The Zeroie Cardinal is a large macro-zeroid. It may be denoted by Ie(0). The Zeroie Cardinal first appeared on September 26th of 2021, as seen in the linked video[1]. It occurs at the 2:48 mark.
Conjectured Properties[]
The exact nature and value of the Zeroie Cardinal is not fully known or understood. In the video it occurs after 0, but before the infinitesimals. This should be impossible! Any number greater than 0, would be an infinitesimal ... normally. This opens up the possibility of a new class of numbers called macro-zeroids. These numbers are larger than 0, yet smaller than any infinitesimal! They are distinguished from infinitesimals in that they retain the properties of 0 and hyper-zeroes rather than those of infinitesimals. For this reason they are believed to be signless like zero, unlike infinitesimals which have signs. They are also believed to exhibit the same rigidity of 0, refusing to get larger or smaller, unlike infinitesimals which can be multiplied or divided to make them larger or smaller.
The meaning of the function Ie is not known, however in the video, Ie(0) occurs after 0*infinity. This implies infinity is not the reciprocal of 0, since multiplying 0 by infinity still returns a number smaller than even the smallest infinitesimal. It is further speculated that any multiple of 0 will necessarily be less than Ie(0), though this is not clear. It is also believed to be greater than sqrt(0), or any root of 0 for that matter.
It is not known whether or not Ie(0) = _(-1). However _(-1) has a cardinal like property in that it can not be made larger or smaller by multiplication or division:
alef0 * 2 = alef0 , alef0 / 2 = alef0
_(-1)*2 = _(-1) , _(-1)/2 = _(-1)
So the idea of "cardinals" hidden between 0 and all positive infinitesimals may be possible. This implies that Ie(0) corresponds to a discrete level corresponding to some _(-a). Ie(0) suggests there is a lot more space getting to 0 than previously thought ... a lot more. It may be that 0*ABSINF is still smaller than Ie(0).
The exact placement of Ie(0) is still an active area of research.
Here are some further conjectured properties based on it's cardinal and macro-zeroid nature:
-Ie(0) = Ie(0)
le(0) + le(0) = le(0)
From the first two it follows that:
Ie(0) - Ie(0) = Ie(0)
"positive real number" * Ie(0) = Ie(0)
Ie(0)/"positive real number" = Ie(0)
Ie(0) * Ie(0) = Ie(0)
0 * Ie(0) = 0
1 <<> 1 + Ie(0), yet both 1+Ie(0) > 1 and 1+Ie(0) < 1 are false. At the same time 1+Ie(0) != 1 is true.
Links[]
[1] "(SNEAK PEAK) -0 to 1" https://www.youtube.com/watch?v=S2z37qCyEDQ