Skip to main content Accessibility help
×
Home
Hostname: page-component-99c86f546-4k54s Total loading time: 0.269 Render date: 2021-12-01T18:57:26.453Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

6 - Isolating Tame Mathematics

from PART II - THE PARADIGM SHIFT

Published online by Cambridge University Press:  19 January 2018

John T. Baldwin
Affiliation:
University of Illinois, Chicago
Get access

Summary

Martin Davis wrote,

Gödel showed us that the wild infinite could not really be separated from the tame mathematical world where most mathematicians may prefer to pitch their tents.

We will now see howmodernmodel theory avoids the Gödel phenomena; the key is to formalize topics locally by axioms which catch the relevant data but avoid accidentally encoding arithmetic and, more generally, pairing functions. We do not attempt a general definition of tame but provide a number of examples of sufficient model theoretic conditions. For more details, see page 160 and [Teissier 1997].

The most basic examples are when mathematicians are already studying definable relations on a class of structures and the natural axiomatization of the area yields a tame theory. In studying real or complex algebraic geometry, the formalization is automatic; Steinitz (ACF) and Artin-Schreier (RCF) defined concepts that happen to be first order; these theories provide the framework for much of the development of the geometries. The theories are ℵ1-categorical (Chapter 3.3; indeed, interpretable in a strongly minimal structure, Example 4.3.1) and o-minimal (Chapter 6.3), respectively.

One of the earliest algebraic discoveries linking algebraic structure with stability properties echoes the Bourbaki assertion of the importance of groups. An ω-stable group cannot have a descending chain of ‘definable subgroups’. This condition extends to stable groups (for uniformly definable chains) and the distinction between the stability classes is signaled by the size of the allowed quotient groups. This principle is now seen to apply to different algebraic structures and gives a unified explanation for finding various kinds of radicals.5 (See [Baldwin 1979] for a very early account of this phenomenon and [Altinel & Baginski 2014], [Aldama 2013], and [Freitag 2015] for recent updates.)

Groups of FiniteMorley Rank

In this section we give a short case study of one research area that serves both as a tool for unifying studies in several areas of mathematics and for isolating the role of basic concepts. In particular, it is seen that finite plays a dual role in the study of finite groups: both as the size of structure and as a dimension on which one can do inductions. In the case at hand, the study is extended to infinite groups by introducing a broader definition of ‘dimension’, Morley rank (Chapter 5.3), and requiring it to be finite.

Type
Chapter
Information
Model Theory and the Philosophy of Mathematical Practice
Formalization without Foundationalism
, pp. 148 - 166
Publisher: Cambridge University Press
Print publication year: 2018

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Send book to Kindle

To send this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • Isolating Tame Mathematics
  • John T. Baldwin, University of Illinois, Chicago
  • Book: Model Theory and the Philosophy of Mathematical Practice
  • Online publication: 19 January 2018
  • Chapter DOI: https://doi.org/10.1017/9781316987216.010
Available formats
×

Send book to Dropbox

To send content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox.

  • Isolating Tame Mathematics
  • John T. Baldwin, University of Illinois, Chicago
  • Book: Model Theory and the Philosophy of Mathematical Practice
  • Online publication: 19 January 2018
  • Chapter DOI: https://doi.org/10.1017/9781316987216.010
Available formats
×

Send book to Google Drive

To send content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive.

  • Isolating Tame Mathematics
  • John T. Baldwin, University of Illinois, Chicago
  • Book: Model Theory and the Philosophy of Mathematical Practice
  • Online publication: 19 January 2018
  • Chapter DOI: https://doi.org/10.1017/9781316987216.010
Available formats
×