模型论与现代微积分（修改稿）

    从模型论的视野里，我们如何看待现代微积分？

    2018年12月6日，我们在“（ε，δ）条件与无穷小方法之比研究”博文小中正式阐明了相关学术立场。

   2008年，Keisler教授，作为塔尔斯基模型论的传人，发表研究论文，题为“Quantiers in Limits”（极限中的量词）站在模型论的视角深入阐述了现代微积分的一些弊端。

    我们的观点就是从这篇文章中“知识共享”出来的。菲氏极限论的信徒们不知作何感想?

袁萌  陈启清  12月7日

附件：极限理论中的量词

Quantiers in Limits

H. Jerome Keisler University of Wisconsin Madison, WI, USA keisler@math.wisc.edu www.math.wisc.edu/∼keisler

Quantiers in limits

1. Robinson’s limit denition

2. Quantier hierarchies

3. Cases with low quantier level

4. Cases with maximum quantier level

5. Innitely long sentences

6. o-minimal structures

7. Summary

1

Robinson’s limit denition

The nonstandard approach to calculus eliminates two quantier blocks in the limit denition. The standard denition of

lim z→∞

F(z) =∞ requires three quantier 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 innite.

2 Quantier hierarchies

Fix an ordered structure M= (M,≤,...) with no greatest element.

Hierarchy of sentences in L(M)∪{F}: Πn: n quantier blocks starting with ∀ Σn: n quantier 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 quantier hierarchy.

For each M, it’s Π3 or lower.

3 Cases with low quantier level

Theorem. For every M, LIM is not B1. Theorem. If M has universe R and a symbol for each function denable 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 quantier 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 innite }, LIM is not Σ3.

So Robinson’s result for standard functions does not extend to arbitrary functions.

5 Innitely 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 denable inMwith parameters is a niteSof 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

Quantier 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,denable) > 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 Innitesimals. Ph. D. Thesis, University of Wisconsin, Madison, 1974