Please tell me the notational meaning of Zermelo-Fraenkel set theory with the axiom of choice.
#ZFC
pale steeple
rigid saddleBOT
pale steeple
In particular, I would like to know what the area circled in red means.
lofty coyote
In particular, I would like to know what the area circled in red means.
top to bottom:
Axiom of Union: This helps us define $A:=\bigcup X$ (union of all the sets in $X$, because it's saying that $t$ in $A$ is equivalent to $t$ being an element of some set $x$ in $X$)
Axiom of Infinity: Typically we say the successor $succ(x)=x\cup{x}$. Simplifying, it just says that there exists a set containing the empty set and closed under successor. (Not necessarily the smallest.)
olive garnetBOT
Element118
lofty coyote
I believe the next ones are
Axiom of Replacement: Firstly we establish that $\psi$ defines a partial function, and we can replace $x$ with $y:=\psi(x)$ should $x$ be in the domain of $\psi$.
Axiom of Foundation: Main idea here is saying that nonempty sets A have an element that isn't contained in any of the elements of A.
olive garnetBOT
Element118
