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Foundation of Mathematics and the
Reducts of the Set-theoretical Universe
杨睿之
Yang, Ruizhi
PKU Week of Mathematical PhilosophyJune 23, 2019
Peking University
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Set theory is widely accepted as the foundation of
mathematics
The structure of set-theoretical universe is denoted by (V, ∈)
The 1st order language of set theory is simply {∈}
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Initial segments of V
V0 = ∅,
V1 = P(V0) = {∅},
V2 = P(V1) ={∅, {∅}},
V3 = P(V2)
={∅, {∅}, {{∅}}, {∅, {∅}}},
V4 = P(V3) = . . . Image originally made by @ErinCarmody
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Initial segments of V
V0 = ∅,
V1 = P(V0) = {∅},
V2 = P(V1) ={∅, {∅}},
V3 = P(V2)
={∅, {∅}, {{∅}}, {∅, {∅}}},
V4 = P(V3) = . . . Image originally made by @ErinCarmody
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Initial segments of V
V0 = ∅,
V1 = P(V0) = {∅},
V2 = P(V1) ={∅, {∅}},
V3 = P(V2)
={∅, {∅}, {{∅}}, {∅, {∅}}},
V4 = P(V3) = . . . Image originally made by @ErinCarmody
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Initial segments of V
V0 = ∅,
V1 = P(V0) = {∅},
V2 = P(V1) ={∅, {∅}},
V3 = P(V2)
={∅, {∅}, {{∅}}, {∅, {∅}}},
V4 = P(V3) = . . . Image originally made by @ErinCarmody
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Initial segments of V
V0 = ∅,
V1 = P(V0) = {∅},
V2 = P(V1) ={∅, {∅}},
V3 = P(V2)
={∅, {∅}, {{∅}}, {∅, {∅}}},
V4 = P(V3) = . . . Image originally made by @ErinCarmody
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The backbone of V
Ordinals (finite)
0 = ∅
α + 1 = α ∪ {α}Therefore
1 = {0}
2 = {0, 1} = {∅, {∅}}
3 = {0, 1, 2}
· · ·
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The backbone of V
Ordinals (finite)
0 = ∅
α + 1 = α ∪ {α}Therefore
1 = {0}
2 = {0, 1} = {∅, {∅}}
3 = {0, 1, 2}
· · ·
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The backbone of V
Ordinals (finite)
0 = ∅
α + 1 = α ∪ {α}Therefore
1 = {0}
2 = {0, 1} = {∅, {∅}}
3 = {0, 1, 2}
· · ·
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The backbone of V
Ordinals (finite)
0 = ∅
α + 1 = α ∪ {α}Therefore
1 = {0}
2 = {0, 1} = {∅, {∅}}
3 = {0, 1, 2}
· · ·
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The backbone of V
Ordinals (finite)
0 = ∅
α + 1 = α ∪ {α}Therefore
1 = {0}
2 = {0, 1} = {∅, {∅}}
3 = {0, 1, 2}
· · ·
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The backbone of V
Ordinals (infinite)
N = ω = {0, 1, 2, . . .}
Vω =∪
n∈ω Vn
n,m ∈ N⇒ (n,m) ={{n}, {n,m}} ∈ Vω, soN × N ⊂ Vω
ω + 1, ω + n, ω + ω,ω ·
ω,ωω, ϵ0, . . .
V
N
ω
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The backbone of V
Ordinals (infinite)
N = ω = {0, 1, 2, . . .}
Vω =∪
n∈ω Vn
n,m ∈ N⇒ (n,m) ={{n}, {n,m}} ∈ Vω, soN × N ⊂ Vω
ω + 1, ω + n, ω + ω,ω ·
ω,ωω, ϵ0, . . .
