MHB Degree of Sentence in Smith's Godel Book: Explained

[agapito]

In Smith's Godel book he presents induction proofs that deal with the degree of a sentence of a formal language. So the base case corresponds to degree 0. The assumption is for sentence of degree k. Then finally proving that sentence of degree k+1 holds. What exactly does the property of "degree" mean in this context? Thanks for any clarification.

 

[HallsofIvy]

Using "Godel numbering" every statement or proof can be assigned a unique positive integer. I believe that is the "degree" you are referring to.

 

[agapito]

Using "Godel numbering" every statement or proof can be assigned a unique positive integer. I believe that is the "degree" you are referring to.

OK many thanks for your input. am

 

[Evgeny.Makarov]

What exactly does the property of "degree" mean in this context?

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]

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."

Thanks for replying. 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? If correct, what might an example of k = 3 look like?

As always, your help is greatly appreciated. am

 

[Evgeny.Makarov]

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?

Yes.

If correct, what might an example of k = 3 look like?

The degree of formula (vi) right before that is 3.

 

[agapito]

Yes.

The degree of formula (vi) right before that is 3.

Understood, many thanks, Evgeny. agapito