A Nonstandard Counterpart of DNR Kojiro Higuchi
joint work with
Keita Yokoyama,
Tokyo Institute of Technology
Diagonally Non-Recursive function Kleene’s recursion theorem (1938) For any recursive function f, we can find e such that {e}={f(e)}. Arslanov’s completeness criterion (1981) For any r.e. set A, A is complete ⇐⇒ (∃g≤T A)(∀e)[{e}6={g(e)}]. In fact,
⇐⇒ (∃h≤T A)(∀e)[{e}(e)6=h(e)].
Definition f is d.n.r. ⇐⇒ (∀e)[{e}(e)6=f(e)]. Definition f is d.n.r.(A) ⇐⇒ (∀e)[{e}A (e)6=f(e)]. On this concept, I was asked by Yokoyama-san...
Question (Yokoyama, March 2010) Given any infinite recursive binary tree T and dnr function f, (∃f0 ≤T f)(∃A∈[T])[f0 is dnr(A)] ?? →Yes In fact, (∃f0 ≤T f)(∃inf f-rec tree T0 ⊂T)(∀A∈[T0 ])[f0 is dnr(A)]. Let us prove this!
Theorem Given any inf. rec. binary tree T and f:dnr, (∃f0 ≤T f)(∃inf f-rec tree T0 ⊂T)(∀A∈[T0 ])[f0 is dnr(A)]. Proof. For x∈ N
Theorem Given any inf. rec. binary tree T and f:dnr, (∃f0 ≤T f)(∃inf f-rec tree T0 ⊂T)(∀A∈[T0 ])[f0 is dnr(A)]. Proof. For x∈ N
Theorem Given any inf. rec. binary tree T and f:dnr, (∃f0 ≤T f)(∃inf f-rec tree T0 ⊂T)(∀A∈[T0 ])[f0 is dnr(A)]. The above theorem can be used to show some theorem concerning on reverse mathematics of second order arithmetic.
Reverse mathematics of second order arithmetic Reverse mathematics is a study to classify “natural” mathematical statements according to equivalence over a base system. Many mathematical concepts and theorems can be formalized in second order arithmetic. We can also formalize “f is dnr(A)” in second order arithmetic.
The strength of DNR Theorems non-provable in RCA0 are quite often equivalent to one of the statements WWKL, WKL, ACA, ATR and Π1 1 -CA over RCA0 . But the statement DNR is neither provable in RCA0 nor equivalent to one of these statements. Here, DNR = (∀A)(∃f)(∀y)[f(y)6={y}A (y)]. It is known that DNR
Although many mathematical concepts and statements can be formalized in second order arithmetic, sometimes they will become too complicated.
Nonstandard arithmetic and analysis In nonstandard arithmetic and analysis, many concepts and theorems can be formalized by a simple notation or formula and can be proved in a simple way. Example f is continuous ⇐⇒ (∀x∗ ,y∗ )[x∗ ≈y∗ ⇒ f∗ (x∗ )≈f∗ (y∗ )].
Example of nonstandard argument Theorem (∀{xn }n∈N ⊂[0,1])[{xn }n has an accumulating point]. Sketch of Proof. Let n∗ > N and a∈ R such that a≈x∗ n∗ . By Transfer Principle, (∀k∈ N)(∃m>k)[|xm -a| <1/k] ∗ -a| <1/k ]. since (∀k∗ ∈ N)(∃m∗ >k∗ )[|x∗ m∗
How about doing reverse mathematics with nonstandard argument??
Review of RCA0 L :={0,1,=,+,·,<,∈} There are two sorts for natural numbers and for sets of natural number. • x,y,· · · for natural numbers, • X,Y,· · · for sets of natural numbers. The axioms of RCA0 consists of 0 BASIC axioms + IΣ0 + ∆ 1 1 -CA.
Lns L :={0,1,=,+,·,<,∈} L∗ :={0∗ ,1∗ ,=∗ ,+∗ ,·∗ ,<∗ ,∈∗ } √ ns ∗ L :=L ∪ L ∪{ }
• x,y,· · · for standard natural numbers, • X,Y,· · · for sets of standard natural numbers. • x∗ ,y∗ ,· · · for nonstandard natural numbers, • X∗ ,Y∗ ,· · · for sets of nonstandard natural numbers.
Structures for L
ns
∗
=L ∪ L ∪{
√
}
N =(N,SN ,0N ,1N ,+N ,·N ,
Terms and Formulas for L
ns
√ =L ∪ L ∪{ } ∗
I hope you can imagine what are Lns -terms and Lns -formulas. Example. 1. “x∗ is a bigger than any standard natural number (x∗ > N)” √ ≡ (∀y)[ (y)<∗ x∗ ]. √ 2. “If A is unbounded, then (A) has an infinitely large number” √ ∞ ∗ ∗ ≡ (∀A)[(∃ x)[x∈A]⇒(∃y > N)[y ∈ (A)]]. 3. “N ⊂e N ∗ ” ∗
≡ (∀x)(∀y <
∗
√ ∗ ∗√ (x))(∃z)[y = (z)].
