Anti-Specker Properties in
Constructive Reverse Mathematics
James Dent
University of Canterbury
Convergent, (Non-)Oscillatory and
Specker Sequences
Consider a sequence (xn ) in [0, 1].
Consider a sequence (xn ) in [0, 1].
v Maybe (xn ) converges.
Consider a sequence (xn ) in [0, 1].
v Maybe (xn ) converges.
v Maybe (xn ) oscillates.
We say that (xn ) is eventually bounded away from the point
x ∈ [0, 1] if there exist N ∈ N+ and δ > 0 such that |xn − x| > δ for
all n > N.
We say that (xn ) is eventually bounded away from the point
x ∈ [0, 1] if there exist N ∈ N+ and δ > 0 such that |xn − x| > δ for
all n > N.
Specker’s theorem
RUSS ` There exists an increasing sequence of rational numbers
in the Cantor set that is eventually bounded away from each point
of R.
We say that (xn ) is eventually bounded away from the point
x ∈ [0, 1] if there exist N ∈ N+ and δ > 0 such that |xn − x| > δ for
all n > N.
Specker’s theorem
RUSS ` There exists an increasing sequence of rational numbers
in the Cantor set that is eventually bounded away from each point
of R.
The Specker property
Speck[0,1] : There exists a sequence in [0, 1] that is eventually
bounded away from each point of [0, 1].
Consider a sequence (xn ) in [0, 1].
v Maybe (xn ) converges.
v Maybe (xn ) oscillates.
Consider a sequence (xn ) in [0, 1].
v Maybe (xn ) converges.
v Maybe (xn ) oscillates.
v Maybe (xn ) is a Specker sequence!
Consider a sequence (xn ) in [0, 1].
v Maybe (xn ) converges.
v Maybe (xn ) oscillates.
v Maybe (xn ) is a Specker sequence!
We can capture some of the spirit of sequential compactness,
without committing to LPO, by ruling out Specker sequences.
The non-Specker property
AS¬
[0,1] : If (xn ) is a sequence in [0, 1], then it is impossible for (xn )
to be eventually bounded away from each point of [0, 1].
The non-Specker property
AS¬
[0,1] : If (xn ) is a sequence in [0, 1], then it is impossible for (xn )
to be eventually bounded away from each point of [0, 1].
The full anti-Specker property
AS[0,1] : If (xn ) is a sequence in [0, 1] ∪ {2} that is eventually
bounded away from each point of [0, 1], then xn = 2 eventually.
The non-Specker property
AS¬
[0,1] : If (xn ) is a sequence in [0, 1], then it is impossible for (xn )
to be eventually bounded away from each point of [0, 1].
The full anti-Specker property
AS[0,1] : If (xn ) is a sequence in [0, 1] ∪ {2} that is eventually
bounded away from each point of [0, 1], then xn = 2 eventually.
The limited anti-Specker property
ASltd
[0,1] : If (xn ) is a sequence in [0, 1] ∪ {2} that is eventually
bounded away from each point of [0, 1], then xk = 2 for some k.
AS[0,1]
ASltd
[0,1]
AS¬
[0,1]
AS[0,1]
FTc
ASltd
[0,1]
AS¬
[0,1]
+MP
AS[0,1]
ASltd
[0,1]
AS¬
[0,1]
FTc
Markov’s principle
MP: For each binary sequence (λn ): if it is impossible for all the
terms of (λn ) to be equal to 0, then there exists a term equal to 1.
+MP
AS[0,1]
+MP
ASltd
[0,1]
AS¬
[0,1]
FTc
Markov’s principle
MP: For each binary sequence (λn ): if it is impossible for all the
terms of (λn ) to be equal to 0, then there exists a term equal to 1.
A seemingly weaker assertion: if (xn ) does not oscillate (in some
positive sense), then it either converges or is a Specker sequence.
A seemingly weaker assertion: if (xn ) does not oscillate (in some
positive sense), then it either converges or is a Specker sequence.
More formally, we rule out oscillation by requiring that (xn ) has at
most one limit, in the sense that for all distinct points a, b ∈ [0, 1],
either (xn ) is e.b.a. from a or (xn ) is e.b.a. from b.
