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Single Idea 10161
[filed under theme 5. Theory of Logic / K. Features of Logics / 4. Completeness
]
Full Idea
Completeness is when, if a sentences holds in every model of a theory, then it is logically derivable from that theory.
Gist of Idea
If a sentence holds in every model of a theory, then it is logically derivable from the theory
Source
Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V)
Book Ref
Feferman,S/Feferman,A.B.: 'Alfred Tarski: life and logic' [CUP 2008], p.281
The
12 ideas
from Feferman / Feferman
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10156
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'Recursion theory' concerns what can be solved by computing machines
[Feferman/Feferman]
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10147
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The Axiom of Choice is consistent with the other axioms of set theory
[Feferman/Feferman]
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10148
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Axiom of Choice: a set exists which chooses just one element each of any set of sets
[Feferman/Feferman]
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10149
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Platonist will accept the Axiom of Choice, but others want criteria of selection or definition
[Feferman/Feferman]
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10150
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The Trichotomy Principle is equivalent to the Axiom of Choice
[Feferman/Feferman]
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10146
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Cantor's theories needed the Axiom of Choice, but it has led to great controversy
[Feferman/Feferman]
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10155
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Both Principia Mathematica and Peano Arithmetic are undecidable
[Feferman/Feferman]
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10158
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A structure is a 'model' when the axioms are true. So which of the structures are models?
[Feferman/Feferman]
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10162
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Tarski and Vaught established the equivalence relations between first-order structures
[Feferman/Feferman]
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10160
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Löwenheim-Skolem says if the sentences are countable, so is the model
[Feferman/Feferman]
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10159
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Löwenheim-Skolem Theorem, and Gödel's completeness of first-order logic, the earliest model theory
[Feferman/Feferman]
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10161
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If a sentence holds in every model of a theory, then it is logically derivable from the theory
[Feferman/Feferman]
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