Anselm described god as “something than which nothing greater can be thought” [1, p. 93], and Descartes viewed him as “a supreme being” [7, p. 122]. I first capture those characterizations formally in a simple language for monadic predicate logic. Next, I construct a model class inspired by Stoic and medieval doctrines of grades of being [8, 20]. Third, I prove the models sufficient for recovering, as internal mathematics, the famous ontological argument of Anselm, and show that argument to be, on this formalization, valid. Fourth, I extend the models to incorporate a modality fit for proving that any item than which necessarily no greater can be thought is also necessarily real. Lastly, with the present approach, I blunt the sharp edges of notable objections to ontological arguments by Gaunilo and by Grant. A trigger warning: every page of this writing flouts the old saw “Existence is not a predicate” and flagrantly.

There seems to be a view that intuitionists not only take the Axiom of Choice (AC) to be true, but also believe it a consequence of their fundamental posits. Widespread or not, this view is largely mistaken. This article offers a brief, yet comprehensive, overview of the status of AC in various intuitionistic and constructivist systems. The survey makes it clear that the Axiom of Choice fails to be a theorem in most contexts and is even outright false in some important contexts. Of the systems surveyed, only intensional type theory renders AC a theorem, but the extent of AC in that theory does not include, for instance, real analysis. Only a small amount of extensionality is required in order for the obvious proof an intuitionist might offer for AC to break down.

Applications of 2-methoxy-3,6-dichlorobenzoic acid (dicamba) and 2,4-dichlorophenoxyacetic acid (2,4-D) controlled top growth the year of treatment, but western ironweed (Vernonia baldwini Torr.) recovered in the following year. One-half lb/A or more of 4-amino-3,5,6-trichloropicolinic acid (picloram) effectively controlled top growth and destroyed roots and rhizomes. Treatment with 2 lb/A of picloram resulted in ironweed-free plots by the fourth season after treatment.

The sedge genus Kyllinga consists of 40 to 45 species distributed in tropical, subtropical, and warm temperate regions around the world (KUkenthal 1936; Tucker 1987). This genus of low rhizomatous perennials or cespitose annuals is classified in the large cosmopolitan family Cyperaceae. Many Kyllinga species are considered weedy (Holm et al. 1979; Tucker 1987), while Kyllinga nervosa Steudel is considered an important forage plant in Africa (McNaughton 1985).

Straightforwardly and strictly intuitionistic inferences show that the Brouwer– Heyting–Kolmogorov (BHK) interpretation, in the presence of a formulation of the recognition principle, entails the validity of the Law of Testability: that the form ¬ f V ¬¬ f is valid. Therefore, the BHK and recognition, as described here, are inconsistent with the axioms both of intuitionistic mathematics and of Markovian constructivism. This finding also implies that, if the BHK and recognition are suitably formulated, then Brouwer’s original weak counterexample reasoning was fallacious. The results of the present article extend and refine those of McCarty, C. (2012). Antirealism and Constructivism: Brouwer’s Weak Counterexamples. The Review of Symbolic Logic. First View. Cambridge University Press.

Strictly intuitionistic inferences are employed to demonstrate that three conditions—the existence of Brouwerian weak counterexamples to Test, the recognition condition, and the BHK interpretation of the logical signs—are together inconsistent. Therefore, if the logical signs in mathematical statements governed by the recognition condition are constructive in that they satisfy the clauses of the BHK, then every relevant instance of the classical principle Test is true intuitionistically, and the antirealistic critique of conventional logic, once thought to yield such weak counterexamples, is seen, in this instance, to fail.

