HallsofIvy said:Using "Godel numbering" every statement or proof can be assigned a unique positive integer. I believe that is the "degree" you are referring to.
Page 75 says, "So let us say that a $\Delta_0$ sentence has degree $k$ iff it is built up from wffs of the form $\sigma=\tau$ or $\sigma\le\tau$ by $k$ applications of connectives and/or bounded quantifiers."agapito said:What exactly does the property of "degree" mean in this context?
Evgeny.Makarov said:Page 75 says, "So let us say that a $\Delta_0$ sentence has degree $k$ iff it is built up from wffs of the form $\sigma=\tau$ or $\sigma\le\tau$ by $k$ applications of connectives and/or bounded quantifiers."
agapito said:To check for my understanding, if you look at example vii on previous page:
Sentence has 1 connective and 1 bounded quantifier. Would this mean that k in this case is 2?
The degree of formula (vi) right before that is 3.agapito said:If correct, what might an example of k = 3 look like?
Evgeny.Makarov said:Yes.
The degree of formula (vi) right before that is 3.
The "Degree of Sentence" refers to the level of complexity or difficulty in understanding a particular sentence in Kurt Godel's work, as explained by philosopher Gregory Smith in his book "Godel's Proof". This concept is important in understanding Godel's incompleteness theorems.
Smith uses the idea of "layers" or "levels" of sentences to explain the Degree of Sentence. He argues that some sentences are more complex and require more layers of understanding to fully comprehend, while others are simpler and require fewer layers. He also discusses the role of symbols and formal systems in determining the Degree of Sentence.
The Degree of Sentence is important because it helps us understand the limitations of formal systems and the concept of "truth" in mathematics. Godel's incompleteness theorems show that there are always statements that are true but cannot be proven within a formal system, and the Degree of Sentence helps us understand why this is the case.
The Degree of Sentence is closely tied to Godel's incompleteness theorems, as it helps us understand the limitations of formal systems and the incompleteness of mathematical truths. The concept of Degree of Sentence also sheds light on the complexity of Godel's proofs and the importance of understanding the underlying layers of meaning in his work.
While the concept of Degree of Sentence was developed specifically to explain Godel's work, it can also be applied to other complex texts or systems. The idea of layers of meaning and varying levels of complexity is relevant to many areas of philosophy, mathematics, and other fields of study.