Science is facing a "replication crisis" in which many experimental findings cannot be replicated and are likely to be false. Does this imply that many scientific facts are false as well? To find out, we explore the process by which a claim becomes fact. We model the community's confidence in a claim as a Markov process with successive published results shifting the degree of belief. Publication bias in favor of positive findings influences the distribution of published results. We find that unless a sufficient fraction of negative results are published, false claims frequently can become canonized as fact. Data-dredging, p-hacking, and similar behaviors exacerbate the problem. Should negative results become easier to publish as a claim approaches acceptance as a fact, however, true and false claims would be more readily distinguished. To the degree that the model reflects the real world, there may be serious concerns about the validity of purported facts in some disciplines.