Fundamental sequences in the second and third ordinal number classes

McBeth, C. B. R.

(1973)

McBeth, C. B. R. (1973) Fundamental sequences in the second and third ordinal number classes.

Our Full Text Deposits

Full text access: Open

10098232.pdf - 5.9 MB

Abstract

The text is concerned with the definition and investigation of some classes of ordinal valued-functions; and with the determining of a particular assignment function O. This function assigns to each limit ordinal alpha smaller than a constant theta and belonging to the second number class a fundamental sequence [diagram] that is, a strictly increasing sequence satisfying [diagram].

The operations used to determine O are derived from a class of functions [diagram]. Since generalizes the standard ordinal arithmetic operations,some of the properties of the are studied, together with the operations gammaalpha which are generalizationsof transfinite sum and product. Also investigated is a related class of number-theoretic functions, in this context O is an arbitrary assignment bounded by o in place of [diagram].

It is proved that the functions are normal in the second argument, and these functions are compared with a hierarchy of normal functions obtained by Veblen's process of iteration. The provide a natural means of extending the notion of epsilon number, and some of the properties of the generalized epsilon numbers are presented. The function of [diagram] is normal, and the notation [diagram] is adopted for the sequence of countable fixed points of the function. The generalized epsilon numbers determine a hierarchical classification of limit numbers, and on this basis a normal form is determined for each, and thence the function O is defined by transfinite recursion.

Information about this Version

This is a Accepted version
This version's date is: 1973
This item is not peer reviewed

Link to this Version

https://repository.royalholloway.ac.uk/items/9fb5f686-e5a5-4e4c-ad12-b806cfe03564/1/

Item TypeThesis (Doctoral)
TitleFundamental sequences in the second and third ordinal number classes
AuthorsMcBeth, C. B. R.
Uncontrolled KeywordsMathematics; Pure Sciences; Classes; Fundamental; Number; Ordinal; Ordinal Number; Ordinal Number; Second; Sequences; Third
Departments

Identifiers

ISBN978-1-339-62295-8

Deposited by () on 31-Jan-2017 in Royal Holloway Research Online.Last modified on 31-Jan-2017

Notes

Digitised in partnership with ProQuest, 2015-2016. Institution: University of London, Bedford College (United Kingdom).


Details