VOrd
N
ω Vω = Hω
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The backbone of V
Ordinals (infinite)
N = ω = {0, 1, 2, . . .}
Vω =∪
n∈ω Vn
n,m ∈ N⇒ (n,m) ={{n}, {n,m}} ∈ Vω, soN × N ⊂ Vω
ω + 1, ω + n, ω + ω,ω ·
ω,ωω, ϵ0, . . .
VOrd
N
ω Vω = Hω
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The backbone of V
Ordinals (infinite)
N = ω = {0, 1, 2, . . .}
Vω =∪
n∈ω Vn
n,m ∈ N⇒ (n,m) ={{n}, {n,m}} ∈ Vω, soN × N ⊂ Vω
ω + 1, ω + n, ω + ω,ω ·
ω,ωω, ϵ0, . . .
VOrd
N
ω Vω = Hω
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The backbone of V
Ordinals (infinite)
N = ω = {0, 1, 2, . . .}
Vω =∪
n∈ω Vn
n,m ∈ N⇒ (n,m) ={{n}, {n,m}} ∈ Vω, soN × N ⊂ Vω
ω + 1, ω + n, ω + ω,ω ·
ω,ωω, ϵ0, . . .
VOrd
N
ω Vω = Hω
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The backbone of V
Ordinals (infinite)
N = ω = {0, 1, 2, . . .}
Vω =∪
n∈ω Vn
n,m ∈ N⇒ (n,m) ={{n}, {n,m}} ∈ Vω, soN × N ⊂ Vω
ω + 1, ω + n, ω + ω,ω ·
ω,ωω, ϵ0, . . .
VOrd
N
ω Vω = Hω
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The backbone of V
Ordinals (infinite)
N = ω = {0, 1, 2, . . .}
Vω =∪
n∈ω Vn
n,m ∈ N⇒ (n,m) ={{n}, {n,m}} ∈ Vω, soN × N ⊂ Vω
ω + 1, ω + n, ω + ω,ω ·
ω,ωω, ϵ0, . . .
VOrd
N
ω Vω = Hω
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The backbone of V
Ordinals (infinite)
N = ω = {0, 1, 2, . . .}
Vω =∪
n∈ω Vn
n,m ∈ N⇒ (n,m) ={{n}, {n,m}} ∈ Vω, soN × N ⊂ Vω
ω + 1, ω + n, ω + ω,ω ·
ω,ωω, ϵ0, . . .
VOrd
N
ω Vω = Hω
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The backbone of V
Ordinals (infinite)
N = ω = {0, 1, 2, . . .}
Vω =∪
n∈ω Vn
n,m ∈ N⇒ (n,m) ={{n}, {n,m}} ∈ Vω, soN × N ⊂ Vω
ω + 1, ω + n, ω + ω,ω ·
ω,ωω, ϵ0, . . .
VOrd
N
ω Vω = Hω
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The backbone of V
Ordinals (infinite, countable)
N = ω = {0, 1, 2, . . .}
Vω =∪
n∈ω Vn
n,m ∈ N⇒ (n,m) ={{n}, {n,m}} ∈ Vω, soN × N ⊂ Vω
ω + 1, ω + n, ω + ω,ω ·
ω,ωω, ϵ0, . . .
VOrd
N
ω Vω = Hω
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The backbone of V
Ordinals (uncountable)
P(ω) ⊂ P(Vω) = Vω+1 areuncountable
Vω+2, . . .
R = the set of Dedekindcuts
ω1 is the first uncountableordinal, then Vω1 , . . .
R
VOrd
N
ω Vω = Hω
Vω+1
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The backbone of V
Ordinals (uncountable)
P(ω) ⊂ P(Vω) = Vω+1 areuncountable
Vω+2, . . .
R = the set of Dedekindcuts
ω1 is the first uncountableordinal, then Vω1 , . . .
R
VOrd
N
ω Vω = Hω
Vω+1
Vω+2
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The backbone of V
Ordinals (uncountable)
P(ω) ⊂ P(Vω) = Vω+1 areuncountable
Vω+2, . . .
