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

[email protected] www.math.wisc.edu/∼keisler
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