Skip to main content

Beginnings

  • Chapter

Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

The beginning threads of the subject are picked up in its early history. §1 discusses weak inaccessibility and Mahloness, concepts that arose in the study of cardinal limit processes, and their strong versions, which led to early speculations about completeness and consistency. §2 describes Ulam’s formulation of measurability, the most prominent of all large cardinal hypotheses, out of a measure problem for sets of reals. In §3 Gödel’s work on L, the beginning of axiomatic set theory as a distinctive field of mathematics, is reviewed since in both reaction and generalization it shaped much of the subsequent work in large cardinals. §4 discusses weak and strong compactness, concepts that emerged from Tarski’s study of infinitary languages, and establishes Hanf’s result, that in a strong sense there are many inaccessibles below a measurable cardinal. The focus of §5 is on elementary embeddings and the ultrapower construction: Scott’s pivotal result that if there is a measurable cardinal, then VL; the characterization of measurability in terms of ultrapowers and elementary embeddings; and the related notion of normality. And finally §6 discusses indescribability, a natural formalization of reflection phenomena in terms of higher-order languages that provided a schematic approach to comparing large cardinals by size.

Keywords

  • Extension Property
  • Weak Compactness
  • Large Cardinal
  • Measurable Cardinal
  • Satisfaction Relation

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-540-88867-3_2
  • Chapter length: 53 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   119.00
Price excludes VAT (USA)
  • ISBN: 978-3-540-88867-3
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   159.99
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Rights and permissions

Reprints and Permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

(2009). Beginnings. In: The Higher Infinite. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88867-3_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-88867-3_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-88866-6

  • Online ISBN: 978-3-540-88867-3

  • eBook Packages: Springer Book Archive