MHB Gödel Numbering - Exercise 3.2.5 - Chiswell and Hodges

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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 $$p_1$$ (15) is involved ... and indeed also $$\neg p_1$$ ( $$2^{15} \times 3^9$$ )

It also seems that $$p_0$$ (13) is involved ...

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

Peter
 
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Hey Peter!

Your intuition

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

It also seems that $$p_0$$ (13) is involved ...

Peter

is correct: $$p_{0}$$, $$p_{1}$$, and $$\neg p_{1}$$ 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
 

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The following is more or less taken from page 6 of C. Smorynski's "Self-Reference and Modal Logic". (Springer, 1985) (I couldn't get raised brackets to indicate codification (Gödel numbering), so I use a box. The overline is assigning a name. The detail I would like clarification on is in the second step in the last line, where we have an m-overlined, and we substitute the expression for m. Are we saying that the name of a coded term is the same as the coded term? Thanks in advance.
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