模型論與現代微積分(修改稿)
模型論與現代微積分(修改稿)
從模型論的視野裡,我們如何看待現代微積分?
2018年12月6日,我們在“(ε,δ)條件與無窮小方法之比研究”博文小中正式闡明瞭相關學術立場。
2008年,Keisler教授,作為塔爾斯基模型論的傳人,發表研究論文,題為“Quantifiers in Limits”(極限中的量詞)站在模型論的視角深入闡述了現代微積分的一些弊端。
我們的觀點就是從這篇文章中“知識共享”出來的。菲氏極限論的信徒們不知作何感想?
袁萌 陳啟清 12月7日
附件:極限理論中的量詞
Quantifiers in Limits
H. Jerome Keisler University of Wisconsin Madison, WI, USA
Quantifiers in limits
1. Robinson’s limit definition
2. Quantifier hierarchies
3. Cases with low quantifier level
4. Cases with maximum quantifier level
5. Infinitely long sentences
6. o-minimal structures
7. Summary
1
Robinson’s limit definition
The nonstandard approach to calculus eliminates two quantifier blocks in the limit definition. The standard definition of
lim z→∞
F(z) =∞ requires three quantifier blocks, ∀x∃y∀z[y ≤z ⇒x≤F(z)]. A. Robinson 1960: For standard functions F, this is equivalent to the universal sentence ∀x[I(x)⇒I(F(x))] where I(x) means x is infinite.
2 Quantifier hierarchies
Fix an ordered structure M= (M,≤,...) with no greatest element.
Hierarchy of sentences in L(M)∪{F}: Πn: n quantifier blocks starting with ∀ Σn: n quantifier blocks starting with ∃ ∆n: Both Πn and Σn Bn: Boolean in Πn. ∆1 ⊂Π1 ⊂B1 ⊂∆2 ⊂Π2 ⊂B2 ⊂∆3 ⊂Π3, ∆1 ⊂Σ1 ⊂B1 ⊂∆2 ⊂Σ2 ⊂B2 ⊂∆3 ⊂Σ3. Problem. Given M, locate LIM ={(M,F) : limz→∞F(z) =∞} in the quantifier hierarchy.
For each M, it’s Π3 or lower.
3 Cases with low quantifier level
Theorem. For every M, LIM is not B1. Theorem. If M has universe R and a symbol for each function definable in (R,≤,+,•,N), LIM is ∆2. Proof: M has a symbol for a function G(x,n) such that RN ={G(x,•) : x∈R}. LIM is equivalent to ∃x∀n∀y[G(x,n)≤y ⇒n≤F(y)]. The negation of LIM is equivalent to ∃m∃x∀n[n≤G(x,n)∧F(G(x,n))≤m].
4 Cases with maximum quantifier level
Theorem. If M is countable, then LIM is not Σ3. Theorem. If M= (R,≤,N) with N = (N,...), then LIM is not Σ3. Theorem. If M is saturated (or special), then LIM is not Σ3.
K. Sullivan, Ph.D. Thesis 1974, showed that LIM is not Π2 and not Σ2 when M saturated. Theorem. If M= (K,I), K saturated and I ={x : x is infinite }, LIM is not Σ3.
So Robinson’s result for standard functions does not extend to arbitrary functions.
5 Infinitely long sentences
Given a set of sentences Q, VQ ={Vnθn : θn ∈Q} WQ ={Wnθn : θn ∈Q}. If M has universe R and a constant for each n∈N, then LIM isVWΠ1, ^ m_ n ∀z[n≤z ⇒m≤F(z)], and LIM isVΣ2, ^ m∃y∀z[y ≤z ⇒m≤F(z)]. Theorem. If M has universe R, LIM is notWVB1.
6 o-minimal structures
Mis o-minimal if every set definable inMwith parameters is a finiteSof intervals and points. (Van den Dries 1984, Pillay and Steinhorn 1986).
Examples of o-minimal structures: (R,≤,+,•) (Tarski 1939). (R,≤,+,•,exp) (Wilkie 1991). Above plus restricted analytic functions (van den Dries and C. Miller 1994).
Theorem. If M is an o-minimal expansion of (R,≤,+,•), LIM is notVΠ2 and notWB2. Proof uses recent results of H. Friedman and C. Miller (2005) on fast sequences.
Conjecture. For every o-minimal expansion M of (R,≤,+,•), LIM is not Σ3.
7 Summary
Quantifier Level of LIM Over M ∆1 ⊂Π1 ⊂B1 ⊂∆2 ⊂Π2 ⊂B2 ⊂∆3 ⊂Π3 ≤Π3 always ≤VΣ2 (R,≤,0,1,2,...) ≤VWΠ1 (R,≤,0,1,2,...) > ∆3 countable > ∆3 (M,I) with M saturated > ∆3 (R,≤,(N,...)) >WB2 o-minimal (R,≤,+,•,...) >VΠ2 o-minimal (R,≤,+,•,...) >WVB1 (R,≤,...) ∆2 (R,≤,+,•,N,definable) > B1 always
8
References
C.C. Chang and H. Jerome Keisler. Model Theory, Third Edition. Elsevier 1990.
Lou van den Dries. Tame Topology and o-minimal Structures. Cambridge 1998.
Lou van den Dries. o-minimal structures. Pp. 137-185 in Logic: From Foundations to Applications. Oxford 1996.
Harvey Friedman and Chris Miller. Expansions of o-minimal structures by fast sequences. Journal of Symbolic Logic 70 (2005), pp. 410-418.
Kathleen Sullivan. The Teaching of Elementary Calculus: An Approach Using Infinitesimals. Ph. D. Thesis, University of Wisconsin, Madison, 1974