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Zenodo
2025
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| Accesso online: | https://doi.org/10.5281/zenodo.17845724 |
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| _version_ | 1866901825495498752 |
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| author | Salvador Loria, Ramir G. Pacheco, Richel B. Rabas, Emma O. Cajurao, Rocel A. Turco |
| author_facet | Salvador Loria, Ramir G. Pacheco, Richel B. Rabas, Emma O. Cajurao, Rocel A. Turco |
| contents | <p>This paper presents a comparative theoretical analysis of several weak forms of the Axiom of Choice (AC) and<br>their logical consequences within the framework of Zermelo–Fraenkel Set Theory (ZF). While the full Axiom of<br>Choice ensures the existence of a global choice function for all families of nonempty sets, weaker variants such<br>as the Axiom of Countable Choice, the Axiom of Dependent Choice, and the Boolean Prime Ideal Theorem<br>capture limited but significant forms of selection. Through model-theoretic and proof-theoretic examinations,<br>this study highlights the relative strength, independence, and interrelations among these axioms. The findings<br>affirm that while these principles are equivalent in ZFC, their logical independence in ZF reveals an order of<br>choice principles. This comparative analysis deepens understanding of how restricted forms of choice operate<br>and clarifies the structural role of AC in modern set-theoretic foundations.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17845724 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
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| spellingShingle | A COMPARATIVE THEORETICAL ANALYSIS OF WEAK FORMS OF THE AXIOM OF CHOICE AND THEIR LOGICAL CONSEQUENCES IN ZERMELO– FRAENKEL SET THEORY Salvador Loria, Ramir G. Pacheco, Richel B. Rabas, Emma O. Cajurao, Rocel A. Turco <p>This paper presents a comparative theoretical analysis of several weak forms of the Axiom of Choice (AC) and<br>their logical consequences within the framework of Zermelo–Fraenkel Set Theory (ZF). While the full Axiom of<br>Choice ensures the existence of a global choice function for all families of nonempty sets, weaker variants such<br>as the Axiom of Countable Choice, the Axiom of Dependent Choice, and the Boolean Prime Ideal Theorem<br>capture limited but significant forms of selection. Through model-theoretic and proof-theoretic examinations,<br>this study highlights the relative strength, independence, and interrelations among these axioms. The findings<br>affirm that while these principles are equivalent in ZFC, their logical independence in ZF reveals an order of<br>choice principles. This comparative analysis deepens understanding of how restricted forms of choice operate<br>and clarifies the structural role of AC in modern set-theoretic foundations.</p> |
| title | A COMPARATIVE THEORETICAL ANALYSIS OF WEAK FORMS OF THE AXIOM OF CHOICE AND THEIR LOGICAL CONSEQUENCES IN ZERMELO– FRAENKEL SET THEORY |
| url | https://doi.org/10.5281/zenodo.17845724 |