The famous diagonal argument plays a prominent role in set theory as well as in the proof of undecidability results in computability theory and incompleteness results in metamathematics. Lawvere (1969) brings to light the common schema among them through a pretty neat fixpoint theorem which generalizes the diagonal argument behind Cantor's theorem and and characterizes self-reference explicitly in category theory. Not until Yanofsky (2003) rephrases Lawvere's fixpoint theorem using sets and functions, Lawvere's work has been overlooked by logicians. This paper will continue Yanofsky's work, and show more applications of Lawvere's fixpoint theorem to demonstrate the ubiquity of the theorem. For example, this paper will use it to construct uncomputable real number, unnameable real number, partial recursive but not potentially recursive function, Berry paradox, and fast growing Busy Beaver function. Many interesting lambda fixpoint combinators can also be fitted into this schema. Both Curry's $Y$ combinator and Turing's $\Theta$ combinator follow from Lawvere's theorem, as well as their call-by-value versions. At last, it can be shown that the lambda calculus version of the fixpoint lemma also fits Lawvere's schema.
In 1950s, Carnap develops inductive logic to express the degree of confirmation of some hypothesis relative to some evidence. In 1960s, Somlomonoff invents the universal induction method to make prediction. In this paper we integrate the two methods to extent universal induction's expressive power and to enhance inductive logic's predictive power. We introduce Solomonoff prior into inductive logic, and prove the monadic first order logic version of Solomonoff completeness theorem. Then we make some comparison of the two quite different induction methods in the framework of the modified "inductive logic". As is well known, Carnap's $\lambda$-continuum fails to confirm universal generalizations. However, in the modified "inductive logic", the proposition "all ravens are black" can be confirmed in any computable world as long as all ravens are really black in that world. If we want to prove the completeness theorem by the method of Solomonoff's universal induction, we have to keep the complete information of the past history in memory. In the modified "inductive logic", we can neglect all of the irrelevant information to concentrate only on some specific pattern, and prove similar convergence results. Even without complete record of relevant information of the specific pattern, we can still build our belief through random sampling.