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Equality under the law (i.e. how government treats its citizens) is still possible
"Equality" is impossible in any society, because no one can agree on the definition when there are so many variables at play. It might be something to shoot for, but since some people don't even care about it, the question is moot.
It inherently relies on an unequal distribution of wealth/goods.
Equality of outcomes is antithetical to capitalism, however equality of opportunity is very well aligned with the premises of capitalism insofar as it enables society to make the most efficient use of human capital.
Capitalism relies on exploitation of inequality to sustain itself
Inequality is natural for humans and an unavoidable state. Some children will inevitably be born with different traits than others.
we associate capitalism with meritocracy - hierarchy and inequality are symbolic measures of one's merit within the system, therefore, egalitarianism and capitalism are antithetical to each other.
Trying to create an equal outcome isn't desirable. It is more important to have a system built upon creating opportunity, and to achieve this individuals must be able to accumulate and invest wealth; otherwise the value of human interaction is set arbitrarily.
Social equality is impossible in general because of culture, tradition, religion, and any other method of passing of beliefs from one person to another.
The premise is so open that it is pretty meaningless thing to argue about.
Inequality of outcome is always impossible, so while technically true the premise is dishonest.
Inequality of rights is only possible in a capitalist society.
This depends entirely on your definition of equality. If you believe that equality means that everyone is treated the same and treated fairly, then it is fair that one man can make more money than another if he works harder.