Zeroid Theory

**Zeroid Theory** is the name given to a loosely defined set of mathematical theories which attempt to propose and deal with certain mathematical entities, called micro-zeroids, which are said to be both "smaller than zero" and "closer to zero than zero", but not "less than zero, "more than zero", or "equal to zero". To achieve this, definitions are established for what it even means for one number to be smaller than another, and what it means for a number to be smaller than 0 in particular, in this specialized context.

In Zeroid Theory, "smaller than zero" is a phrase explicitly distinguished here from being a negative number. Negative numbers are said to be *less than zero* but not *smaller than zero*. In fact, by the postulates of Zeroid Theory every positive and negative number is said to be *larger than zero*. They are larger in the sense that all positive and negative numbers are further away from 0 than 0 is to itself. The set of positive and negative numbers in Zeroid Theory is presumed to include both infinitesimal and infinite quantities without limit, including any limits typically proscribed within mainstream mathematics, such as the size of a Proper Class. By contrast to positive and negative numbers, the micro-zeroids are said to be *closer to 0 than even 0 is,* a statement which common sense would suggest is nonsensical and impossible. Zeroid Theory begins with the assumption that it is sensible and possible, assumes there is a Class of discrete entities for which this is true, and attempts to derive necessary consequences from certain initial premises.

If we think of less than, in terms of removal, then we can think of negative numbers as removing something from 0. Zeroids are *too small* to obtain them by removing something from 0. They can not be reached by removing something from 0, or by adding something to 0.

Numbers which are "strictly smaller than 0" are said to be subfinite. The main property of the subfinite is that multiplying a subfinite number by 0 does not return 0, but returns the same or a smaller subfinite number (depending on the specifics of the particular model).

The Primary Model of Zeroid Theory is called Z\*, pronounced Z Prime or Z Star or Z Root, and it is codified by a set of propositions which are assumed to be reasonable and from which further reasoning can be derived. Within Z\* certain assumptions of the homogeneity of certain notions of size and distance are maintained while others are relaxed to make Zeroid Theory possible. The Primary Model assumes that Multiplication is a more fundamental notion of relative size than that of summation and difference.

Zeroid Theory is a subfield of paragoogology in that it is concerned with entities that would not normally be considered proper numbers in googology, yet it proposes them for the express purpose of denying the existence of a properly smallest number without denying the existence of 0 itself. It does this by proposing a shrinking hierarchy of ever more collapsed Zeroids beginning with 0 but possessing no smallest Zeroid. The phrase *There is No True Zero* is used to encapsulate this notion in contrast with the phrase *There is No True End*, for investigations into entities proposed to be larger than the *Absolutely Infinite.*

## History

Researching the history of numbers "smaller than zero" is difficult because invariably it gets tied up with the more common notion of negative numbers. This is due to mathematicians informal use of the term "smaller than" merely to mean a positional relationship between two numbers, such as on a number line. The notion of size is merely reduced to that of a set having a total order defined on it. This comes ultimately from the idea that if we want something smaller we have to "remove a part" from something. Logically if we remove the whole of something from itself we have the "smallest" possible number we can have, which is to have nothing at all ... right? However, if we generalize this problem we might say that if *we could* remove something from nothing the result, following the pattern, would somehow have to be "smaller than 0". If this is *not* what is meant by the phrase, this begs the question, what do we mean?

## Key Concepts

Z\*

Sanctuary

Haven

Arcanum

Exofinite Number