Thinking the Model of Science: The Problem of Undecidability in the Real

In contemporary science we are always confronted with the conditions of its formalization and the ability to rely on the independence and functionality of its axioms and the validity of its models. Hence, a true scientific thought can be the product of only those sciences that deem themselves worthy of the name. These have the traits of scientifically constructing, as well as applying, their models of sentences, clauses, theories, propositions etc. In the first part of the paper, we follow the understanding of science developed by Gaston Bachelard by revisiting the notion of the “epistemological obstacle”, which we relate to Heidegger's notion of bringing-forth of the “not-to-be-gotten-around”. After that, we introduce and review certain aspects about the problem of undecidability in formal systems or formal languages identified by some of the key logicians of the 20th century – including Tarski, Gödel, Löwenheim and Skolem. The main goal of the analysis is to dissect the epistemological consequences of the problem of undecidability arising in formal systems of scientific inquiry. In the second part of the article, we follow Alain Badiou’s earlier writings on the concept of model, delineating the conditions of possibility of thinking scientific thought – one that is aware of its own thinking, while simultaneously needing an external reference point for it – a point in the real. We conclude that all encounters with epistemological obstacles take place in the field of the real, i.e. beyond the cognitive domain of models and theories, by the inscription of a supplementary point in the real, retroactively imposing a new structure of model in e.g. formal system or language.