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Wikipedia has this definition of "radical feminism":
Radical feminism is a perspective within feminism that calls for a radical reordering of society in which male supremacy is eliminated in all social and economic contexts, while recognizing that women's experiences are also affected by other social divisions such as in race, class, and sexual orientation.
What does this 'radical reordering' include? It is not socialism, as radical feminists aren't Marxist feminists.
Feminism, broadly put, is a political/philosophical movement that seeks to address institutional differences between men and women in society. When feminism began, of course, these institutional differences were far more significant: women were largely excluded from the workforce, completely excluded from positions of authority, discouraged from voting, unable to get credit cards or personal bank accounts, even obliged to give up their names on marriage (i.e., it was expected that Jane Smith would become Mrs Bob Jones as soon as she said "I do"). Great progress has been made these issues — enough so that the average millennial is shocked hearing about them — but there's still a way to go to achieve the kind of institutional equality that feminism asks for.
Radical feminism extends that line of thought from practical (institutional) concerns to philosophical/metaphysical issues of equality. They don't merely want women to have equal pay, or to break down professional glass ceilings. They want the social, political, and economic worlds to be free of gender biases. Sometimes this is (wrongly) interpreted as a 'gender-neutral' society in which socially constructed concepts of masculinity and femininity don't exist, but the more sophisticated (and difficult) understanding is that radical feminists want a society where women have the same rights, privileges, and abilities to determine when an interaction is or is not gendered/sexual that men have traditionally enjoyed.
Mary Wollstonecraft (way back in the 18th century) said: "I do not wish [women] to have power over men; but over themselves." 'Having power over oneself' means, first and foremost, establishing oneself as a metaphysical equal to others in society, and this is the philosophical conundrum that radical feminism tries to work through.
By "radical reordering", they just mean "dismantling of the entrenched patriarchy".
In other words, a radical feminist would tell you that our society is so entrenched with ideals, principles, laws, constructs, etc. designed to keep women down that in order to achieve true equality of the sexes, radical deconstructionist action is required.
The specific goal of this reordering is not specified or implied. It could be Marxist feminism, but not necessarily. It could be calling for gender neutrality, it could be calling for matriarchy, or anything else in between.
The point is merely that things are so bad, and have been so bad for so long, that radical, disruptive change is going to be required in order to achieve the goals of feminism, whatever those goals may be.