A seemingly weaker assertion: if (xn ) does not oscillate (in some
positive sense), then it either converges or is a Specker sequence.
More formally, we rule out oscillation by requiring that (xn ) has at
most one limit, in the sense that for all distinct points a, b ∈ [0, 1],
either (xn ) is e.b.a. from a or (xn ) is e.b.a. from b.
The limit-stability property
LSP[0,1] : If (xn ) is a sequence in [0, 1] that has at most one limit,
then (xn ) either converges in [0, 1] or is eventually bounded away
from each point of [0, 1].
Assume LSP[0,1] , and let (xn ) in [0, 1] ∪ {2} be eventually bounded
away from each point of [0, 1].
Assume LSP[0,1] , and let (xn ) in [0, 1] ∪ {2} be eventually bounded
away from each point of [0, 1]. Define:
(yn ) = 12 x1 , 1, 12 x2 , 1, 12 x3 , 1, . . . in [0, 1].
Assume LSP[0,1] , and let (xn ) in [0, 1] ∪ {2} be eventually bounded
away from each point of [0, 1]. Define:
(yn ) = 12 x1 , 1, 12 x2 , 1, 12 x3 , 1, . . . in [0, 1].
(yn ) has at most one limit, so it either converges or is eventually
bounded away from each point of [0, 1].
Assume LSP[0,1] , and let (xn ) in [0, 1] ∪ {2} be eventually bounded
away from each point of [0, 1]. Define:
(yn ) = 12 x1 , 1, 12 x2 , 1, 12 x3 , 1, . . . in [0, 1].
(yn ) has at most one limit, so it either converges or is eventually
bounded away from each point of [0, 1]. Hence (yn ) converges, and
its limit is 1.
Assume LSP[0,1] , and let (xn ) in [0, 1] ∪ {2} be eventually bounded
away from each point of [0, 1]. Define:
(yn ) = 12 x1 , 1, 12 x2 , 1, 12 x3 , 1, . . . in [0, 1].
(yn ) has at most one limit, so it either converges or is eventually
bounded away from each point of [0, 1]. Hence (yn ) converges, and
its limit is 1. So 12 xn → 1, whence xn = 2 eventually.
Assume LSP[0,1] , and let (xn ) in [0, 1] ∪ {2} be eventually bounded
away from each point of [0, 1]. Define:
(yn ) = 12 x1 , 1, 12 x2 , 1, 12 x3 , 1, . . . in [0, 1].
(yn ) has at most one limit, so it either converges or is eventually
bounded away from each point of [0, 1]. Hence (yn ) converges, and
its limit is 1. So 12 xn → 1, whence xn = 2 eventually.
That is, BISH + LSP[0,1] ` AS[0,1] .
The limit-stability property (I)
LSP[0,1] : If (xn ) is a sequence in [0, 1] that has at most one limit,
then (xn ) either converges in [0, 1] or is eventually bounded away
from each point of [0, 1].
The limit-stability property (II)
LSP[0,1] : If (xn ) is a sequence in [0, 1] that has at most one limit, then
(xn ) converges in [0, 1].
An Application of Limited
Anti-Specker
The countable Heine-Borel property for intervals
HB0 : If (In ) is a sequence of
S inhabited, bounded open intervals
+
(an , bn ) such that [0, 1] ⊆ ∞
i=1 Ii , then there exists k ∈ N such
Sk
that [0, 1] ⊆ i=1 Ii .
The countable Heine-Borel property for intervals
HB0 : If (In ) is a sequence of
S inhabited, bounded open intervals
+
(an , bn ) such that [0, 1] ⊆ ∞
i=1 Ii , then there exists k ∈ N such
Sk
that [0, 1] ⊆ i=1 Ii .
Theorem
0
BISH + ASltd
[0,1] ` HB .
For each n and appropriate > 0, denote by I−
n () the deflated
interval (an + , bn − ).
For each n and appropriate > 0, denote by I−
n () the deflated
interval (an + , bn − ). For each n, let
n := min 2−n ∪ 18 |Ii | : i 6 n ,
For each n and appropriate > 0, denote by I−
n () the deflated
interval (an + , bn − ). For each n, let
n := min 2−n ∪ 18 |Ii | : i 6 n ,
and construct a finite n -approximation Yn ≡ {y1 , y2 , . . . , yν } to
[0, 1].
