On the Partial Respects in Which a Real Valued Arithmetic System Can Verify Its Tableaux Consistency

It is known that Equations (1) – (3) are related to both generalizations of Gödel’s Second Incompleteness Theorem and to its boundary-case exceptions. For instance, Equation (1) will enable the Second Incompleteness Theorem to apply to Hilbert deduction. Also, [36, 38] showed that the semantic tableaux version of the Second Incompleteness Theorem generalizes for essentially all axiom systems that can prove the validity of Equations (1) – (3) for integer arithmetic. On the other hand, [35, 38, 39, 41] showed exceptions to the semantic tableaux version of the Second Incompleteness Theorem do exist when an axiom system fails to support Equation (3). The preceding research naturally raises the question whether or not an analogous phenomenon holds when one changes the venue of application from integer arithmetic to the addition and multiplication operations of a computer’s floating point arithmetic set. Throughout this paper, we will use the term simulated real-arithmetic to refer to an instruction set that is slightly more general and powerful than the common floating point instructions on a digital computer’s hardware. We will prove that simulated real arithmetic is quite unlike integer arithmetic — insofar as an axiom system can simultaneously recognize its semantic tableaux consistency and the validity of Equations (1) – (3) for simulated real arithmetic. This result is significant because a computer’s floating point instruction set has essentially as many practical applications as an integer arithmetic. Moreover, Section 5