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Single Idea 10761
[filed under theme 5. Theory of Logic / K. Features of Logics / 4. Completeness
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Full Idea
All that should be required to get a semantics relative to which a given deductive system is complete is a sufficiently cunning model-theorist.
Gist of Idea
Completeness can always be achieved by cunning model-design
Source
Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §5)
The
10 ideas
from 'First-order Logic, 2nd-order, Completeness'
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10751
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Second-order logic needs the sets, and its consequence has epistemological problems
[Rossberg]
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10753
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Logical consequence is intuitively semantic, and captured by model theory
[Rossberg]
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10752
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Γ |- S says S can be deduced from Γ; Γ |= S says a good model for Γ makes S true
[Rossberg]
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10754
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In proof-theory, logical form is shown by the logical constants
[Rossberg]
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10758
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If models of a mathematical theory are all isomorphic, it is 'categorical', with essentially one model
[Rossberg]
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10756
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A model is a domain, and an interpretation assigning objects, predicates, relations etc.
[Rossberg]
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10759
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There are at least seven possible systems of semantics for second-order logic
[Rossberg]
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10757
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Henkin semantics has a second domain of predicates and relations (in upper case)
[Rossberg]
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10755
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A deductive system is only incomplete with respect to a formal semantics
[Rossberg]
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10761
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Completeness can always be achieved by cunning model-design
[Rossberg]
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