 There have proved to be many places throughout analysis where there are distinctive inconsistent insights. The examples in the remainder of this section are drawn from Mortensen (1995). For example: (1) Robinson’s (1974) non-standard analysis was based on infinitesimals, quantities smaller than any real number, as well as their reciprocals, the infinite numbers. This has an inconsistent version, which has some advantages for calculation in being able to discard higher-order infinitesimals. Interestingly, the theory of differentiation turned out to have these advantages, while the theory of integration did not. A similar result, using a different background logic, was obtained by Da Costa (2000). (2) Another place to find applications of inconsistency in analysis is topology, where one readily observes the practice of cutting and pasting spaces being described as “identification” of one boundary with another. One can show that this can be described in an inconsistent theory in which the two boundaries are both identical and not identical, and it can be further argued that this is the most natural description of the practice. (3) Yet another application is the class of inconsistent continuous functions. Not all functions which are classically discontinuous are amenable of inconsistent treatment; but some are, for example f(x)=0 for all x&lt;0 and f(x)=1 for all x≥0. The inconsistent extension replaces the first &lt; by ≤, and has distinctive structural properties. These inconsistent functions may well have some application in dynamic systems in which there are discontinuous jumps, such as quantum measurement systems. Differentiating such functions leads to the delta functions, applied by Dirac to the study of quantum measurement. (4) Next, there is the well-known case of inconsistent systems of linear equations, such as the system (i) x+y=1, plus (ii) x+y=2. Such systems can potentially arise within the context of automated control. Little work has been done classically to solve such systems, but it can be shown that there are well-behaved solutions within inconsistent vector spaces. (5) Finally, one can note a further application in topology and dynamics. Given a supposition which seems to be conceivable, namely that whatever happens or is true, happens or is true on an open set of (spacetime) points, one has that the logic of dynamically possible paths is open set logic, that is to say intuitionist logic, which supports incomplete theories par excellence. This is because the natural account of the negation of a proposition in such a space says that it holds on the largest open set contained in the Boolean complement of the set of points on which the original proposition held, which is in general smaller than the Boolean complement. However, specifying a topological space by its closed sets is every bit as reasonable as specifying it by its open sets. Yet the logic of closed sets is known to be paraconsistent, ie. supports inconsistent nontrivial theories; see Goodman (1981), for example. Thus given the (alternative) supposition which also seems to be conceivable, namely that whatever is true is true on a closed set of points, one has that inconsistent theories may well hold. This is because the natural account of the negation of a proposition, namely that it holds on the smallest closed set containing the Boolean negation of the proposition, means that on the overlapping boundary both the proposition and its negation hold. Thus dynamical theories determine their own logic of possible propositions, and corresponding theories which may be inconsistent, and are certainly as natural as their incomplete counterparts.