Shahada ya kwanza (juu)FoundationsImepangwa
Completeness, compactness and Löwenheim–Skolem
Gödel's completeness theorem and its consequences: compactness, the Löwenheim–Skolem theorems, nonstandard models and the limits of first-order axiomatisation.
math.found.completeness-theorem
Somo la mada hii bado halijaandikwa. Muhtasari ulio hapa chini ni sehemu ya ramani ya mtaala na haujakaguliwa.
Inajengwa juu ya
Masharti ya awali ya lazima 16 (hadi chini kabisa)
Utajifunza
- Prove the completeness theorem for countable languages by a Henkin construction.
- Use compactness to show that some classes, such as torsion groups or finite structures, are not first-order axiomatisable.
- State the Löwenheim–Skolem theorems and explain Skolem's paradox.
Fasili
- A theory is consistent if it does not prove a contradiction, and satisfiable if it has a model.
- Compactness: a set of sentences has a model whenever every finite subset of it has a model.
Nadharia
- Completeness: a first-order theory is consistent if and only if it has a model.
- The compactness theorem holds for first-order logic.
- Löwenheim–Skolem: a theory in a countable language with an infinite model has a model of every infinite cardinality.
- Peano arithmetic has nonstandard models.
Mawazo potofu ya kawaida
- Thinking the completeness and incompleteness theorems contradict each other; one concerns truth in all models, the other provability of truths about one intended model.
- Reading Skolem's paradox as an inconsistency: a countable model of ZFC contains no bijection between ω and its version of ℝ, although one exists outside the model.
Inaelekea kwa
Ugunduzi wa hiari
- Real analysis Compactness yields ordered fields elementarily equivalent to ℝ that contain infinitesimals, the basis of Robinson's nonstandard analysis.
Usomaji zaidi
- Herbert B. Enderton, A Mathematical Introduction to Logic (1972)
- David Marker, Model Theory: An Introduction (2002)