What is the intuitive meaning of the notion of absolutness in set theory? I would love to know what the notion of absolutness in set theoy mean intuitively. I wonder if anyone can provide me a heuristics about it. I know the formal definition and that it corresponds to bounding the quantifiers so that they range over only the elements of the model.  The heuristics  I'm looking for is something like the following;
While I was reading the essay about forcing in handbook of mathematical logic editted by Barwise, It was mentioned that the notion of absolutness is intuitively the notion of being checkable inside the model without looking outside the model to check it. I do not understand what does this mean exactly.I would love to have an explanation of this. Also,  could you please show me why we consider the notion of being a stationary set in an ordinal to be non-absolute using this notion of checking this from outside the model (since it is not absolute so we must go outside the model to check it)
I would also appreciate making an analogy between  absolutness  in set theory and in any other mathematical theory, say group theory. What does absolutness look like in something as group theory if it is possible to occur. if it does not, What is special about sets?
 A: Absoluteness means that if you know something in one model, you know it in other models. 
So if $\kappa$ is a cardinal in $V$, it is a cardinal in any inner model of $V$. If it is not a cardinal, then there is a function from some $\alpha<\kappa$ onto $\kappa$ witnessing that. Since there are none in $V$, there must be none in inner models, because a set of ordered pairs will be a function in any model of set theory it belongs to. This is called downwards absoluteness.
But using forcing we can make $\kappa$ countable, so it's not true that if $\kappa$ is a cardinal then it remains so in any larger model. But if $\kappa$ is a countable ordinal, then there is a function from $\omega$ onto $\kappa$, and that function is not going to disappear if we extend the universe. So being countable is true in any larger universe. This is called upwards absoluteness.
If a property is both upwards and downwards absolute, we just say it is absolute. In both examples above, we see that "$f$ is a function with domain $\alpha$ and range $\kappa$, with $\alpha$ and $\kappa$ as ordinals" is an absolute property. In any model which knowns about $f$, $f$ is a function. 
This can be useful for actually proving theorems. If you can force a statement which is downwards absolute, then you have proved it [to hold in $V$]. Why? Because you forced it in $V[G]$ and $V$ is an inner model of $V$, it has to hold in $V$. 
As an example from group theory, being in the center of a group is downwards absolute. If $x\in Z(G)$ and $H\leq G$ such that $x\in H$ then $x\in Z(H)$. On the other hand, this property is not upwards absolute since we can always add an element which does not commute with $x$.