R = the set of Dedekindcuts
ω1 is the first uncountableordinal, then Vω1 , . . .
Hω1
R
VOrd
ω1
N
ω Vω = Hω
Vω+1
Vω+2
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The backbone of V
Ordinals (uncountable)
P(ω) ⊂ P(Vω) = Vω+1 areuncountable
Vω+2, . . .
R = the set of Dedekindcuts
ω1 is the first uncountableordinal, then Vω1 , . . .
Hω1
R
VOrd
ω1
N
ω Vω = Hω
Vω+1
Vω+2
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The backbone of V
Ordinals (uncountable)
P(ω) ⊂ P(Vω) = Vω+1 areuncountable
Vω+2, . . .
R = the set of Dedekindcuts
ω1 is the first uncountableordinal, then Vω1 , . . .
Hω1
R
VOrd
ω1
N
ω Vω = Hω
Vω+1
Vω+2
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Set theory as foundation of
mathematics
Every mathematical object / concept can be defined as aset / a class of sets.
Every mathematical theorem can thus be viewed as atheorem proved in set theory, and so true in the universeof sets.
These are done in a natural and elegant way.
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Set theory as foundation of
mathematics
Pros:
Expressive power / Interpretability strength
Natural
Scalability
ZF− +替 换 公 理 模 式
Replacements + AC +更 强 的 无 穷 公 理
Higher Infinities +力 迫 公 理
Forcing Axioms + · · ·
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Set theory as foundation of
mathematics
Cons:
Independency
Consistency statements
Forcing连 续 统 假 设
Continuous Hypothesis (|P(ω)| = ω1) can be forced to
be false, and then true, and then false ...
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Set theory as foundation of
mathematicsThis abundance of set-theoretic possibilities poses a seri-
ous difficulty for the universe view, ... one must explain or
explain away as imaginary all of the alternative universes
that set theorists seem to have constructed. This seems
a difficult task, for we have a robust experience in those
worlds, and they appear fully set theoretic to us.
Joel D. Hamkins
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How about a compromise?
Good modal logics are successful compromises betweenexpressive power and complexity
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How about a compromise?
DefinitionLet A and B be structures (not necessarily for the samefirst-order language). A is a
限 制
reduct of B if its domain ,relations and functions are definable in B.
Clearly, the theory of a reduct is interpretable in the theory ofthe original structure.
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How about a compromise?
反 推 数 学
Reverse mathematics on subsystems of second-orderedarithmetics: RCA0, WKL0, ACA0, ATR0, Π1
1 − CA0,…These are theories of the intended ω-models (N,S,+, ·),which are reducts of (an initial segment of) (V, ∈)内 模 型
Inner models, e.g. (L, ∈), are reducts of (V, ∈)
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How about a compromise?
反 推 数 学
Reverse mathematics on subsystems of second-orderedarithmetics: RCA0, WKL0, ACA0, ATR0, Π1
1 − CA0,…These are theories of the intended ω-models (N,S,+, ·),which are reducts of (an initial segment of) (V, ∈)内 模 型
Inner models, e.g. (L, ∈), are reducts of (V, ∈)
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How about a compromise?
反 推 数 学
Reverse mathematics on subsystems of second-orderedarithmetics: RCA0, WKL0, ACA0, ATR0, Π1
1 − CA0,…These are theories of the intended ω-models (N,S,+, ·),which are reducts of (an initial segment of) (V, ∈)内 模 型
Inner models, e.g. (L, ∈), are reducts of (V, ∈)
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How about a compromise?
反 推 数 学
Reverse mathematics on subsystems of second-orderedarithmetics: RCA0, WKL0, ACA0, ATR0, Π1
1 − CA0,…These are theories of the intended ω-models (N,S,+, ·),which are reducts of (an initial segment of) (V, ∈)内 模 型
Inner models, e.g. (L, ∈), are reducts of (V, ∈)
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How about a compromise?
