackermann functions are used recursively to define the transfinite cardinals of Cantor. Axiom of Monotonicity is defined and used to derive Continuum Hypothesis. Axiom of Fusion is defined and used to split the unit interval into infinitesimals

Probabilistic induction allows us to conclude that the probability of a hypothesis being true increases when evidences in favor of the hypothesis occur. A claim made by Karl Popper and David Miller that probabilistic induction is impossible is shown to be untenable, using an example

Sentient Arithmetic is defined as an extention of Elementary Arithmetic with three more derivation rules and the Incompleteness Theorems are derived within it without using any metalanguage. It is shown that Consistency cannot be chosen as an axiom