Belief in non-material realities incorporating the notion of a
supreme being. Most religions (including the
judaic faiths and
hinduism) teach that religious belief comes as a result of a supreme being revealing itself to the individual. This constitutes
proof that transcends material/logical/mathematical proofs, since a genuine supreme being would be proof itself.
Matrix style objections are innapropriate since the supreme being is also held to unite itself to the individual therefore giving the individual the ability to see as God, even if just for a moment and only of a thin slice of reality (in order not to compromise "
free will")
In non-protestant christianity it is also taught that a belief in the existence of God (not strictly speaking "religious belief") can be attained without divine revelation, but simply by observing nature.
Religious belief is not said to be irrational except by the anti-religious and ignorant.
The word "faith" literally and originally means "trust", and is not the same as "religious belief". However religious believers are expected to have put their trust in God as a consequence of God's self-revelation. Due to free-will this is not inevitable.