In Smith's Godel book, the concept of "degree" in formal languages is defined through induction proofs, where the base case corresponds to degree 0. A sentence of degree k is built from well-formed formulas (wffs) using k applications of connectives and/or bounded quantifiers. Godel numbering assigns a unique positive integer to each statement or proof, which is integral to understanding the degree of a sentence. For example, a sentence with one connective and one bounded quantifier has a degree of 2, while a specific formula discussed has a degree of 3.
PREREQUISITESMathematicians, logicians, and students of formal languages who seek to deepen their understanding of Godel's theories and the structure of logical sentences.
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.