TITLE: why Skolemization?
QUESTION [10 upvotes]: Can someone explain what is the reason for using Skolemization, I clearly understand that its the removal of existencial quantifiers, but whats the use? why is keeping existencial quantifiers not good??

REPLY [3 votes]: Somewhat related to Jason Rute's answer: suppose you give me a formula, say $\varphi\equiv\forall x\exists y\theta(x, y)$ where $\theta$ is quantifier free. Skolemization gives me a way to think about the complexity of $\varphi$ being true: how complicated must a Skolem function for $\varphi$ be?
For example, consider the formula "For every $x$ there is a $y$ such that EITHER $\Phi_x(x)$ halts in exactly $y$ many stages, OR $y=0$ and $\Phi_x(x)$ never halts." This is clearly a true sentence; however, any Skolem function for it must be equivalent to the Halting problem. So the formula is not "recursively true." 
(Note that in the above example, our underlying structure is the natural numbers with $+$ and $\times$. In general, we either need to look at copies of a given structure with domain $\omega$, or adopt some other notion of effectiveness which treats a broader class of structures.)
There are now a bunch of interesting directions one can go with this! We could try to formalize the notion of "recursively true"; depending what aesthetic we bring to the table, this could lead to an intuitionistic logic, or to something like Reverse Mathematics. We could look at set of true $\Pi^0_2$ (or beyond) sentences of arithmetic, partially preordered by "every Skolem function for $\varphi_1$ computes a Skolem function for $\varphi_2$" (or other partial preorders with the sameish motivation, for example the 'degrees of provability' http://projecteuclid.org/download/pdfview_1/euclid.ndjfl/1352383227). We could focus on some particular sentence of interest, and try to understand the family of Skolem functions associated to it. We could move one type higher - to second-order statements like "For every set $X\subseteq\mathbb{N}$, there is a set $Y\subseteq\mathbb{N}$ such that if $X$ codes an infinite linear order then $Y$ codes an infinite ascending sequence or an infinite descending sequence in $X$" and look at the descriptive set-theoretic complexity of the associated higher-order Skolem functions. Or any number of other approaches I haven't thought of!

The bottom line for me: Skolemization lets us turn a statement which may be false or true into a statement which may be false or may be true with a certain complexity; and this makes things infinitely more fascinating, at least for me!