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Single Idea 13529
[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / e. Axiom of the Empty Set IV
]
Full Idea
Empty Set Axiom: ∃x ∀y ¬ (y ∈ x). There is a set x which has no members (no y's). The empty set exists. There is a set with no members, and by extensionality this set is unique.
Gist of Idea
Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists
Source
Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.3)
Book Ref
Wolf,Robert S.: 'A Tour Through Mathematical Logic' [Carus Maths Monographs 2005], p.70
A Reaction
A bit bewildering for novices. It says there is a box with nothing in it, or a pair of curly brackets with nothing between them. It seems to be the key idea in set theory, because it asserts the idea of a set over and above any possible members.
The
19 ideas
from 'A Tour through Mathematical Logic'
13519
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Model theory uses sets to show that mathematical deduction fits mathematical truth
[Wolf,RS]
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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
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Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x)
[Wolf,RS]
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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]
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13525
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Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens
[Wolf,RS]
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13526
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Comprehension Axiom: if a collection is clearly specified, it is a set
[Wolf,RS]
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13529
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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
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Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants'
[Wolf,RS]
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13533
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First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem
[Wolf,RS]
|
13534
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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
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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]
|