Notations Given an L-formula F, F∗ : an Lns -formula obtained by adding * to all L-symbols in F.
RCAns 0 =RCA0 + •
√
: N ,→ N ∗ and N ⊂e N ∗ .
• N ≡Σ1 N ∗ : for any sentence F in Σ1 1 (L) 1
F ⇐⇒ F∗ • N ≺Σ0 N ∗ : for any F(x,X) in Σ0 0 (L) 0 √ ∗ √ (∀x)(∀X)[F(x,X) ⇐⇒ F ( (x), (X))] 0 • Σ0 overspill: for any F(x,y) in Σ 1 1 (L)
√ (∀x)(∃y>x)[F(x,y)]⇒(∃y > N)(∀x)[F ( (x),y∗ )] ∗
∗
• Finite standard part principle: ∗
∗
(∀X )[card(X )∈ N ⇒(∃Y)(∀x)[x∈Y ⇐⇒
√
(x)∈X∗ ]]
WKLns 0 =WKL0 + •
√
: N ,→ N ∗ and N ⊂e N ∗ .
• N ≡Σ1 N ∗ : for any sentence F in Σ1 1 (L) 1
F ⇐⇒ F∗ • N ≺Σ0 N ∗ : for any F(x,X) in Σ0 0 (L) 0 √ ∗ √ (∀x)(∀X)[F(x,X) ⇐⇒ F ( (x), (X))] 0 • Σ0 overspill: for any F(x,y) in Σ 1 1 (L)
√ (∀x)(∃y>x)[F(x,y)]⇒(∃y > N)(∀x)[F ( (x),y∗ )] ∗
• Standard part principle: ∗
(∀X )(∃Y)(∀x)[x∈Y ⇐⇒
∗
√
(x)∈X∗ ]
RCA and WKL It is known that • RCAns 0 is a conservative extension of RCA0 . • WKLns 0 is a conservative extension of WKL0 .
WKL0 ⊂con WKLns 0 It is clear that every theorem in WKL0 is a theorem in WKLns 0 . To see the converse, note that it suffices to prove: ∼ (∀cntb nonst M |=WKL0 )(∃N |=WKLns )[N L = M]. 0
∼ (∀cntb nonst M |=WKL0 )(∃N |=WKLns 0 )[N L = M] Theorem (Tanaka, 1997) (∀cntb nonst M |=WKL0 )(∃I (e M)[I ∼ = M and SI =Code(I/M)]. Given cntb nonst M |=WKL0 , take I (e M as in the above theorem. Consider N =(I,M,Id(-like)).
∼ (∀cntb nonst M |=WKL0 )(∃N |=WKLns 0 )[N L = M] Theorem (Tanaka, 1997) (∀cntb nonst M |=WKL0 )(∃I (e M)[I ∼ = M and SI =Code(I/M)]. Given cntb nonst M |=WKL0 , take I (e M as in the above theorem. Consider N =(I,M,Id(-like)). N satisfies: • I ≡ 1 M since I ∼ = M; Σ1
• I ≺Σ0 M since I ⊂e M; 0
0 • Σ0 overspill since I ( M | =IΣ e 1 1; √ ∗ • (∀X )(∃Y)(∀x)[x∈Y⇔ (x)∈X∗ ] since SI =Code(I/M).
DNR and DNRns DNR0 = RCA0 + (∀A)(∃f)(∀y)[f(y)6={y}(A;y)]. ns ∗ ∗ DNRns = RCA + (∀x )(∃f)(∀y)[f(y)6 = {y}(Code(x );y)]. 0 0
Main Theorem DNR0 ⊂con DNRns 0 .
We see an idea of proof and end this talk.