反 推 数 学
Reverse mathematics on subsystems of second-orderedarithmetics: RCA0, WKL0, ACA0, ATR0, Π1
1 − CA0,…These are theories of the intended ω-models (N,S,+, ·),which are reducts of (an initial segment of) (V, ∈)内 模 型
Inner models, e.g. (L, ∈), are reducts of (V, ∈)
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How about a compromise?
A good compromise should be
Preserving enough expressive power (interpretsarithmetic, etc.)
Immune from forcing
Global
Natural
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How about a compromise?
Second-order arithmetic:
Can be restricted to be immune from forcing
limited but still enough expressive power
Local, a very initial segment of V
ZF + V = L (theory of (L, ∈))
Immune from forcing
Strong expressive power, not compatible with large
cardinals
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How about a compromise?
Woodin’s program: characterise and justify
ZF + V = Ultimate-L + higher infinities
How about reducts of the form (V , Pi, F j)i∈I, j∈J
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How about a compromise?
Woodin’s program: characterise and justify
ZF + V = Ultimate-L + higher infinities
How about reducts of the form (V , Pi, F j)i∈I, j∈J
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How about a compromise?
Woodin’s program: characterise and justify
ZF + V = Ultimate-L + higher infinities
How about reducts of the form (V , Pi, F j)i∈I, j∈J
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Set-theoretic mereology
In contrast to set theory based on the membership relation ∈,集 合 论 的 部 分 论
set-theoretic mereology is the theory of the parthood relation⊂ between sets.
Venn diagram
∅, X ∪ Y , X ∩ Y , X \ Y , X △ Y , |X| = n (for every naturalnumber n), . . .
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Question (Hamkins and Kikuchi 2016)
Is parthood relation ∈-complete in the sense that ∈ isdefinable in its reduct (V,⊂)?
Can set-theoretic mereology serve as a foundation ofmathematics?
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Set-theoretic mereology
Observation (Hamkins and Kikuchi 2016)(V,⊂) is not ∈-complete
Proof sketch: Let θ : V → V be a nontrivial permutation.Define
τ(a) ={θ(b)
∣∣∣ b ∈ a}
Then τ : V → V is a non-trivial automorphism on (V,⊂)
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Set-theoretic mereology
Theorem (Hamkins and Kikuchi 2016)Set-theoretic mereology, namely the theory of (V,⊂) isprecisely the theory of an atomic unbounded relativelycomplemented distributive lattice. This finitely axiomatizabletheory is complete and decidable.
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Set-theoretic mereology
Key factThe above theory admits quantifier elimination based on thelanguage {⊂, ∅, x ∩ y, x ∪ y, x \ y, |x| = n, |x| ≥ n}n∈N.
Proof sketch:
A单 格 词
cell term c(X) is of the form ±X ∩ ±A ∩ ±B ∩ ±C
General cases can be reduced to the simple case
∃X(∧
i∈I|ci(X)| = ni ∧
∧j∈J
|c j(X)| ≥ n j)
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Set-theoretic mereology
Mathematical thinking is, and must remain, essen-tially creative.
Emil L. Post 1944
Gödel’s ThesisA decidable complete theory can not serve as a foundation ofmathematics.
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Expanding the subset relation
Observation (Hamkins and Kikuchi 2016)(V,⊂, {x}) is ∈-complete
a ∈ b ⇔ {a} ⊂ b
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Expanding the subset relation
Definition
a ⊂∗ b iff a \ b is finite
|a| = ∞ iff there is a surjection from a proper subset of a
to a itself
ObservationThe theory of (V,⊂ ,⊂∗) is mutually interpretable with thetheory of (V,⊂ , |x| = ∞). The latter has a complete decidabletheory.
