1 Biological Information from Practicing the Incompleteness Theorem Positively Koichiro Matsuno Nagaoka University of Technology Nagaoka 940-2188, Japan Abstract A theoretical basis of reasoning the origin and development of biological information can be founded on the second incompleteness theorem discovered by Kurt Gödel in 1931. The incompleteness theorem stating that there should be a statement or a theorem that could be right but not proved right within the given axiomatic system carries a positive implication as focusing upon the role of inductive judgment on the part of mathematicians going beyond the stipulation of the system. Likewise, atoms and molecules in the empirical world exercise their capacity of induction on their own. When two hydrogen atoms form a hydrogen molecule empirically, no computation for obtaining a hydrogen molecule can halt insofar as one sticks to the axiomatic formalism preserving the hydrogen atom as a nonnegotiable element. What underlies the transition from hydrogen atoms to a hydrogen molecule is transformation of a quantum. In view of the fact that each quantum is characterized by a set of infinities or infinite varieties it can allow within itself, transformation of a quantum assumes the act of transition from one class of infinities to another one. In fact, the class of infinities unique to a hydrogen atom is different from another one belonging to a hydrogen molecule. The origin of biological information can be associated with the occurrence of transformation of a quantum closed to material flow to another type of quantum open to material flow, in which each quantum is taken to be a robust material unit admitting within itself a class of infinities. A concrete implementation of a quantum open to material flow is in the genesis of a robust heat engine operating in the presence of temperature gradients. Empirically, the evolutionary likelihood of expecting such temperature gradients must have been near hydrothermal vents on the sea floor in the primitive ocean. Theoretically, on the other hand, the origin and development of biological information could be built upon the transformation of the robust class of infinities that no axiomatic system can internalize into itself. Naturalization of the incompleteness theorem underlies the physical origin and development of biological information. Keywords: Heat engine, Incompleteness theorem, Induction, Infinities, Quantum, Temperature gradients.