_ is a hypothetical number whose existence has not been proven, but is none the less said to be so small that even multiplying it by 0 does not return 0 but rather _. Namely:
_ * 0 = 0 * _ = _
This would only be possible if _ was a number that was somehow strictly smaller than 0. In fact, if every number no matter how small times 0 was 0, this would imply that 0 was in fact the smallest possible number, thus in order for there to be a smaller number there must be something which violates the 0 property on account of it's smallness.
Technical details[]
The existence of _ is hypothetical. There is no known way to demonstrate its existence. Unlike the transfinite cardinals which Cantor was able to demonstrate form a hierarchy of ever larger and larger kinds of infinity, the existence of smaller and smaller "zeroes" is not based on a demonstration of their existence, but rather on an examination of the properties they would necessarily have to possess if they did exist.
Their hypothetical existence is derived from the observation that if we multiply any two numbers whose magnitude is strictly smaller than 1, their product is always strictly smaller than or equal to the smaller of the two numbers. This may be written succinctly as:
|m1| * |m2| ≤ min(|m1|,|m2|) { |m1| < 1 , |m2| < 1 }
The min function here always returns the smaller of it's two arguments, ignoring sign, and returns either argument in the case when they are both the same size. Numbers that are smaller than 1, in this theory, are known as small numbers.
Small numbers have the property that multiplying something by them generally makes the product either smaller or the same size as the original number, but never larger. The easiest way to see this is to take two real numbers in the interval (0,1) and multiply them together. For example, 0.2 * 0.3 = 0.06. This follows the rule because:
0.2 * 0.3 ≤ min(0.2,0.3)
0.06 ≤ 0.2
The equal sign comes into play when dealing with 0. For example:
0 * 0.2 ≤ min(0.2,0)
0 ≤ 0
This is interpreted as being the result of the fact that 0 is too small to be made smaller by multiplying by 0.2. This is analogous to how ℵ0 is so large that multiplying it by 2 does not make it larger:
alef0 * 2 = alef0
In fact, even multiplying 0 by itself can not give us something smaller than 0:
0 * 0 = 0
just as:
ℵ0 * ℵ0 = ℵ0
Cantor demonstrated the existence of infinities larger than ℵ0, by showing that the powerset of any set is always of strictly larger cardinality than the original set. Thus if a set has cardinality ℵ0, then its power set has cardinality strictly greater than ℵ0. ℵ 1 by definition is the first cardinal number after ℵ0. We can not reach it from below by multiplication. Instead if ℵ1 is the largest cardinal in the product than ℵ1 is the result. So ℵ1 x ℵ1 = ℵ1, and even ℵ1 x ℵ1 = ℵ1.
We have no means to demonstrate the existence of magnitudes strictly smaller than 0, unlike the larger cardinals, none the less if we assume their existence we know that _ can not be reached "from above". No product involving ordinary numbers can get us to it, yet if some product involves _ then the result is _.
If we assume the existence of some _ and we apply our Axiom of smallness we obtain the following:
_ * 0 ≤ min(_,0)
Since _ by definition is strictly smaller than 0, min(_,0) = _ by definition, thus we obtain:
_ * 0 ≤ _
Since _ is strictly smaller than 0 by definition we obtain:
_ * 0 ≤ _ < 0 ⇒ _ * 0 < 0
So _ ≠ 0. Yet _ can not be negative either, because for any negative number -x, -x * 0 = 0. We therefore interpret _ as a number which is somehow smaller than 0 without being less than 0.
_ can not be made "larger" by multiplying by any real number, including 0. Just as 0 times any transfinite cardinal is 0, such as alef0 * 0 = 0, ℵ1* 0 = 0, _ times any cardinal number is still _. What this means is that 0/_ is undefined just like 1/0 is undefined, because there is no number (even of infinite size) that we can multiply _ by to get 0. It is incomparably smaller. Because of this property we can say _ is absolutely infinitely smaller than 0. Therefore:
_ * ℵn = _ for any ordinal number n
_ << 0/ℵn for any ordinal number n
Another remarkable property of _ is that _ - _ = _. This aspect of _ is not yet fully understood, but what it means is that _ has the property of being closer to itself than 0 is close to 0. This is of course only possible if magnitudes smaller than 0, called hyper-0s, already exist.
_ is believed to be sign-less like 0, since it can neither be said to be less than nor greater than 0, which is what signs are based on. _/0 is held to be equal to _, being the only known value for which division by 0 is not undefined. _/_ is indeterminate just like 0/0 since _*x = _ does not have a unique solution, just as 0*x = 0.
Even though their existence is unproven, their properties can still be reasoned about hypothetically.