| _version_ | 1866901280017874944 |
|---|---|
| author | Revista, Zen MATH, 10 |
| author_facet | Revista, Zen MATH, 10 |
| contents | The continuum function, mapping a cardinal number $kappa$ to $2^kappa$, stands as a cornerstone of set theory, particularly for its role in the Continuum Hypothesis (CH) which posits $2^{aleph_0} = aleph_1$. Since Paul Cohen's groundbreaking work in 1963 demonstrating the independence of CH from Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), the method of forcing has emerged as the principal tool for exploring the consistent values of $2^kappa$. This paper investigates the "forcing spectrum" of the continuum function, defined as the collection of all cardinal assignments to $2^kappa$ for various $kappa$ that are consistent with ZFC, typically achieved through specific forcing extensions. We delve into how different forcing notions, ranging from simple Cohen forcing to more complex constructions like those used in Easton's Theorem and the study of singular cardinals, can yield a diverse array of cardinalities for $2^kappa$. The paper provides a comprehensive review of established results, including the profound implications of Easton's Theorem for regular cardinals and the intricate challenges posed by singular cardinals. Furthermore, it explores the impact of forcing axioms such as Martin's Axiom on the value of $2^{aleph_0}$ and discusses the philosophical ramifications of the continuum's variability. By examining the theoretical landscape shaped by forcing, this study elucidates the profound flexibility of the set-theoretic universe and the limitations of ZFC in uniquely determining the sizes of power sets. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17707497 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Forcing Spectrum of the Continuum Function Revista, Zen MATH, 10 The continuum function, mapping a cardinal number $kappa$ to $2^kappa$, stands as a cornerstone of set theory, particularly for its role in the Continuum Hypothesis (CH) which posits $2^{aleph_0} = aleph_1$. Since Paul Cohen's groundbreaking work in 1963 demonstrating the independence of CH from Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), the method of forcing has emerged as the principal tool for exploring the consistent values of $2^kappa$. This paper investigates the "forcing spectrum" of the continuum function, defined as the collection of all cardinal assignments to $2^kappa$ for various $kappa$ that are consistent with ZFC, typically achieved through specific forcing extensions. We delve into how different forcing notions, ranging from simple Cohen forcing to more complex constructions like those used in Easton's Theorem and the study of singular cardinals, can yield a diverse array of cardinalities for $2^kappa$. The paper provides a comprehensive review of established results, including the profound implications of Easton's Theorem for regular cardinals and the intricate challenges posed by singular cardinals. Furthermore, it explores the impact of forcing axioms such as Martin's Axiom on the value of $2^{aleph_0}$ and discusses the philosophical ramifications of the continuum's variability. By examining the theoretical landscape shaped by forcing, this study elucidates the profound flexibility of the set-theoretic universe and the limitations of ZFC in uniquely determining the sizes of power sets. |
| title | The Forcing Spectrum of the Continuum Function |
| url | https://doi.org/10.5281/zenodo.17707497 |