Universe set and nullary intersection

Question

Let $$\mathbf{B}=\{B_i : i \in I_1\} \subseteq 2^\Omega$$ and suppose that $$\cup(\mathbf{B}) \neq \Omega$$ If $I_2 \subseteq I_1$ then $$\cap\{B_i : i \in I_1\} \subseteq \cap\{B_i : i \in I_2\}$$ therefore the operator $$\cap : 2^{2^\Omega} \mapsto 2^\Omega$$ is, loosely speaking, decreasing. Perhaps, this could be a naif reasoning in favor of $$\cap \varnothing = \Omega$$ but it raises a paradox: given that each $B_i$ "doesn't remember" what set it has been cutted out from, we can also conceivably conjecture $$\cap \varnothing = \cup(\mathbf{B})$$ and, by hypothesis $$\cap \varnothing = \cup(\mathbf{B}) \neq \Omega = \cap \varnothing$$ Obiviously, something is dead wrong. I bumped into this pitfall because I'm studying general topology, and I suspect that it can lead to major misunderstandings.

Answer 1

In SET THEORY by Kunen, he defines $\cap F=\{x:\forall y\in F (x\in y)\}$ only for $F\ne \phi$ and says this "should" make $\cap \phi$ equal to the set of all sets (if applied when $F=\phi$). The convention is that $\cap \phi =\phi$ to avoid this. Then your formula for "monotonicity" only applies to non-empty $B_1,B_2$.

Answer 2

You should reject the conjecture that $\bigcap\varnothing=\bigcup\mathbf B$.

Working in universe $\Omega$ and $\mathbf B\subseteq\wp(\Omega)$ think of $\bigcap\mathbf B$ defined as:$$\bigcap\mathbf{B}:=\left\{ x\in\Omega:\forall B\in\mathbf{B}\left[x\in B\right]\right\}\subseteq\Omega$$ Then: $$\bigcap\varnothing=\Omega$$

This because for every $x\in\Omega$ no $B\in\varnothing$ exists with $x\notin B$, or equivalently because $x\in B$ is (vacuously) true for every $B\in\varnothing$.

Answer 3

I worried about this kind of stuff too when I was younger. It turns out that the weirdness you're observing is the result of taking a materialistic view. If we take a structuralist viewpoint, the weirdness goes away, and the empty intersection is indeed $\Omega$, as it should be. I say "as it should be" because from the perspective of lattice theory, the empty meet is always the top element. Note that when $\mathcal{P}(\Omega)$ is viewed a poset, the top element is $\Omega$ and the meet operation is intersection.

For a great book on set theory from a structural standpoint, check out Sets for Mathematics. You'll also want to look into category theory.