A hypothetical number possessing the following two properties (1) multiplying by 2 returns 0 (2) It is not equal to 0. Since multiplying it by 2 returns zero, it implies that it is somehow "half of zero". This should be understood as distinct from zero halves, which is 0. It may be symbolized as 0/2, however this can be misleading as 0/2 = 0 conventionally. The key here is that we hypothetically postulate a new solution to 2x = 0 where x is not 0. We further assume that this implies that it is smaller than 0, and that it is smaller by a factor of exactly 2. 0/2 = 5*0.0
The Existential Axiom ensures that for any property, definition, or concept, there exists as many objects possessing that property as desired. Thus we may postulate another solution to 2x=0 other than 0.
This concept of dividing 0 into two equal parts and actually obtaining a smaller object, which we will call half-null, and symbolize (/), or occasionally as h0, has certainly been contemplated before. It actually occurs in the following video:
https://www.youtube.com/watch?v=S2z37qCyEDQ&t=68s
Such expressions as 0/2, 0/10, etc. are sortable using the concept of infinitesimal asymptotics. Here we compare two limiting processes which approach 0, and we assign the smaller size to the expression which approaches zero faster. In this case it is clear that x/2 < x as x approaches 0 from above. So we conclude that 0/2 < 0, in this technical sense. For clarity an asterisk can be used. *0 represents the limit of approaching 0 from above. In this way we can say *0/2 < *0. This can be used to order many expressions which would normally all be considered equivalent and equal to 0. So for example we may say *0/10 < *0/2.
(/) is the largest hyper-zero (number smaller than 0) that has currently been explicitly expressed within a video, and is the largest hyper-zero on the wiki.
Technical Details[]
We may say that 2*0=0 and 2*(/)=0. It also seems reasonable to say 0+0=0 and (/)+(/)=0.
This immediately leads to further questions. For example. If we allow the existence of such an entity we might ask what happens when multiplying it by other numbers.
(/)*4 = (/)+(/)+(/)+(/) = 0+0 =0 , (/)*6 = (/)*2*3 = 0*3 = 0
If we group it into pairs, multiplying by any even number just gives us 0. We run into a problem with odd numbers because we haven't defined how to add 0+(/). To investigate this we multiply this by 2
2*(0+(/)) = 2*0+2*(/) = 0+0 = 0
This implies that if 0 and (/) is a solution to 2x = 0, then so is 0+(/). One may assume that if 2(0+(/))=0 then it implies that it must be equal to 0. Let's assume that is the case. In that case we have 0+(/) = 0. This implies (/) = 0 - 0. This yields (/) = 0. However by definition (/) != 0. Therefore this can not equal 0. Remember we no longer are assuming 0 is the only solution to 2x=0, so 0+(/) does not necessarily equal 0.
Let's consider another solution. If 2*(0+(/)) = 0 then by dividing both sides of equation by 2 we obtain 0+(/) = 0/2 which by definition equals (/). This gives us 0+(/) = (/) which actually makes sense since adding zero to any value doesn't change that value. In that case the conclusion is that (/) multiplied by any odd number equals (/).
Other questions may arise from this number, such as the product 0*(/). Hyper-Zero Theory would imply this could not equal 0 but would be equal to (/) or smaller.