introduction

In the spirit of Robert Munafo's "Notable Properties of Specific Numbers" and Jonathan Bowers' "Infinity Scrapers", and very similar to the Pointless Gigantic List of Numbers which was inspired in part by my list, I present my own Ultimate Finite Numbers List (the UFNL) as a reference for comparing real numbers of various finite sizes, from the very smallest numbers one could argue are germane to a large numbers list to some of the largest finite numbers ever devised!

The UFNL is currently neither as far reaching as Bowers' page or as thorough as Munafo's, but it still manages to go into greater depth and reach further than many other similar lists on the internet. Feel free to browse through it to learn about all sorts of large numbers.

A Large Number List is essential to any meaningful discussion of large numbers because of the vast number of systems used in "googology". Because the numbers generated by these systems are so vast, direct comparison is not possible. Even with today's computers we can only compute numbers with billions of digits, but most of the numbers googologist's study have so many digits that even the number of digits can't be computed! It therefore becomes necessary to prove that one number is larger than another using mathematics. In this way googologist's can define a continuum of large number sizes. In order to keep track of this developing continuum it is essential to maintain a ordered catalog of Large Numbers, sorted by size. Numbers can then easily be compared and a "largest number" determined. Of coarse once a largest number has been determined larger numbers can easily be defined, but with qualifications such as "Largest Number with official entry on this list" a precise meaning to "Largest Number" can be achieved. Currently the largest well defined number with an official entry on this web page is Jonathan Bowers' kungulus.

Googology sometimes gives the impression of a "forest" of numbers due to the fact that many googological functions are not related in an obvious way, and one can feel a little lost. Although we can think of functions as forming a branching tree structure we mustn't forget that googology is primarily about positive integers. Despite appearances all the systems are talking about the same mathematical object (integers) and therefore rather than being a forest it is more accurate to think of the objects of googological study as existing along a infinite perfectly straight railroad with a definite beginning station but no final destination (infinity is NOWHERE along the track. It isn't part of the track, it describes the endless nature of the track). The main purpose of the ULNL is to make sense of the numbers of googology and establish a definite order to them. Because my other pages are largely concerned with introducing numbers and notations, it is important to have a list where all of these discussions can be put into some kind of coherent order. That is the purpose of this page.

I do not claim this list to be anything near complete or authoritative, and I selectively choose numbers which I find interesting or important to googology. Over time I will be expanding this List, to match up along with the expansion of my site. For this reason I will not discuss numbers on this list that are non-constructible from the methods I have presented so far on my site.

Since this is a "large numbers list" and not merely a "numbers list" I've had to consider what should be meant by a "Large Number". In keeping with Robert Munafo's tradition to include a few examples of non-positive reals on his list, I have followed suit by considering this as a comprehensive list of "interesting" real numbers, both negative, positive, and zero, with an emphasis on googologically large positive numbers. None the less, we can consider everything below and equal to 1 as merely a prelude. The "real" list can be thought of as beginning as soon as we pass 1. Stuff in the interval (-inf,1] is simply for reference purposes, while (1,inf) is considered the large numbers.

I have concluded that only the positive reals should be considered along a large-small axis, with zero being designated "infinitely small", and infinity being "infinitely big". This is meant to be a sequentially determined list so imaginary numbers , complex, quaternions, octonions, etc. are not even up for consideration on this list. Although zero is important in googology as the smallest argument for some functions (such as the Ackermann function for example), it is rarely an output of ANY large number function. Furthermore, since zero is the smallest number, it can't be called a large number, since a number needs to be large relative to something else in order to be large. In the case of zero however there is nothing for it to be larger than, and therefore zero is not a "large number". Furthermore, numbers less than 1 I don't consider to be large since they are all "small numbers". I have made the case on my blog entry Very Small Very Large Numbers that positive reals less than 1 are "small numbers" and positive reals greater than 1 are "large numbers". This however leaves the status of 1 ambiguous. It seems to be neither large nor small, but average. It is at least, the smallest non-small number. It can also be thought of as an "infinitely small large number". An important reason for including 1 however, is that it provides a definite starting point, and can be called the "smallest large number" in a certain sense, since it is the limit of all small large numbers. I was pretty divided as to whether to include 1 or not, but the swing vote is on account that "1" is the smallest positive integer, and one of the goals of the UFNL is to list as many positive integers as possible. It would therefore be a shame not to begin with the very first positive integer, and the natural starting point of a count to infinity. It is also the first member of a lot of sequences. For this reason, I've included it despite it's very untenable status as a large number.

For real numbers less than 1, you can look in the "googologically small epoch" and "ordinarily small numbers epoch". Some of these numbers actually have relevance to googology, so I have provided a space for them, which is still sectioned off from the numbers of primary interest to googologist's, namely the finite reals greater than 1. Numbers greater than any finite number you can check out my new Infinite Numbers List. Again, some infinities, especially the ordinals, are closely related to topics in googology.

The format I use for this list is simple:

Entries are separated by bars. At the top of each entry is the simplest mathematical expression that defines the value of the entry. If the number is "small" enough that the digits can be listed out on a single line, then I write it out in decimal. Otherwise a mathematical expression is used. If there is a short name for the number, this is listed underneath the mathematical expression. After this there is an optional description that may follow. This is used either when I have something I really want to say about the number, or when it satisfies a lot of properties. Not every entry needs a description, but I'll try to include one for the more "important" numbers. If there is a lot I have to say about a particular entry there will be a link to further content on that specific number. The rule of thumb is, if the description for an entry would take up more viewing space than available at any given time, then it should be linked to instead.

Entries are broken up into Epochs, delineated by color, in an analogy with cosmological time periods. This is to help to identify what kinds of numbers your currently looking at. First the Epoch name is listed, followed by a range of values, followed by a count of the number of entries for that class.

Now sit back and get ready for a journey through the world of EXTREMELY Large (and small) Numbers!!! ...