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Expanding the subset relation
Define a - b if the size of a is less or equal to the size of b
Note: The infinity predicate |x| = ∞ and the equinumerousrelation are definable in (V,⊂,-). Yet,
Theorem (Y. 2019)(V,⊂,-) has a complete decidable theory.
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Expanding the subset relation
We add the the unary union ∪ or the unary intersection ∩instead.
ObservationBoth (V,⊂,∪) and (V,⊂,∩) are ∈-complete
y = {x} ⇔ ∪y = x ∧ |y| = 1
⇔ ∩y = x ∧ |y| = 1
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Expanding the subset relation
Observation(V,∩,∩), (V,∪,∪), and (V,
∩,∪
) are all ∈-complete
x ⊂ y⇔ ∃z (∩
z = x ∧∪z = y)
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Expanding the subset relation
The power set operation P is “coarser” than the unary union
Observation(V,⊂, P) is ∈-comlete
We can define y = {x} as follow: Define z to be the ⊂-leastsuch that
∀w((w ⊂ x ∧ w , x)→ P(w) ⊂ z
)Then z = P(x) \ {x}, so y = P(x) \ z
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QuestionWhat about the unary intersection ∩, the unary union ∪, orthe power set operation P on themselves?
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The unary intersection structure
The reduct (V,∩
) is just a proper-class-branching “tree” ofheight ℵ0 if (V, ∈) is well-founded
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The unary union structure
Theorem (Hamkins and Y. 2017)There is a computable complete axiomatization of the theoryof (V,
∪).
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The unary union structure
There are exactly two covers of ∅:
∪ ∅ = ∅∪{∅} = ∅Note that ∅ is the only finite set x such that ∪
x = x.
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The unary union structure
There are exactly two covers of {∅}:
∪{{∅}} = {∅}∪{∅, {∅}} = {∅}Note that it is the case for any singleton {a}.
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The unary union structure
There are exactly two covers of {a}:
∪{{a}} = {a}∪{∅, {a}} = {a}Note that it is the case for any singleton {a}.
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The unary union structure
How many covers are there for a set of size two {a, b}?{{a, b}}; {∅, {a, b}}; {{a}, {a, b}}; {{b}, {a, b}}; {{a}, {b}};{∅, {a}, {a, b}}; {∅, {b}, {a, b}}, {∅, {a}, {b}}; {{a}, {b}, {a, b}};{∅, {a}, {b}, {a, b}}.
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The unary union structure
How many covers are there for a set of size two {a, b}?{{a, b}}; {∅, {a, b}}; {{a}, {a, b}}; {{b}, {a, b}}; {{a}, {b}};{∅, {a}, {a, b}}; {∅, {b}, {a, b}}, {∅, {a}, {b}}; {{a}, {b}, {a, b}};{∅, {a}, {b}, {a, b}}.
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The unary union structure
How many covers are there for a set of size three {a, b, c}?There are 218 covers of a set containing three elements!
The number of covers of 4 is 64594.
The number of covers of 5 is 4294642034.......
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The unary union structure
How many covers are there for a set of size three {a, b, c}?There are 218 covers of a set containing three elements!
The number of covers of 4 is 64594.
The number of covers of 5 is 4294642034.......
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The unary union structure
How many covers are there for a set of size three {a, b, c}?There are 218 covers of a set containing three elements!
The number of covers of 4 is 64594.
The number of covers of 5 is 4294642034.......
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The unary union structure
FactThere is a recursive function C, given a finite setA = {a1, . . . an}, there are exactly C(n) many covers of A.
C(n) = 22n − ( n−1∑i=0
(ni
)·C(i)
).
C(n) =n∑
i=0
(−1)n−i ·(ni
)· 22i
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The unary union structure
FactThere is a recursive binary function c such that, given a finiteset A of size n, there are c(n,m) many covers of A of size m.
c(n,m) =(2n
m
)− ( n−1∑
i=0
(ni
)· c(i,m)
).
c(n,m) =n∑
i=0
(−1)n−i ·(ni
)·(2i
m
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The unary union structure
The infinite caseLet κ be infinite, λ ≥ 2. There are C(κ) = 22κ many covers ofκ, among them, there are c(κ, λ) = [2κ]λ many covers ofcardinality λ.
