Decidable Fragments of the Simple Theory of Types with Infinity and NF

Dawar, Anuj; Förster, Thomas; McKenzie, Zachiri

doi:10.1215/00294527-2017-0009

Cited by 3 publications

(3 citation statements)

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“…Similar versions of these last two corollaries can also be found in [, Corollaries 4.18 and 4.20]. In addition, contains an interesting decidability result on existential‐universal sentences that cannot be derived from our results, namely that

{TST}^{\infty}

decides all

APTARAMOPREFIX {scriptL}_{sans-serifTST}

‐sentences of the form

\exists \bar{x} \forall \bar{y} φ false(\overset{x}{true}, \overset{y}{true} false)

, where φ is quantifier‐free and all variables

\overset{y}{true}

are of the same type (Corollary 4.19).…”

Section: Existential‐universal Sentencessupporting

confidence: 71%

“…In general, there are a lot of important unanswered questions related to decidability. For example, a question that we discuss below is whether

sans-serifNFO

decides all stratified existential‐universal sentences (the reader interested in this problem is strongly encouraged to see the recent work by Dawar, Forster, and McKenzie, , especially since some of the results there are relevant to the ones we present below). For a comprehensive list of open problems regarding decidabability, we refer the reader to .…”

Section: Introductionmentioning

confidence: 99%

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Decreasing sentences in Simple Type Theory

Rouvelas

2017

Mathematical Logic Qtrly

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We present various results regarding the decidability of certain sets of sentences by Simple Type Theory (sans-serifTST). First, we introduce the notion of decreasing sentence, and prove that the set of decreasing sentences is undecidable by Simple Type Theory with infinitely many zero‐type elements (TST∞); a result that follows directly from the fact that every sentence is equivalent to a decreasing sentence. We then establish two different positive decidability results for a weak subtheory of TST∞. Namely, the decidability of ∃¯((∀normalb∃¯)normalbtrue∀¯normalb)∧,∨ (a subset of Σ1) and ∃¯STDEC (the set of all sentences ∃x¯φfalse(truex¯false), where φ is strictly decreasing). Finally, we present some consequences for the set of existential‐universal sentences. All the above results have direct implications for Quine's theory of “New Foundations” (sans-serifNF) and its weak subtheory sans-serifNFO.

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{TST}^{\infty}

decides all

APTARAMOPREFIX {scriptL}_{sans-serifTST}

‐sentences of the form

\exists \bar{x} \forall \bar{y} φ false(\overset{x}{true}, \overset{y}{true} false)

, where φ is quantifier‐free and all variables

\overset{y}{true}

are of the same type (Corollary 4.19).…”

Section: Existential‐universal Sentencessupporting

confidence: 71%

“…In general, there are a lot of important unanswered questions related to decidability. For example, a question that we discuss below is whether

sans-serifNFO

Section: Introductionmentioning

confidence: 99%

Decreasing sentences in Simple Type Theory

Rouvelas

2017

Mathematical Logic Qtrly

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“…In [3], Dawar, Forster, and McKenzie show that T ST I decides all universalexistential sentences of the form:…”

Section: The Color Constructionmentioning

confidence: 99%

Ambiguity in Typed Set Theory and the Universal-Existential Conjecture

Pireva

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<p><strong>Typed set theory deals with the traditional paradoxes of set theory by implementing the notion of a type at the level of syntax. This gives rise to the Simple Theory of Types (TST) and the Theory of Positive and Negative types (TZT). New Foundations NF is a theory that explicitly gets rid of the typing discipline used in TST and TZT but still essentially codes it into the formation of sets. The scheme of typical ambiguity asserts that all the types look the same. It turns out that TZT together with the scheme of typical ambiguity is consistent if and only if NF is consistent. Consequently, certain subsystems of NF are consistent if and only if TZT together with a corresponding subclass of the scheme of ambiguity is consistent. The consistency of NF has been an open problem since its conception and so ambiguity for certain classes of formulae have generated great interest. One such class is the class of universal-existential sentences. The main result of this thesis makes progress on what is known as the universal-existential conjecture. This is done by showing that TZT decides a larger subclass of the universal-existences than previously known. This has the consequence of showing that NF decides all universal-existential sentences that have a stratification belonging to this subclass. There is also a new result showing that ambiguity holds in the arithmetic of TZTI and the first verification of the canonicity of the term model for TZTO, a relativisation of a known result for NFO.</strong></p>

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Increasing sentences in Simple Type Theory

Rouvelas

2017

Annals of Pure and Applied Logic

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Decidable Fragments of the Simple Theory of Types with Infinity and NF

Abstract: We identify complete fragments of the Simple Theory of Types with Infinity (TSTI) and Quine's NF set theory. We show that TSTI decides every sentence φ in the language of type theory that is in one of the following forms:

Cited by 3 publications

References 4 publications

Decreasing sentences in Simple Type Theory

Decreasing sentences in Simple Type Theory

Ambiguity in Typed Set Theory and the Universal-Existential Conjecture

Increasing sentences in Simple Type Theory

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