Is this the Axiom of Infinity? While studying elementary Set Theory, I came across the Axiom of Infinity. This comes before the book introduces ZFC, so I'm not convinced that it is necessarily the same as the typical definition.
My notes use the following definition (edit to add in the logical not $\lnot$ symbol in order to make the statement correct as pointed out by Eric Wofsey in the comments):

Weak Axiom of Infinity: $\small \exists x \lnot (\forall w (( \exists y \space y \in w \land \forall y (y \in w \rightarrow \forall z (z \in y \rightarrow z \in x ))) \rightarrow$
$\exists y (y \in w \land \forall v (( v \in w \land \forall z (z \in y \rightarrow z \in v )) \rightarrow y = v )))$

On the other hand, the more standard notion of the Axiom of Infinity that I have come across (whilst attempting to understanding the (difficult to follow) definition above, is very different. Wikipedia uses the following notion:

Axiom of Infinity: $\exists I (\emptyset \in I \land \forall x \in I ((x \cup ${$x$}$) \in I ))$

I'm struggling to properly understand the first definition well enough to know if these are essentially saying the same thing. If they are different in some way, then what is the difference and why is this difference present?
 A: To parse your "Weak Axiom of Infinity", it says there exists a set $x$ such that if $w$ is any nonempty set of subsets of $x$ (that's $\exists y \space y \in w \land \forall y (y \in w \rightarrow \forall z (z \in y \rightarrow z \in x ))$), then there exists $y\in w$ such that no element $v\in w$ strictly contains $y$.  In other words, every nonempty set of subsets of $x$ has a maximal element (with respect to inclusion).
This is an alternate definition of finiteness (due to Tarski, if I remember correctly): a set $x$ is finite iff every nonempty set of subsets of $x$ has a maximal element.  The forward direction is easy (for instance, if $w$ is a nonempty set of subsets of a finite set $x$, there is an element of $w$ of maximal cardinality which then must be maximal with respect to inclusion).  For the reverse direction, if $x$ is infinite, then the set $w$ of all finite subsets of $x$ has no maximal element.
Now, of course, there is a problem here: your Weak Axiom of Infinity states the existence of a finite set, not an infinite set!  It should be corrected with a negation after the $\exists x$ so that it is saying there exists $x$ which does not satisfy the finiteness condition, i.e. an infinite set.  Then it will be equivalent to the usual axiom of infinity (to sketch the proof, if $x$ is an infinite set, then by replacement there is a set of all the cardinalities of finite subsets of $x$, and this is in inductive set).
