# Question about Godel's Incompleteness Theorems

• galoisjr
In summary: But I think that it's a promising start.In summary, the statement says that if you have a set of axioms that establish the intuition of arithmetic, then there are statements about the set or sets that this system acts on that are unprovable. However, it is not what is assumed to be false that limits provability, but the fact that we require these statements to be true. This makes sense if you think about it in terms of deduction from process of elimination, or trial and error.
galoisjr

I guess that a very brief sketch of this would be that since T limits provability then this implies that not T does not limit provability. If we assume that the double negative law from English holds in this statement then the phrase "limits provability" should be logically equivalent to "does not limit unprovability" and thus by the the same reasoning "does not limit provability" is logically equivalent to the statement "limits unprovability." Here I am assuming that the set of all conclusions are either provable or unprovable so this should hold. Then the second statement can be rewritten as F limits unprovability.

I have no idea what you mean by "Godel assumes that the system of axioms have been assigned the truth value of being true". That's pretty much the definition of "axiom" isn't it?

Yes and no. Our axioms contain only true statements that is true. But an axiom can be anything that is considered necessary to the subject. False statements are just as essential to logic as true statements are.

But I thought about it a little bit more and I think that in setting a system of statements that are false we are actually accepting that it is true that they are false. So my theory needs a little work.

Thank you for your insights on Godel's Incompleteness Theorems. I am always interested in exploring different perspectives on complex concepts. While your idea of approaching provability through deduction rather than assumption is intriguing, it is important to note that Godel's theorems are based on mathematical logic and not just intuition. The axioms in question are not simply assigned a truth value, but are carefully constructed to form a consistent and complete system. Additionally, Godel's theorems have been extensively studied and verified by mathematicians, so any alternative approach would need to be thoroughly examined and tested before it could be considered a valid alternative. Nevertheless, your thoughts on the subject are a valuable contribution to the ongoing discussion and exploration of Godel's Incompleteness Theorems.

## 1. What are Godel's Incompleteness Theorems?

Godel's Incompleteness Theorems are two theorems in mathematical logic that were discovered by Kurt Godel in 1931. They state that any consistent formal system that is capable of expressing basic arithmetic cannot prove all true statements about arithmetic.

## 2. How do Godel's Incompleteness Theorems impact mathematics?

Godel's Incompleteness Theorems have had a profound impact on mathematics and logic. They have shown that there are inherent limitations in our ability to prove all true statements about a formal system. This has led to further research and advancements in logic and the foundations of mathematics.

## 3. What is the significance of Godel's Incompleteness Theorems?

Godel's Incompleteness Theorems are significant because they challenge the idea that mathematics is a complete and consistent system. They also demonstrate that there will always be statements that are true but cannot be proven within a given formal system.

## 4. Can Godel's Incompleteness Theorems be applied to other fields?

Yes, Godel's Incompleteness Theorems have been applied to other fields such as computer science, philosophy, and linguistics. They have also had an impact on the study of artificial intelligence and the limitations of formal systems in representing human thought.

## 5. Are there any criticisms of Godel's Incompleteness Theorems?

While Godel's Incompleteness Theorems are widely accepted, there have been some criticisms. Some argue that the theorems rely on self-reference, which is problematic for some formal systems. Others argue that the theorems do not apply to all formal systems, and that there may be other ways to prove all true statements about a given system.

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