Chris Bird has done a lot of work on large numbers, including much original work that goes beyond that of others. He has created a system of fast-growing recursive functions that rivals or exceeds the functions of Jonathan Bowers and others. The work presented here has been of great benefit to the large and growing community of people with interest in large numbers and related maths; of particular note is his Array Notation and the extensive work he has done to explain its relation to the (earlier and related) Bowers notation and to the function hierarchies of various mathematicians over the past 120 years or more.

Of the following files, the first describes the relationship between his linear arrays (the first and slowest-growing of his recursive functions) and the well-known Conway chained arrow notation.

The numbered chapters are a series describing Bird's functions and notation step by step.

The Fast-Growing Hierarchy in Terms of Bird’s Array Notations, as the title indicates, correlates Bird's work with the Fast-growing hierarchy and the Veblen hierarchy, and also to a few other well-known notations and functions — this includes Conway's chained arrows, the Reuben Goodstein sequences, fast-growing functions from the work of Harvey Friedman (block subsequence theorem, TREE(), etc.), and more. (Some of the other documents also reference some of these, but most is in the Fast-Growing Hierarchy article).

These specific articles were begun in 2012, with substantial updates to the later chapters in later years; much of the material is similar to earlier work by Chris Bird going back to 2006 or earlier.

Chris lives in the United Kingdom, and holds a Master's degree in mathematics despite suffering from Asperger's Syndrome (a form of autism).


Chris Bird's Super Huge Numbers  

Here is a proof that Bird's Linear Array Notation with 5 or more entries goes beyond Conway's Chained Arrow Notation.


1. Linear Array Notation (last updated 1 Apr 2012)

2. Multi-Dimensional Array Notation (last updated 2 Apr 2012)

3. Hyper-Dimensional Array Notation (last updated 2 Aug 2017 with minor corrections on pages 4-5)

4. Nested Array Notation (last updated 31 Aug 2013 with additions on pages 9-10)

5. Beyond Nested Arrays I (last updated 6 Sep 2017 with minor corrections on page 12; corrections on pages 6-8 made on 13 Dec 2013)

6. Beyond Nested Arrays II (last updated 6 Sep 2017 with minor corrections on page 6; minor corrections on pages 18-19 made on 10 Feb 2015; subscripts of subscripts made on 14 Mar 2014)

7. Beyond Nested Arrays III (last updated 14 Mar 2014 to allow subscripts of subscripts)

8. Beyond Nested Arrays IV (last updated 14 May 2018 with minor corrections on page 12; subscripts of subscripts made on 25 Mar 2014; minor changes to ‘counter’ variables on 25 Feb 2014)

9. Beyond Nested Arrays V (Last updated 12 May 2014; adapted from the original Beyond Nested Arrays IV)


The Fast-Growing Hierarchy in Terms of Bird’s Array Notations (last updated 28 Mar 2014)


Bowers' Named Numbers

This is an auxiliary document listing many of the "named" big numbers of Jonathan Bowers, and giving values of these in terms of Bird's notation.

Bowers' Named Numbers (last updated 19 Mar 2012)


These files are written by and are copyright © 2006-2022 Chris Bird.

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