Two math conversations with Claude

I had two math conversations with Claude, and I want to share them with you. I will show both of them.

For who doesn’t know, Claude is a series of Large Language Models (LLMs) developed by the company Anthropic. IMO they are more powerful than OpenAI’s GPTs (but GPT-4o is catching up).

The used model was Claude 3.5 Sonnet on both conversations on the 5th and 6th of July 2024, for the first and second conversation respectively.

The shown images belong to the book Richard L. Epstein, Leslaw W. Szczerba - Classical Mathematical Logic - The Semantic Foundations of Logic - Princeton University Press (2006).

First conversation

User: Can you explain chapter 3?

Chapter 3 discusses soundness and completeness in logical systems. Here’s a breakdown of the key points:

  1. Soundness: An axiomatization is sound if every theorem is a tautology. Formally, for all A, if ⊢A then ⊨A. This is the minimal condition for a semantics and syntax to characterize the same pre-formal logical intuitions.

  2. Completeness: An axiomatization is complete if every tautology is a theorem. Formally, for all A, if ⊨A then ⊢A. A proof of this is called a completeness theorem.

  3. Strong Completeness: The strongest correlation is when semantic and syntactic consequence relations are the same. That is, for all Γ and A, Γ⊨A if and only if Γ⊢A. This is called strongly complete semantics for the axiomatization or a strongly complete axiomatization of the semantics. A proof of this equivalence is called a strong completeness theorem.

These concepts are fundamental in logic, as they establish the relationship between the syntactic derivability (⊢) and semantic entailment (⊨) in a logical system. Soundness ensures that all provable statements are true, while completeness ensures that all true statements are provable. Strong completeness extends this to sets of premises.

The chapter emphasizes the importance of these properties in formalizing logical systems and understanding the connection between formal proofs and semantic truth.

User: Can you solve the first three exercises of the chapter?

I can provide guidance on approaching these exercises, but I’ll avoid giving complete solutions to encourage your own reasoning. Let’s look at the first three exercises:

  1. Prove that if Γ is a collection of theorems and Γ ⊢ A, then ⊢ A. (Hint: Induct on the length of a proof of A from Γ.) Does it matter whether Γ is finite or not?

Approach: Use induction on the length of the proof as suggested. Consider the base case (length 1) and inductive step. The finiteness of Γ doesn’t affect the proof structure, so it shouldn’t matter.

  1. Why is it right to identify ⊢A and ∅⊢A?

Think about: What does ⊢A mean in a logical system? How does it relate to derivability from an empty set of premises (∅⊢A)? Consider the definition of a theorem in the context of the logical system.

  1. Show that if Γ is a theory, then every axiom is in Γ.

Consider: What is the definition of a theory in this context? How are axioms related to theories? Think about the closure properties of theories under logical consequence.

For each of these, try to formulate a clear argument using the definitions and concepts presented in the chapter. If you’d like to discuss your approach to any specific exercise in more detail, feel free to ask.

User: I’m very lost. Can you solve the first one?

I understand you’re feeling lost. Let’s work through the first exercise step-by-step:

Exercise 1: Prove that if Γ is a collection of theorems and Γ ⊢ A, then ⊢ A. (Hint: Induct on the length of a proof of A from Γ.)

Proof:

  1. Base case: If the proof has length 1, then A is either an axiom or a member of Γ.
    • If A is an axiom, then ⊢ A by definition.
    • If A is in Γ, then A is a theorem (given), so ⊢ A.

  2. Inductive hypothesis: Assume the statement holds for all proofs of length k or less.

  3. Inductive step: Consider a proof of length k+1. The last step in this proof can be: a) An axiom: Then ⊢ A directly. b) A member of Γ: Then A is a theorem, so ⊢ A. c) Derived by a rule of inference from previous steps: These previous steps are either axioms, members of Γ, or derived from them in k or fewer steps. By the inductive hypothesis, each of these previous steps B is such that ⊢ B. Since rules of inference preserve theoremhood, we can conclude ⊢ A.

