set. Imagine this process repeated infinitely, so that truth is defined for The barn is big; then for "The barn is big is true"; then for "'The barn is big is true' is true", and so on. Truth never gets defined for sentences like This sentence is false, since it was not in the original subset and does not predicate truth of any sentence in the original or any subsequent set. In Kripke's terms, these are "ungrounded." Since these sentences are never assigned either truth or falsehood even if the process is carried out infinitely, Kripke's theory implies that some sentences are neither true nor false. This contradicts the principle of bivalence: every sentence must be either true or false. Since this principle is a key premise in deriving the liar paradox, the paradox is dissolved. However, it has been shown by Gödel that self-reference cannot be avoided naively, since propositions about seemingly unrelated objects can have an informal self-referential meaning; in Gödel's work, these objects are integer numbers, and they have an informal meaning regarding propositions. In fact, this idea — manifested by the diagonal lemma—is the basis for Tarski's theorem that truth cannot be consistently defined. It has thus been claimed that Kripke's system indeed leads to contradiction: while its truth predicate is only partial, it does give truth value (true/false) to propositions such as the one built in Tarski's proof, and is therefore inconsistent. While there is still a debate on whether Tarski's proof can be implemented to every similar partial truth system, none have been shown to be consistent by acceptable methods used in mathematical logic. Kripke's semantics are related to the use of topoi and other concepts from category theory in the study of mathematical logic. They provide a choice of formal semantics for intuitionistic logic. Folk beliefs The truth predicate "P is true" has great practical value in human language, allowing efficient endorsement or impeaching of claims made by others, to emphasize the truth or falsity of a statement, or to enable various indirect (Gricean) conversational implications. Individuals or societies will sometime punish "false" statements to deter falsehoods; the oldest surviving law text, the Code of Ur-Nammu, lists penalties for false accusations of sorcery or adultery, as well as for committing perjury in court. Even four-year-old children can pass simple "false belief" tests and successfully assess that another individual's belief diverges from reality in