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The unary union structure
Some definable sets in (V,∪
):
x = ∅ ⇔ ∪x = x ∧ ∃=2y
∪y = x
x = {∅} ⇔ ∪x = ∅ ∧ x , ∅
x = {{∅}} ⇔ ∪x = {∅} ∧ ∃=2y
∪y = x
x = {∅, {∅}} ⇔ ∪x = {∅} ∧ ∃=10y
∪y = x
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The unary union structure
ObservationNot every hereditarily finite set in Vω is definable.
Recall that there are four covers of {∅, {∅}}, namely {∅, {∅, {∅}}},{{∅}, {∅, {∅}}}, {{{∅}}, {∅, {∅}}}, and {{∅}, {{∅}}}. All of them haveisomorphic upward structures and the same downwardstructure. By swapping the upward substructures, we find anautomorphism.Thus(V,
∪) is not ∈-complete
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The unary union structure
ObservationNot every hereditarily finite set in Vω is definable.
Recall that there are four covers of {∅, {∅}}, namely {∅, {∅, {∅}}},{{∅}, {∅, {∅}}}, {{{∅}}, {∅, {∅}}}, and {{∅}, {{∅}}}. All of them haveisomorphic upward structures and the same downwardstructure. By swapping the upward substructures, we find anautomorphism.Thus(V,
∪) is not ∈-complete
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The unary union structure
ObservationNot every hereditarily finite set in Vω is definable.
Recall that there are four covers of {∅, {∅}}, namely {∅, {∅, {∅}}},{{∅}, {∅, {∅}}}, {{{∅}}, {∅, {∅}}}, and {{∅}, {{∅}}}. All of them haveisomorphic upward structures and the same downwardstructure. By swapping the upward substructures, we find anautomorphism.Thus(V,
∪) is not ∈-complete
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The unary union structure
Observationrank(
∪x) ≤ rank x for all x. In particular,
if rank x = α + 1, then rank(∪
x) = α;
else if rank x = α is limit, then rank(∪
x) = α.
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The unary union structure
ObservationFor each 1 ≤ k < ω there are infinitely many sets x ∈ Vω+1
such that x have exactly k many ∪-successors in Vω+1 (itfollows ∪k x = x) constituting a k-loop.
Example{∅, {∅}, {{∅}}, . . . }{∅, {∅}, {{∅}, ∅}, {{{∅}, ∅}}, . . . }{∅, {{∅}}, {{{{∅}}, ∅}}, . . . }
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Axioms of union
Definition
D0 |x| = 0↔ ∪x = x ∧ ∃=2y
∪y = x
Dn |x| = n↔ ∪x , x ∧ ∃=C(n)y
∪y = x (for n ≥ 1)
E1 |x| ≥ 0↔ x = x
En |x| ≥ n↔ ∃≥C(n)y∪
y = x ∧ y , x (for n ≥ 1)
Note: |x| = 0 ∨ . . . ∨ |x| = n − 1 ∨ |x| ≥ n is not valid
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Axioms of union
U1∃=1x |x| = 0
U2 For n, k ∈ ω,
∃≥nx (x =∪k x ∧
∧1≤l<k
x ,∪l x)
U3 For n ≥ 1,∀x (
n−1∨i=0
|x| = i ∨ |x| ≥ n)
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Axioms of union
U4 For n, k ≥ 1,
∀x (|x| = n→ x ,∪k x)
U5 For n,m ∈ ω,
∀x(|x| ≥ n→ ∃≥c(n,m)y (
∪y = x ∧ |y| = m)
)U6
∀x∃=1y (∪
y = x ∧ |y| = 1)
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Quantifier elimination
Let L∗U = {∪, ∅, |x| = n, |x| ≥ n}. Let Λ∗U be constituted by U1
- U6, Dn s, En s, and |∅| = 0
LemmaFor every L∗U-formula θ, there is a quantifier-free L∗U-formulaψ such that
Λ∗U ⊢ θ ↔ ψ
CorollaryΛU = U1 - U6 is complete
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Quantifier eliminationExampleφ is ∪
x = t ∧ x , s0 ∧ . . . ∧ x , sm ∧ |x| = n,
where t, s0, . . . , sm are terms do not contain x
We call si a competitor if ∪si = t =
∪x, a competitor is a
threat if |si| = n. Then, by (U3), (U5) and (U6) ∃xφ isequivalent to
k′∨k=⌈log n⌉
∣∣∣{si
∣∣∣ ∪ si = t ∧ |si| = n}∣∣∣ < c(k, n) ∧ |t| ≥ k.