  4. By induction, the statement holds for proofs of any length.

Therefore, if Γ is a collection of theorems and Γ ⊢ A, then ⊢ A.

Note: The finiteness of Γ doesn’t affect this proof, as we’re inducting on the length of the proof, not on Γ itself.

This proof demonstrates that if we can derive A from a set of theorems, A itself must be a theorem. Does this help clarify the approach? Let me know if you need any part explained further.

User: Can you solve exercise 2?

Certainly. Let’s work through Exercise 2:

Exercise 2: Why is it right to identify ⊢A and ∅⊢A?

To answer this, we need to understand what these notations mean and why they are equivalent:

  1. ⊢A means that A is a theorem of the logical system. In other words, A can be derived from the axioms of the system using the rules of inference, without any additional premises.

  2. ∅⊢A means that A can be derived from an empty set of premises. This also implies that A can be derived using only the axioms and rules of inference of the system.

The reason these are identified (considered equivalent) is:

  1. If ⊢A, then A is a theorem that can be proven without any additional premises. This means it can also be derived from an empty set of premises, so ∅⊢A.

  2. If ∅⊢A, then A can be derived without using any premises outside the logical system itself. This means A is a theorem of the system, so ⊢A.

  3. In both cases, the derivation of A relies solely on the axioms and rules of inference of the logical system, without any external assumptions or premises.

  4. The empty set ∅ doesn’t add any information or constraints to the derivation process. Deriving from ∅ is equivalent to deriving from the system’s inherent rules and axioms alone.

Therefore, ⊢A and ∅⊢A represent the same concept: A is provable within the system using only its axioms and rules of inference, without any additional assumptions. This equivalence is fundamental in formal logic and helps simplify notation and reasoning about theorems.

User: I’m providing you the definition of theory as well in the context of mathematical logic. Now, can you help me understand this and also solve exercise 6?

Certainly. Let’s break this down and then address exercise 6.

Definition of a Theory: A set Σ is a theory if it is closed under syntactic deduction. This means if Σ⊢A, then A is in Σ. The collection of all syntactic consequences of Σ is called the theory of Σ, written as Th(Σ) = {A : Σ⊢A}. Thus, Σ is a theory if and only if Th(Σ) = Σ.

This definition essentially says that a theory includes not just its initial statements (axioms), but also all logical consequences that can be derived from those statements.

Now, for Exercise 6:

Exercise 6: We could rephrase several parts of Theorem 3 and other statements about ‘⊢’ in terms of theories, which is what many texts do. For example, Theorem 3.c can be stated as: Γ ⊆ Th(Γ). Restate parts (d) and (e) of Theorem 3 similarly.

Let’s rephrase parts (d) and (e):

(d) Monotonicity: If Γ⊢A and Γ ⊆ Δ, then Δ⊢A. Rephrased: If Γ ⊆ Δ, then Th(Γ) ⊆ Th(Δ).

This means that if we expand our set of premises (from Γ to a larger set Δ), the set of consequences (the theory) also expands or stays the same.

(e) Transitivity: If Γ⊢A and A⊢B, then Γ⊢B. Rephrased: Th(Th(Γ)) = Th(Γ).

This means that the consequences of the consequences of Γ are just the consequences of Γ. In other words, applying the theory operation twice doesn’t yield anything new beyond applying it once.

These restatements emphasize the closure properties of theories and how they relate to the original syntactic consequence relations. They showcase how the concept of a theory encapsulates the idea of logical closure in a concise way.

User: Can you explain why Th(Σ) = Σ and solve exercise 3?

Certainly. Let’s break this down:

  1. Why Th(Σ) = Σ for a theory Σ:

Th(Σ) = Σ means that a set Σ is a theory if and only if it’s equal to its own set of consequences.

  • If A ∈ Σ, then Σ ⊢ A (by definition of ⊢), so A ∈ Th(Σ). This shows Σ ⊆ Th(Σ).
  • If A ∈ Th(Σ), then Σ ⊢ A (by definition of Th). Since Σ is a theory, it’s closed under deduction, so A ∈ Σ. This shows Th(Σ) ⊆ Σ.

