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 Sep 25 awarded Notable Question May 9 awarded Popular Question Dec 9 awarded Notable Question Oct 13 awarded Notable Question Sep 24 awarded Autobiographer Aug 19 awarded Popular Question Jul 2 awarded Curious May 16 awarded Popular Question Feb 26 awarded Popular Question Nov 26 awarded Popular Question Sep 26 awarded Popular Question Sep 17 accepted Let $C$ be a set of sets defined as follows, Sep 17 comment Let $C$ be a set of sets defined as follows, Oh yeah, I'm really smart. So the elements of $C$ either have cardinality of 1 due to (1), 2 due to (2), or powers of 2 due to (3). Sep 17 comment Let $C$ be a set of sets defined as follows, Ok good, because now it makes sense. So clearly there are no infinite sets in $C$ because for $S, T\in C$ $|\{S, T\}| = |S|+|T|$ and $|S\times T|=|S|*|T|$. But it's uncountable because there are infinitely many ways to combine elements of $C$ using either (2) or (3), right? I can't think of a way in which $C$ could be countable. Sep 17 comment Let $C$ be a set of sets defined as follows, Oh okay that makes sense. But do we know anything other than constructions based on the empty set? Sep 17 comment Let $C$ be a set of sets defined as follows, Thank you, that was part a) actually. I figured that out similarly to how you've done it. What I'm struggling with is understanding how ordered pairs can come into play. If we let $S=\{(a,b),(c,d)\}$ for example, we can have $\{S,\emptyset\}$, but how do I know that's in $C$? Sep 17 asked Let $C$ be a set of sets defined as follows, Feb 15 awarded Yearling Nov 1 comment Aristotle's Axiom in Hyperbolic Geometry @Berci Yes, that is the Angle Unboundedness Axiom in Euclidean/Neutral geometry, but I need to figure out a way to prove that statement holds in Hyperbolic geometry. Nov 1 asked Aristotle's Axiom in Hyperbolic Geometry