Skip to main content Accessibility help
×
Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-22T21:03:57.025Z Has data issue: false hasContentIssue false

3 - Challenges to the Iterative Conception

Published online by Cambridge University Press:  09 January 2020

Luca Incurvati
Affiliation:
Universiteit van Amsterdam
Get access

Summary

The main purpose of the chapter is to defend the iterative conception against three objections. The first objection, the missing explanation objection, is that if the iterative conception is correct, one cannot explain the intuitive appeal of the Naïve Comprehension Schema. The chapter provides plausible explanations of this fact which are compatible with the correctness of the conception. The second objection, the circularity objection, is that the iterative conception presupposes the notion of an ordinal, and since ordinals are treated in set theory like certain kinds of sets, this means that the conception is vitiated by circularity. The chapter shows that this objection can be defeated by constructing ordinals using a trick that goes back to Tarski and Scott or dispensing with the notion of well-ordering altogether in the formulation of the conception. The third objection, the no semantics objection, is that the iterative conception prevents us from giving a semantics for set theory. The chapter defends the approach that this problem can be overcome by doing semantics in a higher-order language. The chapter concludes by discussing the status of the Axiom of Replacement on the iterative conception.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2020

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.)

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] 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 saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved 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.

Available formats
×

Save book to Dropbox

To save 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 saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save 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 saving content to Google Drive.

Available formats
×