DNR0 ⊂con DNRns 0 . • DNR0 = RCA0 + (∀A)(∃f)(∀y)[f(y)6={y}(A;y)]. ns ∗ ∗ • DNRns = RCA + (∀x )(∃f)(∀y)[f(y)6 = {y}(Code(x );y)]. 0 0
It is easy to see that every theorem in DNR0 is a theorem in DNRns 0 . To show the converse, note that it suffices to prove: ∼ (∀cntb nonst M |=DNR0 )(∃N |=DNRns )[N L = M]. 0
∼ (∀cntb nonst M |=DNR0 )(∃N |=DNRns 0 )[N L = M]. • DNR0 = RCA0 + (∀A)(∃f)(∀y)[f(y)6={y}(A;y)]. ns ∗ ∗ • DNRns = RCA + (∀x )(∃f)(∀y)[f(y)6 = {y}(Code(x );y)]. 0 0
∼ (∀cntb nonst M |=DNR0 )(∃N |=DNRns 0 )[N L = M]. • DNR0 = RCA0 + (∀A)(∃f)(∀y)[f(y)6={y}(A;y)]. ns ∗ ∗ • DNRns = RCA + (∀x )(∃f)(∀y)[f(y)6 = {y}(Code(x );y)]. 0 0
Definition M0 ⊂d M1 iff M0 ⊂ω M1 and (∀A∈SM1 )(∃f∈SM0 )[M1 |=f is dnr(A)].
∼ (∀cntb nonst M |=DNR0 )(∃N |=DNRns 0 )[N L = M]. • DNR0 = RCA0 + (∀A)(∃f)(∀y)[f(y)6={y}(A;y)]. ns ∗ ∗ • DNRns = RCA + (∀x )(∃f)(∀y)[f(y)6 = {y}(Code(x );y)]. 0 0
Definition M0 ⊂d M1 iff M0 ⊂ω M1 and (∀A∈SM1 )(∃f∈SM0 )[M1 |=f is dnr(A)]. Recall the following theorem holds, Given any inf. rec. binary tree T and f:dnr, (∃f0 ≤T f)(∃inf f-rec tree T0 ⊂T)(∀A∈[T0 ])[f0 is dnr(A)]. By the same argument of proof, we can modify the above theorem to...
∼ (∀cntb nonst M |=DNR0 )(∃N |=DNRns 0 )[N L = M]. • DNR0 = RCA0 + (∀A)(∃f)(∀y)[f(y)6={y}(A;y)]. ns ∗ ∗ • DNRns = RCA + (∀x )(∃f)(∀y)[f(y)6 = {y}(Code(x );y)]. 0 0
Definition M0 ⊂d M1 iff M0 ⊂ω M1 and (∀A∈SM1 )(∃f∈SM0 )[M1 |=f is dnr(A)]. Given any inf. binary tree T∈SM1 and A∈SM1 , (∃inf tree T0 ⊂T in SM1 )(∀B∈[T0 ])(∃f∈SM1 )[M1 |=f is dnr(A⊕B)].
∼ (∀cntb nonst M |=DNR0 )(∃N |=DNRns 0 )[N L = M]. • DNR0 = RCA0 + (∀A)(∃f)(∀y)[f(y)6={y}(A;y)]. ns ∗ ∗ • DNRns = RCA + (∀x )(∃f)(∀y)[f(y)6 = {y}(Code(x );y)]. 0 0
Definition M0 ⊂d M1 iff M0 ⊂ω M1 and (∀A∈SM1 )(∃f∈SM0 )[M1 |=f is dnr(A)]. Given any inf. binary tree T∈SM1 and A∈SM1 , (∃inf tree T0 ⊂T in SM1 )(∀B∈[T0 ])(∃f∈SM1 )[M1 |=f is dnr(A⊕B)]. Theorem (Harrington, 1977) (∀cntb M0 |=RCA0 )(∃M1 ⊃ω M0 )[M1 |=WKL0 ]. Combining the technique of these theorems, we have...
∼ (∀cntb nonst M |=DNR0 )(∃N |=DNRns 0 )[N L = M]. • DNR0 = RCA0 + (∀A)(∃f)(∀y)[f(y)6={y}(A;y)]. ns ∗ ∗ • DNRns = RCA + (∀x )(∃f)(∀y)[f(y)6 = {y}(Code(x );y)]. 0 0
Definition M0 ⊂d M1 iff M0 ⊂ω M1 and (∀A∈SM1 )(∃f∈SM0 )[M1 |=f is dnr(A)]. Theorem (∀cntb M |=DNR0 )(∃M0 ⊃d M)[M0 |=WKL0 ].
∼ (∀cntb nonst M |=DNR0 )(∃N |=DNRns 0 )[N L = M]. • DNR0 = RCA0 + (∀A)(∃f)(∀y)[f(y)6={y}(A;y)]. ns ∗ ∗ • DNRns = RCA + (∀x )(∃f)(∀y)[f(y)6 = {y}(Code(x );y)]. 0 0
Definition M0 ⊂d M1 iff M0 ⊂ω M1 and (∀A∈SM1 )(∃f∈SM0 )[M1 |=f is dnr(A)]. Theorem (∀cntb M |=DNR0 )(∃M0 ⊃d M)[M0 |=WKL0 ]. From this theorem, we can conclude our claim by the same argument for WKL0 ⊂con WKLns 0 .
fin.