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Single Idea 13526
[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / n. Axiom of Comprehension
]
Full Idea
The comprehension axiom says that any collection of objects that can be clearly specified can be considered to be a set.
Gist of Idea
Comprehension Axiom: if a collection is clearly specified, it is a set
Source
Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.2)
Book Ref
Wolf,Robert S.: 'A Tour Through Mathematical Logic' [Carus Maths Monographs 2005], p.62
A Reaction
This is virtually tautological, since I presume that 'clearly specified' means pinning down exact which items are the members, which is what a set is (by extensionality). The naïve version is, of course, not so hot.
The
19 ideas
from 'A Tour through Mathematical Logic'
13519
|
Model theory uses sets to show that mathematical deduction fits mathematical truth
[Wolf,RS]
|
13518
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Modern mathematics has unified all of its objects within set theory
[Wolf,RS]
|
13520
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A 'tautology' must include connectives
[Wolf,RS]
|
13524
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Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof
[Wolf,RS]
|
13522
|
Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x)
[Wolf,RS]
|
13521
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Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance
[Wolf,RS]
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13523
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Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P
[Wolf,RS]
|
13525
|
Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens
[Wolf,RS]
|
13526
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Comprehension Axiom: if a collection is clearly specified, it is a set
[Wolf,RS]
|
13529
|
Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists
[Wolf,RS]
|
13530
|
An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive
[Wolf,RS]
|
13531
|
Model theory reveals the structures of mathematics
[Wolf,RS]
|
13532
|
Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants'
[Wolf,RS]
|
13533
|
First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem
[Wolf,RS]
|
13534
|
In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide
[Wolf,RS]
|
13535
|
First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation
[Wolf,RS]
|
13537
|
An 'isomorphism' is a bijection that preserves all structural components
[Wolf,RS]
|
13538
|
If a theory is complete, only a more powerful language can strengthen it
[Wolf,RS]
|
13539
|
The LST Theorem is a serious limitation of first-order logic
[Wolf,RS]
|