Abstract
Using the ideas of Henkin, Gödel’s Completeness Theorem is proved, showing that our proof system completely captures the notion of logical consequence.
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Notes
- 1.
Though using the Downward Löwenheim-Skolem Theorem 2.19, one needs to only look at all structures of cardinality at most \(\max (|\mathcal {L}|,\aleph _0)\), still a daunting task.
- 2.
In [36], Henkin discusses the discovery of this proof.
- 3.
Strictly speaking, this exercise should be postponed until we have formalized the notion of computability in Part III, but it can be attempted now using the intuitive notion of an algorithm.
References
Henkin, L.: The discovery of my completeness proofs. Bull. Symbolic Logic 2(2), 127–158 (1996)
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Marker, D. (2024). Gödel’s Completeness Theorem. In: An Invitation to Mathematical Logic. Graduate Texts in Mathematics, vol 301. Springer, Cham. https://doi.org/10.1007/978-3-031-55368-4_4
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DOI: https://doi.org/10.1007/978-3-031-55368-4_4
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