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We discuss a notion of forcing that characterizes enumeration -genericity, and we investigate the immunity, lowness, and quasiminimality properties of enumeration -generic sets and their degrees. We construct an enumeration operator such that, for any , the set is enumeration -generic and has the same jump complexity as . We deduce from this and other recent results from the literature that not only does every degree bound an enumeration -generic degree such that , but also that, if is nonzero, then we can find such satisfying . We conclude by proving the existence of both a nonzero low and a properly nonsplittable enumeration -generic degree, hence proving that the class of -generic degrees is properly subsumed by the class of enumeration -generic degrees.
The liar and kindred paradoxes show that we can derive contradictions if our language possesses sentences lending themselves to paradox and we reason classically from schema (T) about truth: where the letter is to be replaced with a sentence and the letter with a name of that sentence. This article presents a theory of truth that keeps (T) at the expense of classical logic. The theory is couched in a language that possesses paradoxical sentences. It incorporates all the instances of the analogue of (T) for that language and also includes other platitudes about truth. The theory avoids contradiction because its logical framework is an appropriately constructed nonclassical propositional logic. The logic and the theory are different from others that have been proposed for keeping (T), and the methods used in the main proofs are novel.
The logic of paradox, , is a first-order, three-valued logic that has been advocated by Graham Priest as an appropriate way to represent the possibility of acceptable contradictory statements. Second-order is that logic augmented with quantification over predicates. As with classical second-order logic, there are different ways to give the semantic interpretation of sentences of the logic. The different ways give rise to different logical advantages and disadvantages, and we canvass several of these, concluding that it will be extremely difficult to appeal to second-order for the purposes that its proponents advocate, until some deep, intricate, and hitherto unarticulated metaphysical advances are made.
This is in part a reply to a recent work of Vidal-Rosset, which expresses various mistaken beliefs about Core Logic. Rebutting these leads us further to identify, and argue against, some mistaken core beliefs about logic.
Reverse mathematics is a program in the foundations of mathematics. It provides an elegant classification in which the majority of theorems of ordinary mathematics fall into only five categories, based on the “big five” logical systems. Recently, a lot of effort has been directed toward finding exceptional theorems, that is, those which fall outside the big five. The so-called reverse mathematics zoo is a collection of such exceptional theorems (and their relations). It was previously shown that a number of uniform versions of the zoo theorems, that is, where a functional computes the objects stated to exist, fall in the third big five category, arithmetical comprehension, inside Kohlenbach’s higher-order reverse mathematics. In this paper, we extend and refine these previous results. In particular, we establish analogous results for recent additions to the reverse mathematics zoo, thus establishing that the latter disappear at the uniform level. Furthermore, we show that the aforementioned equivalences can be proved using only intuitionistic logic. Perhaps most surprisingly, these explicit equivalences are extracted from nonstandard equivalences in Nelson’s internal set theory, and we show that the nonstandard equivalence can be recovered from the explicit ones. Finally, the following zoo theorems are studied in this paper: (existence of uniformly -generics), (finite intersection principle), 1-GEN (existence of -generics), OPT (omitting partial types principle), AMT (atomic model theorem), SADS (stable ascending or descending sequence), AST (atomic model theorem with subenumerable types), NCS (existence of noncomputable sets), and KPT (Kleene–Post theorem that there exist Turing incomparable sets).
A significant open problem in inner model theory is the analysis of as a strategy premouse, for a Turing cone of reals . We describe here an obstacle to such an analysis. Assuming sufficient large cardinals, for a Turing cone of reals there are proper class -small premice , with Woodin cardinals , respectively, such that and are in , and are countable in , and the pseudo-comparison of with succeeds, is in , and lasts exactly stages. Moreover, we can take , the minimal iterable proper class inner model with a Woodin cardinal, and take to be -like and short-tree-iterable.
We study the uniform computational content of different versions of the Baire category theorem in the Weihrauch lattice. The Baire category theorem can be seen as a pigeonhole principle that states that a complete (i.e., “large”) metric space cannot be decomposed into countably many nowhere dense (i.e., small) pieces. The Baire category theorem is an illuminating example of a theorem that can be used to demonstrate that one classical theorem can have several different computational interpretations. For one, we distinguish two different logical versions of the theorem, where one can be seen as the contrapositive form of the other one. The first version aims to find an uncovered point in the space, given a sequence of nowhere dense closed sets. The second version aims to find the index of a closed set that is somewhere dense, given a sequence of closed sets that cover the space. Even though the two statements behind these versions are equivalent to each other in classical logic, they are not equivalent in intuitionistic logic, and likewise, they exhibit different computational behavior in the Weihrauch lattice. Besides this logical distinction, we also consider different ways in which the sequence of closed sets is “given.” Essentially, we can distinguish between positive and negative information on closed sets. We discuss all four resulting versions of the Baire category theorem. Somewhat surprisingly, it turns out that the difference in providing the input information can also be expressed with the jump operation. Finally, we also relate the Baire category theorem to notions of genericity and computably comeager sets.
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