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No, a Catholic cannot be an anarchist.
Romans 13:1-5 says we are to obey our "princes" (political leaders) because their authority comes from God; we obey them out of obedience to God:
- Let every soul be subject to higher powers. For there is no power but from God: and those that are ordained of God.
- Therefore, he that resisteth the power resisteth the ordinance of God. And they that resist purchase to themselves damnation.
- For princes are not a terror to the good work, but to the evil. Wilt thou then not be afraid of the power? Do that which is good: and thou shalt have praise from the same.
- For he is God's minister to thee, for good. But if thou do that which is evil, fear: for he beareth not the sword in vain. For he is God's minister: an avenger to execute wrath upon him that doth evil.
- Wherefore be subject of necessity: not only for wrath, but also for conscience' sake.
cf. Pope Leo XIII's encyclical Diuturnum on the origin of civil power.
Rebellion is not Christian. Satan, tempting Adam and Eve, was the first rebel; he said "Non serviam" ("I will not serve [God]."); cf. Revolution and Counter-Revolution by Plinio CorrΓͺa de Oliveira.
The State is not a voluntary society (like a chess club, university, trade union, or corporation) but a necessary society. Pope Pius XI, Divini illiud magistri Β§11:
there are three necessary societies, distinct from one another and yet harmoniously combined by God, into which man is born: two, namely the family and civil society, belong to the natural order; the third, the Church, to the supernatural order.
cf. Integralism ch. 1, Β§ "Division of societies".