  This paper outlines new paradigms for real analysis and computability theory
in the recently proposed non-Aristotelian finitary logic (NAFL). Constructive
real analysis in NAFL (NRA) is accomplished by a translation of diagrammatic
concepts from Euclidean geometry into an extension (NPAR) of the NAFL version
of Peano Arithmetic (NPA). Such a translation is possible because NPA proves
the existence of every infinite proper class of natural numbers that is
definable in the language of NPA. Infinite sets are not permitted in NPAR and
quantification over proper classes is banned; hence Cantor's diagonal argument
cannot be legally formulated in NRA, and there is no `cardinality' for any
collection (`super-class') of real numbers. Many of the useful aspects of
classical real analysis, such as, the calculus of Newton and Leibniz, are
justifiable in NRA. But the paradoxes, such as, Zeno's paradoxes of motion and
the Banach-Tarski paradox, are resolved because NRA admits only closed
super-classes of real numbers; in particular, open/semi-open intervals of real
numbers are not permitted. The NAFL version of computability theory (NCT)
rejects Turing's argument for the undecidability of the halting problem and
permits hypercomputation. Important potential applications of NCT are in the
areas of quantum and autonomic computing.
