In intuitionist logic, the negation of a proposition holds on the largest open set contained in the Boolean complement of the original proposition's set, which is in general smaller than the Boolean complement itself, yielding incompleteness
Open set(describing how negation works in intuitionist logic)
A mathematical concept describing a collection of points or values where you can always find a little space around any point that still belongs to the collection—think of it as 'not including the edges.'
incompleteness(Philosophy of mathematics / formal systems)
The property of a formal system whereby there exist true statements within the system that cannot be proven by the system itself.
negation(Standard truth conditions for logical negation, used as the basis for arguments against dialetheism)
¬A is true if and only if A is not true
proposition(Used in the context of a semantic theory sensitive to differences in subject matter.)
The content expressed by a sentence, individuated at least in part by the subject matter of the sentence and the contents of its subsentential expressions.
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. Inter