I agree with the other answers, but there is another consideration at play here, which is that beliefs constitute parts of mental models of the world, and a person can use multiple mutually-contradictory mental models at different times.
This is like the way a physicist can 'believe' in Newtonian classical physics in one context, with idealisations of rigid bodies, infinite frictionless planes, point particles, and so on, and yet know perfectly well that in a relativistic quantum world those constructs are not just false but impossible. This does not lead to trivialisation because belief is model-dependent. When using a classical Newtonian model of the world, rigid bodies are possible. When using a relativistic model, they are not. There is no contradiction between the two, because they are statements in different axiomatic systems. Similarly with Euclidean geometry and spherical geometry, the statement that the angles of a triangle add up to 180 degrees is true in one and false in the other. A Euclidean geometer will be happy to say without qualification they believe the total of a triangles angles is always 180, because the axiomatic context is assumed. (i.e. We assume the axioms are true.) But if asked specifically about non-Euclidean geometries, they will be equally happy to say it is not generally true.
So when the logician says be believes all the theorems stated in the book are true, he is operating in the context of the beliefs of a prover, in which constructing an apparently valid proof of a statement should be taken to mean it is true. Those are the rules of the game. When talking about the fallibility of provers meaning that some of the statements will be false, he is using a different model, with different rules.
For a logician to believe that they are themselves a fallible prover, and thus some of the things they believe true are actually false, sounds like a version of the liar paradox. But the paradox can be avoided by keeping the contradictory beliefs in separate boxes. If you are required to keep all your beliefs in one box, then knowledge that you are fallible does indeed mean you cannot know anything. But you can define belief/truth relative to a particular fallible prover, and keep knowledge of the fallibility in an enclosing model, and get a limited form of belief.
The problem is unavoidable in sufficiently expressive systems, because of Godel's theorems. For any consistent proof system expressive enough to model arithmetic, there are statements in it that are true-but-unprovable in that system. We are only allowed to prove that these Godel statements are actually true because we are using a different proof system from the one under study, that encloses it. No sufficiently expressive proof system can prove itself consistent, (unless it is actually inconsistent!) - they must all accept the possibility of fallibility.