Since intuitionistic sets are not generally stable – their membership relations are not always closed under double negation – the open sets of a topology cannot be recovered from the closed sets of that topology via complementation, at least without further ado. Dana Scott asked, first, whether it is possible intuitionistically for two distinct topologies, given as collections of open sets on the same carrier, to share their closed sets. Second, he asked whether there can be intuitionistic functions that are closed continuous in that the inverse of every closed set is closed without being continuous in the usual, open sense. Here, we prove that, as far as intuitionistic set theory is concerned, there can be infinitely-many distinct topologies on the same carrier sharing a single collection of closed sets. The proof employs Heyting-valued sets, and demonstrates that the intuitionistic set theory IZF [4, 624], as well as the theory IZF plus classical elementary arithmetic, are both consistent with the statement that infinitely many topologies on the set of natural numbers share the same closed sets. Without changing models, we show that these formal theories are also consistent with the statement that there are infinitely many endofunctions on the natural numbers that are closed continuous but not open continuous with respect to a single topology.

For each prime k ∈ ω, let Ak be this ω-sequence of sets open in the standard topology on the closed unit interval: for each n ∈ ω,

We call a logic regular for a semantics when the satisfaction predicate for at least one of its nontheorems is closed under double negation. Such intuitionistic theories as second-order Heyting arithmetic HAS and the intuitionistic set theory IZF prove completeness for no regular logics, no matter how simple or complicated. Any extensions of those theories proving completeness for regular logics are classical, i.e., they derive the tertium non datur. When an intuitionistic metatheory features anticlassical principles or recognizes that a logic regular for a semantics is nonclassical, it proves explicitly that the logic is incomplete with respect to that semantics. Logics regular relative to Tarski, Beth and Kripke semantics form a large collection that includes propositional and predicate intuitionistic, intermediate and classical logics. These results are corollaries of a single theorem. A variant of its proof yields a generalization of the Gödel-Kreisel Theorem linking weak completeness for intuitionistic predicate logic to Markov's Principle.

It follows constructively from weak versions of Markov's principle (MP) and of Church's thesis (WCT) that logical validity for single sentences is not arithmetically definable. This is a direct consequence of the constructive model-theoretic fact that, using MP and WCT, one can prove the categoricity of first-order Heyting arithmetic. By a straightforward refinement of this proof, we then obtain generalizations of the incompleteness theorem of Kreisel [K 2] as presented by van Dalen [Da] and improved by Leivant [L 1]. An analogous result for classical validity in r.e. models appears in [V].

Our methods of proof are new in that they rely not upon variants of the ideas of [K2] but upon a constructive proof idea inspired by the classical result of Tennenbaum ([Te], [E&K], [S]) on recursive models of arithmetic. This line of reasoning was suggested by the applications of readability to recursive mathematics described in [McC1], [McC2] and [McC3].

Markov's principle is more than a convenience in constructive arithmetic and analysis; it is absolutely essential to significant areas of constructive cardinal arithmetic. In turn, logical relations among intuitively appealing principles of constructive cardinal arithmetic parallel relations between MPS and other “problematic axioms” for constructive mathematics, such as the limited principle of omniscience. Finally, simple closure properties on the Dedekind finite sets provide ready examples of statements which are strictly weaker than Markov's principle and yet are independent of extensions of IZF.

MPS, Markov's principle with variables over sets, is equivalent to each of these elementary properties of ⊿, the class of Dedekind finite sets:

1. ∀A(A Є ⊿ ↔ CP(A)).

2. [*]A Є ⊿ ↔ ∀B (A + B is infinite ↔ B is infinite).

3. [**]A Є ⊿ ↔ ∀B((A + 1) x B is infinite ↔ B is infinite).

CP is Tarski's cancellation property. Consequently, MPS is tantamount, in constructive mathematics, to the standard classical characterizations of ⊿ in terms of cardinality.

[*] and [**] imply that ⊿ is closed under addition and multiplication. Closure under addition is, in turn, constructively equivalent to the closure of ⊿ under all combinatorial functions and to the fact that ⊿ is closed under each strict combinatorial function individually. Generally speaking, a function on P(ω) is combinatorial whenever it preserves finiteness, respects cardinality and has a “moduluslike” associate function. It is strict when it is strictly increasing relative to the subset ordering, [cf. §6 for precise definitions.] From this, we see that MPS is foundational for cardinal arithmetic on the constructive Dedekind finite sets.