I have been thinking of a problem that I could not get the most satisfying answer yet.
The problem I have been thinking of:
We know that the Null law says:$$A\cap \varnothing = \varnothing$$but why$$\bigcap \varnothing = X$$ where $X$ is a universal set.
You may ask me, under what theory, since I believe there might be another theory I do not know? I am still unable to answer that, sorry. This question came to mind when I read the given link above. How is this even possible? Or did I miss something very obvious?
The reason I ask this problem is because my uncerebral, misguided thinking is that I could do this:
$$\begin{align}\bigcap_{i=1}^{n} \varnothing_i &= \varnothing_1 \cap \varnothing_2 \cap \bigcap_{i=3}^{n} \varnothing_i\\&= \varnothing \cap \bigcap_{i=3}^{n} \varnothing_i \tag{by Null law}\\&= \varnothing \cap \varnothing_3 \cap \cdots \cap \varnothing_{n-1} \cap \bigcap_{i=n}^{n} \varnothing_i \tag{by Null law}\\&= \varnothing\end{align}$$
which I believe my thinking is absolutely incorrect and is a misguided thinking. Mayhap?
Could you elaborate your answer with mentioning the ZF as well?, since I read it from the link above that ZF does not have a universal set. I am sure I am just misunderstood and misguided. I am currently studying without a teacher, so it has been difficult for me. I added the "general topology" as the tag since I am reading a topology book by James R. Munkres where I am still at chapter 1. Please enlighten my confusion.
I did read this interesting topic:
Universe set and nullary intersection
However, I am afraid I am still confused as the answerer mentioned $\bigcap\varnothing = \varnothing$ in Set Theory by Kunen which I know nothing about.