A Modern Rigorous Approach to Stratification in NF/NFU

Adlešić, Tin; Čačić, Vedran

doi:10.1007/s11787-022-00310-y

Cited by 3 publications

(3 citation statements)

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“…In the first section, we introduce the necessary syntax and notation. This is done in more detail in [1].…”

Section: Overviewmentioning

confidence: 99%

“…In this section, we introduce the syntax of NFU as well as some other necessary notions. Most results are stated without proof; the proofs can be found in [1].…”

Section: Syntaxmentioning

confidence: 99%

“…When defining sets, we will usually not check whether the defining formulas are stratified. That can be easily done using type assignments in the extended language (again, details about this procedure can be found in [1]). Explicit checking will only be done in some proofs.…”

Section: Syntaxmentioning

confidence: 99%

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The Cardinal Squaring Principle and an Alternative Axiomatization of NFU

Adlešić,

Čačić

2023

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In this paper we rigorously prove the existence of type-level ordered pairs in Quine's New Foundations with atoms, augmented by the axiom of infinity and the axiom of choice (NFU+Inf+AC). The proof uses Tarski's theorem about choice, which is a theorem of NFU+Inf+AC. Therefore, we have a justification for proposing a new axiomatic extension of NFU, in order to obtain type-level ordered pairs almost from the beginning. This axiomatization is NFU+Inf+AC+Tarski, a conservative extension of NFU+Inf+AC.

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“…In the first section, we introduce the necessary syntax and notation. This is done in more detail in [1].…”

Section: Overviewmentioning

confidence: 99%

“…In this section, we introduce the syntax of NFU as well as some other necessary notions. Most results are stated without proof; the proofs can be found in [1].…”

Section: Syntaxmentioning

confidence: 99%

See 1 more Smart Citation

The Cardinal Squaring Principle and an Alternative Axiomatization of NFU

Adlešić,

Čačić

2023

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A Modern Rigorous Approach to Stratification in NF/NFU

Adlešić

2022

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A Modern Rigorous Approach to Stratification in NF/NFU

Adlešić

2022

Preprint

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The main feature of NF/NFU is the notion of stratification, which sets it apart from other set theories. We define stratification and prove constructively that every stratified formula has the (unique) least assignment of types. The basic notion of stratification is concerned only with variables, but we extend it to abstraction terms in order to simplify further development. We reflect on nested abstraction terms, proving that they get the expected types. These extensions enable us to check whether some complex formula is stratified without rewriting it in the basic language. We also introduce natural numbers and a variant of the axiom of infinity, in order to precisely introduce type level ordered pairs, which are crucial in simplifying the definitions in the last part of the article. Using these notions we can easily define the sets of ordinal and cardinal numbers, which we show at the end of the article. The same approach can be readily applied to NF.

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A Modern Rigorous Approach to Stratification in NF/NFU

Cited by 3 publications

References 5 publications

The Cardinal Squaring Principle and an Alternative Axiomatization of NFU

The Cardinal Squaring Principle and an Alternative Axiomatization of NFU

A Modern Rigorous Approach to Stratification in NF/NFU

A Modern Rigorous Approach to Stratification in NF/NFU

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