J u m p t o c o n t e n t
M a i n m e n u
M a i n m e n u
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( T o p )
1
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2
S i l v e r m a c h i n e
3
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A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Type of mathematical object
In set theory , Silver machines are devices used for bypassing the use of fine structure in proofs of statements holding in L . They were invented by set theorist Jack Silver as a means of proving global square holds in the constructible universe .
Preliminaries
[ edit ]
An ordinal
α
{\displaystyle \alpha }
is *definable from a class of ordinals X if and only if there is a formula
ϕ
(
μ
0
,
μ
1
,
…
,
μ
n
)
{\displaystyle \phi (\mu _{0},\mu _{1},\ldots ,\mu _{n})}
and ordinals
β
1
,
…
,
β
n
,
γ
∈
X
{\displaystyle \beta _{1},\ldots ,\beta _{n},\gamma \in X}
such that
α
{\displaystyle \alpha }
is the unique ordinal for which
⊨
L
γ
ϕ
(
α
∘
,
β
1
∘
,
…
,
β
n
∘
)
{\displaystyle \models _{L_{\gamma }}\phi (\alpha ^{\circ },\beta _{1}^{\circ },\ldots ,\beta _{n}^{\circ })}
where for all
α
{\displaystyle \alpha }
we define
α
∘
{\displaystyle \alpha ^{\circ }}
to be the name for
α
{\displaystyle \alpha }
within
L
γ
{\displaystyle L_{\gamma }}
.
A structure
⟨
X
,
<
,
(
h
i
)
i
<
ω
⟩
{\displaystyle \langle X,<,(h_{i})_{i<\omega }\rangle }
is eligible if and only if:
X
⊆
O
n
{\displaystyle X\subseteq On}
.
< is the ordering on On restricted to X.
∀
i
,
h
i
{\displaystyle \forall i,h_{i}}
is a partial function from
X
k
(
i
)
{\displaystyle X^{k(i )}}
to X, for some integer k(i ).
If
N
=
⟨
X
,
<
,
(
h
i
)
i
<
ω
⟩
{\displaystyle N=\langle X,<,(h_{i})_{i<\omega }\rangle }
is an eligible structure then
N
λ
{\displaystyle N_{\lambda }}
is defined to be as before but with all occurrences of X replaced with
X
∩
λ
{\displaystyle X\cap \lambda }
.
Let
N
1
,
N
2
{\displaystyle N^{1},N^{2}}
be two eligible structures which have the same function k. Then we say
N
1
◃
N
2
{\displaystyle N^{1}\triangleleft N^{2}}
if
∀
i
∈
ω
{\displaystyle \forall i\in \omega }
and
∀
x
1
,
…
,
x
k
(
i
)
∈
X
1
{\displaystyle \forall x_{1},\ldots ,x_{k(i )}\in X^{1}}
we have:
h
i
1
(
x
1
,
…
,
x
k
(
i
)
)
≅
h
i
2
(
x
1
,
…
,
x
k
(
i
)
)
{\displaystyle h_{i}^{1}(x_{1},\ldots ,x_{k(i )})\cong h_{i}^{2}(x_{1},\ldots ,x_{k(i )})}
Silver machine
[ edit ]
A Silver machine is an eligible structure of the form
M
=
⟨
O
n
,
<
,
(
h
i
)
i
<
ω
⟩
{\displaystyle M=\langle On,<,(h_{i})_{i<\omega }\rangle }
which satisfies the following conditions:
Condensation principle. If
N
◃
M
λ
{\displaystyle N\triangleleft M_{\lambda }}
then there is an
α
{\displaystyle \alpha }
such that
N
≅
M
α
{\displaystyle N\cong M_{\alpha }}
.
Finiteness principle. For each
λ
{\displaystyle \lambda }
there is a finite set
H
⊆
λ
{\displaystyle H\subseteq \lambda }
such that for any set
A
⊆
λ
+
1
{\displaystyle A\subseteq \lambda +1}
we have
M
λ
+
1
[
A
]
⊆
M
λ
[
(
A
∩
λ
)
∪
H
]
∪
{
λ
}
{\displaystyle M_{\lambda +1}[A ]\subseteq M_{\lambda }[(A\cap \lambda )\cup H]\cup \{\lambda \}}
Skolem property. If
α
{\displaystyle \alpha }
is *definable from the set
X
⊆
O
n
{\displaystyle X\subseteq On}
, then
α
∈
M
[
X
]
{\displaystyle \alpha \in M[X ]}
; moreover there is an ordinal
λ
<
[
s
u
p
(
X
)
∪
α
]
+
{\displaystyle \lambda <[sup(X )\cup \alpha ]^{+}}
, uniformly
Σ
1
{\displaystyle \Sigma _{1}}
definable from
X
∪
{
α
}
{\displaystyle X\cup \{\alpha \}}
, such that
α
∈
M
λ
[
X
]
{\displaystyle \alpha \in M_{\lambda }[X ]}
.
References
[ edit ]
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Silver_machine&oldid=1222012697 "
C a t e g o r y :
● C o n s t r u c t i b l e u n i v e r s e
H i d d e n c a t e g o r i e s :
● A r t i c l e s w i t h s h o r t d e s c r i p t i o n
● S h o r t d e s c r i p t i o n m a t c h e s W i k i d a t a
● A r t i c l e s w i t h t o p i c s o f u n c l e a r n o t a b i l i t y f r o m J u n e 2 0 2 0
● A l l a r t i c l e s w i t h t o p i c s o f u n c l e a r n o t a b i l i t y
● W i k i p e d i a a r t i c l e s t h a t a r e t o o t e c h n i c a l f r o m J u n e 2 0 2 0
● A l l a r t i c l e s t h a t a r e t o o t e c h n i c a l
● A r t i c l e s w i t h m u l t i p l e m a i n t e n a n c e i s s u e s
● T h i s p a g e w a s l a s t e d i t e d o n 3 M a y 2 0 2 4 , a t 1 0 : 1 7 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
● P r i v a c y p o l i c y
● A b o u t W i k i p e d i a
● D i s c l a i m e r s
● C o n t a c t W i k i p e d i a
● C o d e o f C o n d u c t
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● S t a t i s t i c s
● C o o k i e s t a t e m e n t
● M o b i l e v i e w