What does the incompleteness theorem say?

…the 20th century: his famous incompleteness theorem, which states that within any axiomatic mathematical system there are propositions that cannot be proved or disproved on the basis of the axioms within that system; thus, such a system cannot be simultaneously complete and consistent.

What did Gödel believe?

In his philosophical work Gödel formulated and defended mathematical Platonism, the view that mathematics is a descriptive science, or alternatively the view that the concept of mathematical truth is objective.

What is the difference between Gödel’s theorem and Gödel sentence?

One should not get confused here: “Gödel’s theorem” is the general incompleteness result of Gödel which concerns a large class of formal systems, while the “Gödel sentence” is the constructed, formally undecidable sentence which varies from one formal system to another.

Can we apply Gödel’s theorems in other fields of Philosophy?

There have also been attempts to apply them in other fields of philosophy, but the legitimacy of many such applications is much more controversial. In order to understand Gödel’s theorems, one must first explain the key concepts essential to it, such as “formal system”, “consistency”, and “completeness”.

Does Gödel’s theorem claim that undecidable statements exist?

Gödel’s theorem does not merely claim that such statements exist: the method of Gödel’s proof explicitly produces a particular sentence that is neither provable nor refutable in F ; the “undecidable” statement can be found mechanically from a specification of F

What are the paradoxes of self-reference?

Most paradoxes of self-reference may be categorised as either semantic, set-theoretic or epistemic. The semantic paradoxes, like the liar paradox, are primarily relevant to theories of truth. The set-theoretic paradoxes are relevant to the foundations of mathematics, and the epistemic paradoxes are relevant to epistemology.