Discussion Overview
The discussion revolves around the Henkin theory, specifically addressing questions related to the formulation of T* and the implications of adding constants to existential sentences in T. Participants explore the nature of existential formulas, the axiom set of T*, and the conditions under which T* qualifies as a Henkin theory.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
Main Points Raised
- Some participants question why the red part holds in T* and how lemma 3.1.8 applies to the existential formulas after adding constants.
- There is a discussion about whether the axiom set of T* should include a different set, denoted as Γ, instead of T in definition 3.1.6.
- One participant expresses interest in examples of theories T such that T* is not a Henkin theory, noting that existing textbooks do not provide motivating examples.
- Clarifications are sought regarding the nature of existential formulas, with some participants asserting that formulas of the form (∃x A(x) → A(c)) for a constant c are not considered existential formulas in the traditional sense.
- Another participant emphasizes that for T* to be a Henkin theory, all existential formulas in the language of T* must have witnesses, raising concerns about the implications of adding new constants.
Areas of Agreement / Disagreement
Participants generally agree on the definition of existential formulas but disagree on the implications of adding constants and the requirements for T* to be a Henkin theory. The discussion remains unresolved regarding the specific examples of theories that do not meet the Henkin criteria.
Contextual Notes
Limitations include the lack of concrete examples of theories where T* is not a Henkin theory, as well as the dependence on the definitions of existential formulas and the language used in T and T*.