(That is: Yn is a finite, inhabited subset of [0, 1] such that for each
x ∈ [0, 1], there exists y ∈ Yn with |x − y| < n .)
Each y ∈ Yn either belongs to
S
n
−
i=1 Ii (2n ).
Sn
−
i=1 Ii (n )
or does not belong to
Each y ∈ Yn either belongs to
S
n
−
i=1 Ii (2n ).
Sn
−
i=1 Ii (n )
or does not belong to
Define (zn ) in [0, 1] ∪ {2} so that:
v if znS∈ [0, 1], then there exists y ∈ Yn such that
−
y∈
/ n
i=1 Ii (2n ), and zn = y; and
S
−
v if zn = 2, then y ∈ n
i=1 Ii (n ) for all y ∈ Yn .
Fix any x ∈ [0, 1]. There exists k such that x ∈ Ik .
Fix any x ∈ [0, 1]. There exists k such that x ∈ Ik . Since Ik is open,
we can find η > 0 such that (x − η, x + η) ⊆ Ik .
Fix any x ∈ [0, 1]. There exists k such that x ∈ Ik . Since Ik is open,
we can find η > 0 such that (x − η, x + η) ⊆ Ik .
Now choose a number N > k such that 2−N+1 < η, and let
δ := min {η − 2N , 1} > 0.
Fix any x ∈ [0, 1]. There exists k such that x ∈ Ik . Since Ik is open,
we can find η > 0 such that (x − η, x + η) ⊆ Ik .
Now choose a number N > k such that 2−N+1 < η, and let
δ := min {η − 2N , 1} > 0.
Take any n > N and suppose zn ∈ [0, 1].
Fix any x ∈ [0, 1]. There exists k such that x ∈ Ik . Since Ik is open,
we can find η > 0 such that (x − η, x + η) ⊆ Ik .
Now choose a number N > k such that 2−N+1 < η, and let
δ := min {η − 2N , 1} > 0.
Take any n > N and suppose zn ∈ [0, 1]. Then zn ∈
/
and so zn ∈
/ I−
(2
)
=:
J.
N
k
Sn
−
i=1 Ii (2n )
Fix any x ∈ [0, 1]. There exists k such that x ∈ Ik . Since Ik is open,
we can find η > 0 such that (x − η, x + η) ⊆ Ik .
Now choose a number N > k such that 2−N+1 < η, and let
δ := min {η − 2N , 1} > 0.
S
−
Take any n > N and suppose zn ∈ [0, 1]. Then zn ∈
/ n
i=1 Ii (2n )
−
and so zn ∈
/ Ik (2N ) =: J. But (x − δ, x + δ) ⊆ J, and hence
|zn − x| > δ.
, we see that there exists k such that zk = 2: that is,
Applying ASltd
Sk −[0,1]
y ∈ i=1 Ii (k ) for all y ∈ Yk .
, we see that there exists k such that zk = 2: that is,
Applying ASltd
Sk −[0,1]
y ∈ i=1 Ii (k ) for all y ∈ Yk .
For every x ∈ [0, 1], we can choose y ∈ Yk such that |x − y| < k .
Then y ∈ I−
` (k ) for some ` 6 k, and it follows that x ∈ I` .
, we see that there exists k such that zk = 2: that is,
Applying ASltd
Sk −[0,1]
y ∈ i=1 Ii (k ) for all y ∈ Yk .
For every x ∈ [0, 1], we can choose y ∈ Yk such that |x − y| < k .
Then y ∈ I−
` (k ) for some ` 6 k, and it follows that x ∈ I` .
+MP
ASX
FTc
+MP
ASltd
X
AS¬
X
+MP
ASX
FTc
+MP
ASltd
X
AS¬
X
FT∆
Thanks to:
Thanks to:
v Douglas Bridges and Maarten McKubre-Jordens
Thanks to:
v Douglas Bridges and Maarten McKubre-Jordens
v The University of Canterbury
Thanks to:
v Douglas Bridges and Maarten McKubre-Jordens
v The University of Canterbury
v The University of Niš