Skip to main content
×
Home
    • Aa
    • Aa

On notation for ordinal numbers

Abstract

Consider a system of formal notations for ordinal numbers in the first and second number classes, with the following properties. Given a notation for an ordinal, it can be decided effectively whether the ordinal is zero, or the successor of an ordinal, or the limit of an increasing sequence of ordinals. In the second case, a notation for the preceding ordinal can be determined effectively. In the third case, notations for the ordinals of an increasing sequence of type ω with the given ordinal as limit can be determined effectively.

Are there systems of this sort which extend farthest into the second number class? When the conditions for the systems have been made precise, the question will be answered in the affirmative. There is an ordinal ω1 in the second number class such that there are systems of notations of the sort described which extend to all ordinals less than ω1, but none in which ω1 itself is assigned a notation.

1. An effective or constructive operation on the objects of an enumerable class is one for which a fixed set of instructions can be chosen such that, for each of the infinitely many objects (or n-tuples of objects), the operation can be completed by a finite process in accordance with the instructions. This notion is made exact by specifying the nature of the process and set of instructions. It appears possible to do so without loss of generality.

Copyright
Corresponding author
The University of Wisconsin
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

Alonzo Church , The constructive second number class, Bulletin of the American Mathematical Society, vol. 44 (1938), pp. 224232

S. C. Kleene , General recursive functions of natural numbers, Mathematische Annalen, vol. 112 (1936), pp. 727742

S. C. Kleene , λ-definability and recursiveness, Duke mathematical journal, vol. 2 (1936), pp. 340353

Alonzo Church , An unsolvable problem of elementary number theory, American journal of mathematics, vol. 58 (1936), pp. 345363

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 4 *
Loading metrics...

Abstract views

Total abstract views: 186 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 28th June 2017. This data will be updated every 24 hours.