The aim of this paper is to consider the question about the reasons of the indefinability of truth. We note at the start that a formula with one free variable can function as a truth predicate for a given set of sentences in two different (although related) senses: relative to a model and relative to a theory. By methods due to Alfred Tarski it can be shown that some sets of sentences are too large to admit a truth predicate (in any of the above senses); the limit case being the set of all sentences. The key question considered by us is: what does 'too large' mean, i.e. which exactly sets of sentences don't have a truth predicate. We give a partial answer to this question: a set of sentences 'K' has a truth predicate in an axiomatizable, consistent theory 'T' if for some natural number 'n', all the sentences belonging to 'K' are equivalent (in 'T') to the sum of 'n' sentences. Here the notion of a 'too large' set receives a clear and definite sense. However, the case of a model-theoretic truth predicate seems to be more complicated: this second problem we leave as open, indicating only some possible directions of future research.