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imjustcurious
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One of Godels Incompleteness Theorems says that a system cannot demonstrate its own consistency. This made me question human consciousness. We can say we are real, so does that make us incomplete?
Honestly, I don't know either lol.micromass said:And I don't know what "we are incomplete" even means in this case.
The theorems of incompleteness, also known as Gödel's incompleteness theorems, are two theorems in mathematical logic that were proven by mathematician Kurt Gödel in 1931. They state that in any formal axiomatic system that is strong enough to express basic arithmetic, there will always be statements that are true but cannot be proven within the system itself.
The theorems of incompleteness have had a profound impact on the fields of mathematics, logic, and philosophy. They have shown that there are limits to what can be proven within a formal system, and have raised questions about the nature of truth and the foundations of mathematics.
The theorems of incompleteness rely heavily on the concept of self-reference, which is when a statement refers to itself or makes a claim about itself. Gödel used this concept to construct a statement that essentially says "this statement cannot be proven." This statement is true, but cannot be proven within the system, leading to incompleteness.
Some attempts have been made to avoid the limitations of the theorems of incompleteness, such as using a stronger axiomatic system or adding additional axioms. However, these approaches often lead to other problems and do not fully solve the issue of incompleteness.
The theorems of incompleteness have had a significant impact on fields outside of mathematics and logic, such as computer science, linguistics, and artificial intelligence. They have also sparked philosophical debates about the nature of truth and the limits of human knowledge.