"The ability to achieve" part is tricky. Its very easy to argue, but very difficult to prove (on either side) the difference between "impossible" conditions and merely "difficult" conditions in which individuals much achieve. These details, however, become less relevant when the focus is moved to the general structure in which class is defined in a Western, capitalist-based culture.
Class divisions are a constant. One can argue how a class is and/or should be viewed. One can argue how a class is and/or should be treated. But any logical attempt to perceive class as something "wrong" (and therefore something we, as people, have some form of power over) will simply not work.
Take divisions along economic lines for example. Though it can be defined using a variety of complex specifics, poverty- in the relativist system of economics- always boils down simply to the lowest X percentile of people in the system. Let's say we draw the line at the lowest twelve percent of people. "Lowest" can mean many things, and depending on the criteria different people enter this class. But, regardless- everyone agrees there is a scale, and there is a line.
The logical unfeasibility occurs when we assume that merely because an individual is below that line, they are in an unfair predicament or "wronged" in some way. We forget that it is, in reality, impossible not to have this division, and not to have this division populated. Simple math dictates that there will always be a lowest twelve percent! If through some concoction of logic we imagine moving this lowest twelve percent "upwards," we don't change anything other than the range of the scale (which, in economics, always moves wildly upwards anyway.) That is to say, the lowest twelve percent will always be the lowest twelve percent.
So the question becomes: is this wrong?
- ENTER, PERSONAL RESPONSIBILITY -
As a culture the goal is not equality, but fair and unforced inequality. It is unfair and wrong to classify someone lower than another based on, for example, race or gender. For all the good the movements against racism, sexism and other forms of discrimination have done, the one misrepresentation of reality that has been extrapolated from such movements is this notion that all inequality is inherently bad.
As we become a more evolved and advanced culture, we don't lose our discrimination- we make it more just. While racism and sexism certainly still exist (though obviously at a lesser degree than decades or centuries ago), our inherent desire to distinguish between people has moved into a somewhat less specific "classism." And into classism we can inject one important positive factor- one which cannot be injected into racism, sexism, or other such practices. That factor is personal responsibility.
When we admit that the environment of classism and inequalities does, should, and must exist in an advanced society, and focus on individual achievement (i.e. action) within this constant environment- solutions to society's "problems" become more obvious. (And, in many situations, these "problems" are no longer actual problems.)