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Quantifier eliminationExampleφ is ∪
x = t ∧ x , s0 ∧ . . . ∧ x , sm ∧ |x| = n,
where t, s0, . . . , sm are terms do not contain x
We call si a competitor if ∪si = t =
∪x, a competitor is a
threat if |si| = n. Then, by (U3), (U5) and (U6) ∃xφ isequivalent to
k′∨k=⌈log n⌉
∣∣∣{si
∣∣∣ ∪ si = t ∧ |si| = n}∣∣∣ < c(k, n) ∧ |t| ≥ k.
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More reducts
Theorem(V,
∩) has a decidable theory:
I1 ∃=1x∩
x = x
I2 ∀x (∩k x = x→ ∩
x = x) (for k > 1)
I3 ∀x∃≥ny∩
y = x (for n < ω)
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More reducts
Theorem(V,
∪, P) has a decidable theory: U1 - U6 together with
P1 ∀x (|Px| � n→ ∨m≤⌊log n⌋ |Px| = 2m) (for n < ω)
P2 ∀x (|x| = n↔ |Px| = 2n) (for n < ω)
P3 ∀x∪
Px = x;
P4 ∀x (Plx ,∪k x) (for l > 1 and k < ω)
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Dichotomy among reducts of (V, ∈)
∈-complete:(V,⊂, {x}), (V,⊂,∪), (V,⊂,∩), (V,
∪,∩
), (V,⊂, P) etc.
completely axiomatizable:(V,⊂), (V,⊂,⊂∗), (V,⊂,-), (V,
∪), (V,
∩), (V,
∪, P), etc.
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Dichotomy among reducts of (V, ∈)
∈-complete:(V,⊂, {x}), (V,⊂,∪), (V,⊂,∩), (V,
∪,∩
), (V,⊂, P) etc.
completely axiomatizable:(V,⊂), (V,⊂,⊂∗), (V,⊂,-), (V,
∪), (V,
∩), (V,
∪, P), etc.
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Further Observation
DefinitionA stratification of a set theory formula φ is a function σon the variables occurring in φ to the natural numberssuch that
if x = y occurs in φ, then σ(x) = σ(y)
if x ∈ y occurs in φ, then σ(x) + 1 = σ(y)
A formula φ is stratifiable if there is a stratification suchthat all free variables in φ have the same σ value.
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Further Observation
Observation
X ⊂ Y, X ∪ Y, X ∩ Y, X \ Y, X △ Y, |X| = ∞, X ∼ Y canbe defined by a stratifiable formula
while, ∪X, ∩
X, P(X) are not stratifiable.
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Further Observation
Observation (McKenzie Zachiri)If relations and functions in reduct (V,R1, . . . ,Rn, f1, . . . , fm)
are all stratifiable, then it is not ∈-complete
QuestionMust every “natural”, “global” and stratifiable reduct(V,R1, . . . ,Rn, f1, . . . , fm) admits a complete axiomatization?
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Thank you!
