Gödel Numbering - Exercise 3.2.5 - Chiswell and Hodges

[Math Amateur]

I am reading the book Mathematical Logic by Ian Chiswell and Wilfred Hodges ... and am currently focused on Chapter 3: Propositional Logic ...

I need help with Exercise 3.2.5 which reads as follows:View attachment 5026Can someone please help me with reconstructing the formula of the Gödel number that is given ...

Thoughts ... it seems that \(\displaystyle p_1\) (15) is involved ... and indeed also \(\displaystyle \neg p_1\) ( \(\displaystyle 2^{15} \times 3^9\) )

It also seems that \(\displaystyle p_0\) (13) is involved ...

... ... BUT ... where to from here ...Hope someone can help ...

Peter

 

[GJA]

Hey Peter!

Your intuition

Thoughts ... it seems that \(\displaystyle p_1\) (15) is involved ... and indeed also \(\displaystyle \neg p_1\) ( \(\displaystyle 2^{15} \times 3^9\) )

It also seems that \(\displaystyle p_0\) (13) is involved ...

Peter

is correct: \(\displaystyle p_{0}\), \(\displaystyle p_{1}\), and \(\displaystyle \neg p_{1}\) are all involved for precisely the reasons you provided. You're clearly thinking about this correctly and you're very close to solving the problem, so I don't want my initial post to give away the answer. What I have attached is what the tree should look like once it's fully decomposed (c.f. (3.10) on page 35 of the text you're studying), and am hoping you'll see how to place all of the pieces correctly.

Let me know how it goes. Good luck!

View attachment 5027