Final answer:
Gödel's incompleteness theorem is not based on premises bearing a contradiction; rather, it states that sufficiently complex, consistent formal systems cannot be both complete and consistent. This is a logical determination about mathematical systems, separate from theological or philosophical arguments regarding the existence of God which may involve circular reasoning or logical fallacies.
Step-by-step explanation:
The question at hand involves assessing whether Gödel's incompleteness theorem is based on premises containing a contradiction. To address this, it is essential to understand that Gödel's theorem emerges from a mathematical and logical ground, which is distinct from philosophical arguments for the existence of a deity or the nature of reality, as discussed by Leibniz, Anselm, Kant, and others.
Gödel's theorem indeed revolves around formal systems within the domain of mathematics and logic. It posits that any sufficiently complex formal system that is consistent (meaning it does not contain contradictions) and capable of expressing basic arithmetic cannot be both complete and consistent. Essentially, there will always be true statements within that system that cannot be proven using the system's own rules. Thus, far from relying on contradictory premises, the theorem underscores the limitations inherent within logical and mathematical systems.
When discussing the proofs of God's existence, as with Anselm's ontological argument or Leibniz's theodicy, one can identify instances where arguments may contain, as critics like Kant argue, logical fallacies or circular reasoning where premises may assume the conclusion. However, these discussions are philosophical and theological and not directly associated with the mathematical foundations of Gödel's work.