This as a post was long due.
What exactly is ? It is the single most confusing thing I have come across. I am going to try and elaborate its properties. I’m going to build the theory in stages.
1. Nothingness. Let us give it a symbol- . This is an element of EVERY set. is the absence of elements, but it is present in everything. This is a little non-intuitive, if one were to take the physical world as a reference point. For to be present inside , there has to be some trance, some sign of the presence of in .
Maybe even a space ” “, showing an absence of elements. But really in set theory, we don’t have spaces of this kind to show an absence of elements. We have no such trace, which is fairly irritating. But one has to get used to it.
2. Now we come to . This is NOT equivalent to . is not the absence of elements. It is a set containing no element, the operating word being “set”. Technically, , which makes non-empty. However, we work around this by translating to English. When we say , what we’re really saying is no element belongs to , thereby making it an emptyset. See? Clever!
3. Say we have a set . Actually, . Hence, one element of is , as stated before (by clever linguistics). Hence, for every set . In fact . Hence, also.
4. Say we have to construct the set . This set contains no element from . However, even THIS set has to contain the element . Hence, .
I have sort of developed my own concepts here, and have made set theory more understandable for myself. I hope it helps other readers too.