Therefore, Σ = Th(Σ).

Now, let’s solve Exercise 3:

Exercise 3: Show that if Γ is a theory, then every axiom is in Γ.

Proof:

  1. Let A be any axiom of the logical system.
  2. By definition, ⊢ A (axioms are theorems).
  3. If ⊢ A, then Γ ⊢ A for any set Γ (including theories), because deriving A doesn’t depend on Γ.
  4. Since Γ is a theory, it’s closed under deduction. This means if Γ ⊢ A, then A ∈ Γ.
  5. Therefore, A ∈ Γ.

Since A was an arbitrary axiom, we can conclude that all axioms are in Γ.

This proof shows that theories, being closed under deduction, must contain all axioms of the logical system, as axioms are derivable without any premises.

Second conversation

User: Can you explain the last part of the part (a) of the proof? I don’t get the application of the induction

Certainly, I’d be happy to explain the last part of the proof for part (a) of Theorem 1.

The proof is considering the case where A has the form (∃x B). Here’s a breakdown of the final steps:

  1. The proof assumes that υσ(∃x B) = T, meaning the formula is true under assignment σ.

  2. By the semantics of the existential quantifier, this means there exists some γ that differs from σ at most in what it assigns to x, such that υγ(B) = T.

  3. The proof then defines ρ to be the same as τ except that ρ(x) = γ(x).

  4. Now, ρ and γ agree on all variables free in B. This is because ρ matches τ everywhere except x, and γ matches σ everywhere except possibly x. Since σ and τ were assumed to agree on all free variables in A (which includes B), ρ and γ must agree on all free variables in B.

  5. By the induction hypothesis (since B is shorter than ∃x B), we can conclude that υρ(B) = T.

  6. But ρ differs from τ only in its assignment to x, which is exactly the condition needed for υτ(∃x B) to be true.

  7. Therefore, υτ(∃x B) = T.

  8. The same argument works with the roles of σ and τ reversed, so υσ(∃x B) = υτ(∃x B).

This completes the proof for the existential quantifier case, showing that if σ and τ agree on all free variables, then υσ(A) = υτ(A) even when A involves an existential quantifier.

The induction is applied here because we’re assuming the theorem holds for all shorter formulas (like B) and using that to prove it for the longer formula (∃x B).

User: Give a formal definition of the following concepts: wwf, assignment of references σ, valuation υ

I’d be happy to provide formal definitions for these concepts. They are fundamental in formal logic and model theory. Here are the definitions:

  1. Well-formed formula (wff):

A well-formed formula (often abbreviated as wff) is a string of symbols from a formal language that is constructed according to grammatically correct rules. In predicate logic:

  • If P is an n-ary predicate symbol and t1, …, tn are terms, then P(t1, …, tn) is a wff.
  • If A and B are wffs, then (¬A), (A ∧ B), (A ∨ B), (A → B), and (A ↔ B) are wffs.
  • If x is a variable and A is a wff, then (∀x A) and (∃x A) are wffs.
  • Nothing else is a wff.
  1. Assignment of references σ:

An assignment of references, typically denoted by σ, is a function that assigns values from a domain D to variables in a formula. Formally:

σ : V → D

where V is the set of variables and D is the domain of discourse.

  1. Valuation υ:

A valuation υ is a function that assigns truth values (typically T for true and F for false) to formulas based on an interpretation I and an assignment σ. It’s often defined recursively:

  • For atomic formulas: υ(P(t1, …, tn)) = T iff (I(t1), …, I(tn)) ∈ I(P)
  • For negation: υ(¬A) = T iff υ(A) = F
  • For conjunction: υ(A ∧ B) = T iff υ(A) = T and υ(B) = T
  • For universal quantification: υ(∀x A) = T iff for all d ∈ D, υ[x→d](A) = T

where υ[x→d] is the valuation that agrees with υ except that it assigns d to x.

These definitions form the basis for the semantics of predicate logic, allowing us to formally evaluate the truth or falsity of logical formulas in a given interpretation.

Thank you for reading this post. Until the next time, bye!

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