My first post here, just wanted to add my thoughts.
First of all the definition I use for capitalism is, the private ownership of the means of production. That being the "right", allowed by the state, for private individuals use their property economically by hiring wage labour, and profiting off that labour through surplus value created.
Socialism is the workers ownership of the means of production, however it is organised politically or socially.
Private property means 'economic' private property, not personal property. Something a lot of people are confused about.
Capitalism is not consistent with anarchism simply because capitalism creates an authority. The capitalist owner has authority over the worker. The private property owner has authority over the non-property owner (e.g. rent).
Private ownership means a small class of people monopolizing the means to produce, at the expense of the majority.
Capitalism keeps resources artificially scarce in order to maintain profits. Anarchism is about producing for our needs, everyone's needs.
Capitalism is liberty for capitalists. Liberty for some is not liberty for all.
Anarchism traditionally was socialist, and apposed to capitalism. Capitalism was the problem that socialism/anarchism was created to solve. Anarchism can be nothing but socialist because that is the only way for any kind of equality.
Capitalism, has always been the problem, more so than government. The early Anarchists, Bakunin etc., were more apposed to capitalism than government, because they new that the government we have is a result of the economic system we have. If the workers owned the means of production the state would lose it's power, and would eventually disappear altogether. The state/government gets it's power because of capitalism, because of the mass inequality of wealth. Money is the tool capitalists use to control, the state and you.
"Anarchism is stateless socialism" - Mikhail Bakunin
"Politically we are anarchists, and economically, communists or socialists." Adolph Fischer, Anarchism: Its Philosophy and Scientific Basis as Defined by Some of its Apostles (1887)