Skip to content
DraconiticResearch
Search Sources API
186,123 studies31 sourcesUpdated 12 min ago
PreprintCC BY-NC-SA 4.0arXiv

The Iterative Conception Reconsidered

  • Bokai Yao
Open at sourcePDFDOICite

Abstract

We investigate the iterative conception of set in its most general form, allowing urelements without assuming that they form a set. We formulate this conception in two ways, as stage theory and as level theory, and develop a general theory of levels with urelements. Unlike their pure-set counterparts, the resulting stage and level theories are not set-theoretically equivalent; moreover, second-order level theory with urelements is not weakly quasi-categorical. We then consider further principles governing stages and levels, motivated by directedness, unboundedness, and reflection. Some of these principles restore set-theoretic equivalence between the corresponding theories, while their level-theoretic versions yield forms of quasi-categoricity. These principles form strict implication hierarchies, thereby revealing distinct stronger conceptions of set beyond the basic iterative conception.

Related

  1. Grad-div stabilized finite element Picard-Newton iterative methods for the Navier-Stokes equations
    Yueqiang Shang, Christian Engström
    HALPreprintPublisher termsFull textNumerical analysis & scientific computingHAL
  2. Iterative Methods for Solving New Generalized Absolute Value Equations
    Shubham Kumar, Joydip Dhar
    arXivPreprintarXiv licenceFull textOptimisation & controlarXiv
  3. Oscillation of a Second-Order Differential Equation as a Consequence of Iterative Monotonicities
    Blanka Baculikova, Jozef Dzurina
    MathematicsPeer-reviewedCC BY 4.0General mathematicsCrossref
  4. An existence result of a functional integral equation via Darbo type theorem and an iterative algorithm to solve it
    Bhuban Chandra Deuri, Anupam Das
    arXivPreprintarXiv licenceFull textAnalysis & PDEsNumerical analysis & scientific computingarXiv
  5. A highly efficient iterative approach for inverse acoustic obstacle scattering problems in three dimensions
    Zhiyong Cheng, Heping Dong, Lu Zhao
    arXivPreprintarXiv licenceFull textNumerical analysis & scientific computingarXiv

Cite

BibTeXRIS
BibTeX
@misc{yao2026iterative,
  title = {{The Iterative Conception Reconsidered}},
  author = {Bokai Yao},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.30422},
  eprint = {2608.30422},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.30422},
  note = {CC BY-NC-SA 4.0},
}
RIS
TY  - UNPB
TI  - The Iterative Conception Reconsidered
AU  - Bokai Yao
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.30422
UR  - https://arxiv.org/abs/2608.30422
AB  - We investigate the iterative conception of set in its most general form, allowing urelements without assuming that they form a set. We formulate this conception in two ways, as stage theory and as level theory, and develop a general theory of levels with urelements. Unlike their pure-set counterparts, the resulting stage and level theories are not set-theoretically equivalent; moreover, second-order level theory with urelements is not weakly quasi-categorical. We then consider further principles governing stages and levels, motivated by directedness, unboundedness, and reflection. Some of these principles restore set-theoretic equivalence between the corresponding theories, while their level-theoretic versions yield forms of quasi-categoricity. These principles form strict implication hierarchies, thereby revealing distinct stronger conceptions of set beyond the basic iterative conception.
ER  -