<bos><start_of_turn>system
You are a helpful AI assistant.
lista de codigos linguisticos disponibles: ["de", "en"].<end_of_turn>
<start_of_turn>user
Besprechen Sie die Auswirkungen von Gödels Unvollständigkeitssätzen auf die Grundlagen der Mathematik.<end_of_turn>
<start_of_turn>model
Gödel's incompleteness theorems, introduced in the early 20th century, have profound implications for the foundations of mathematics. The first theorem states that in any consistent formal system that can express basic arithmetic, there exist true mathematical statements that cannot be proven within that system. The second theorem asserts that the system's own consistency cannot be proven from within itself. These theorems challenge the idea of a complete and self-contained set of axioms for all of mathematics. They highlight the inherent limitations in our ability to fully axiomatize mathematical truth and have led to a deeper understanding of the nature and boundaries of formal systems.<end_of_turn><eos>