模型论与现代微积分(修改稿)
2018-12-07 20:35:15
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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?

    2018126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  127

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, xyz[y z xF(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,·) : xR}. LIM is equivalent to xny[G(x,n)y nF(y)]. The negation of LIM is equivalent to mxn[nG(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 nN, then LIM isVWΠ1, ^ m_ n z[nz mF(z)], and LIM isVΣ2, ^ myz[y z mF(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


 
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