Gödel Numbering - Exercise 3.2.5 - Chiswell and Hodges

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SUMMARY

The discussion centers on Exercise 3.2.5 from "Mathematical Logic" by Ian Chiswell and Wilfred Hodges, specifically regarding Gödel numbering. Participants confirm that the Gödel numbers involved include $$p_0$$ (13), $$p_1$$ (15), and $$\neg p_1$$ ($$2^{15} \times 3^9$$). The exercise requires reconstructing a formula using these Gödel numbers, and a visual representation of the decomposition tree is referenced to aid in understanding the solution process.

PREREQUISITES
  • Understanding of Gödel numbering concepts
  • Familiarity with propositional logic as outlined in "Mathematical Logic" by Chiswell and Hodges
  • Basic knowledge of mathematical notation and operations
  • Ability to interpret and construct logical formulas
NEXT STEPS
  • Study the decomposition tree as referenced in section 3.10 of "Mathematical Logic"
  • Review Gödel's incompleteness theorems for deeper insights into Gödel numbering
  • Practice additional exercises from Chapter 3 to reinforce understanding of propositional logic
  • Explore the implications of Gödel numbering in computational theory
USEFUL FOR

Students of mathematical logic, particularly those studying Gödel numbering and propositional logic, as well as educators seeking to enhance their teaching of these concepts.

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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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