# Tag Info

1 vote

### Proving a set is a Borel Set

That set is neither closed nor open. On the other hand, if $n\in\Bbb N$ and$$A_n=\left\{(x,y,z)\in\Bbb R^3\,\middle|\,0<x<1+\frac1n,0<y^2+z^2<x+\frac1n\right\},$$then each $A_n$ is open ...
1 vote
Accepted

• 51.7k
Accepted

### In forcing, would a generic extension $M[G]$ also extend sentences satisfiable by $M$ that are not provable in ZF?

Certainly not, even if we don't change the language at all! In fact the whole utility of forcing (as Daniel Schepler points out below) is based on the fact that many things aren't "forcing-stable....
• 223k
Accepted

### problem 3.48 of Enderton's "Elements of Set Theory"

Your answer and analysis is correct and shows that you have a good understanding of the definitions being used by the author in your textbook. In particular that of ordered pairs and set operations ...
• 72.9k

### Cannot there be more Partitions or Equivalence Relations than the other?

In set theory, we say that two sets are of equal size if you can construct a bijection between them, i.e. if we can exactly match each element in one set with a unique element from the other and vice ...
• 15.5k

• 173

• 1,165
### Is $\{z: (z = u \cup v) \wedge (u \in U) \wedge (v \in V) \}$ a set?
I believe I have found a way to prove what I intended via the ZFC axioms I'm allowed to use: Using the axiom of union $\cup U$ and $\cup V$ are both sets (because $U$ and $V$ are sets by hypothesis) ...