ε ε x x + ε ε cos(ε) = 1, sin(ε) = ε [6] [5] nonstandard analysis 1974 [4] We shoud add that, to logical positivist, a discussion o

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1 dif engine 2017/12/08 Math Advent Calendar 2017( 12/8 IST(Internal Set Theory; ) (nonstandard analysis, NSA) ε ε (a) ε 0. (b) r > 0 ε < r. (a)(b) ε sin(x) d sin(x) dx = sin(x + ε) sin(x) ε (1) = sin(x) cos(ε) + cos(x) sin(ε) sin(x) ε (2) ε cos(ε) = 1, sin(ε) = ε = cos(x). (3) 1

2 ε ε x x + ε ε cos(ε) = 1, sin(ε) = ε [6] [5] nonstandard analysis 1974 [4] We shoud add that, to logical positivist, a discussion of the ontological significance of infinitary notions of any kind is meaningless. However, presumably even a positivist would concede the historical importance of expressions involving the term infinity and of the (possibly, subjective) ideas associated with such terms. 2

3 However, from a formalist point of view we may look at our theory syntactically and may consider that what we have done is to introduce new deductive procedures rather than new mathematical entities. 1.2 R [7][8][9][10] [11][12][13] IST(Internal Set Theory; ) ZFC HST,KST,RIST,FRIST,RBST,GRIST ( [18] Hrbáček ) IST IST 1977 [1] ZFC IST [14] IST [16] IST [15] IST HST BST [17] IST ZFC IST ZFC IST IST 3

4 X P(X) IST Loeb IST 2 IST 2.1 IST IST 1977 [1] ZFC IST IST ZFC st() ZFC {n N st(n)} {n N st(n)} N N IST ZFC IST st() (2.5) 2.2 IST st() st() ZFC x y x y x y st() x st(x) x st(x) x IST IST st(t) (t x) t x st(t) (t x) t x 4

5 (Hrbáček) [19] RBST IST x st(x) x (observable) IST A st() A - A st() A st- 2.3 x Fin(x) ZFC Fin(x) Fin(x) def n N φ: x 1-1 n. (4) N ZFC SUCC(x) := x {x} ( 0) 0(= ) ( 1) 1(= SUCC(0)) ( 2) 2(= SUCC(1))... N 0, 1, 2,... [20] ZFC N IST N 5

6 2.4 S S S := { x } st(x). (5) S S S. (6) S S S - A A S (A ) : A, (A B) : (B S ), (A B C) : B S C S, A S := (A B C) : B S C S, (A x B(x)) : x ( st(x) = B S (x) ), (A x B(x)) : x ( st(x) B S (x) ). (7) S st x A(x) x (st(x) = A(x)) x A(x) st x A(x) x (st(x) A(x)) x A(x) st fin x A(x) st x (Fin(x) = A(x)) x A(x) st fin x A(x) st x (Fin(x) A(x)) x A(x) fin x A(x) x (Fin(x) = A(x)) x A(x) fin x A(x) x (Fin(x) A(x)) x A(x) 1: - A S A S st st 2.5 IST IST ZFC st() 6

7 (Transfer Principle) A(t 1,..., t k ) - (k 0) (T) st t 1 st t k ( A S A ). (Idealization Principle) B(x, y, u 1,..., u k ) - (k 0) (I) u 1 u k (( st fin F y x F B(x, y, u 1,..., u k ) ) ( y st x B(x, y, u 1,..., u k ) )). (Standarization Principle) C(z, u 1,..., u k ) st- (k 0) (S) u 1 u k ( st x st y st z (z y z x C(z, u 1,..., u k )) ). (Hrbáček) [1] ZFC S, (8) x S = x S, (9) x S = P(x) S, (10) x, y S = {x, y} S, (11) x, y S = x y S, (12) x, y S = y x S, (13) A(x, y, a, t 1,..., t k ) - a, t 1,..., t k S x a!y A(x, y, a, t 1,..., t k ) = {y x a A(x, y, a, t 1,..., t k )} S. (14) V = S [1] K Fin(K) (S K). (15) K S 2.3 K IST 7

8 IST [3] Perhaps it is fair to say that finite does not mean what we have always thought it to mean. What have we always thought it to mean? I used to think that I knew what I had always thought it to mean, but I no longer think so. In any case, intuition changes with experience. 2.6 IST IST r r : i-small def st ε > 0 ( r < ε) (16) r : i-small r r ε r x, y x y def x y : i-small (17) x y x y 2.7 IST IST ZFC ZFC IST ZFC IST IST ZFC ([1] ) IST ZFC ZFC IST ZFC 2.8 IST IST Q st 1 x 1 Q st mx m Q m+1 x m+1 Q m+n x m+n A(x 1,..., x m, x m+1,..., x m+n ) (18) Q 1,..., Q m+n (18) 2 m+n 8

9 st A absolute - st ( absolute A ) st() * [1] 2.9 (1) 0 x R (x 0 x : i-small ). (19) x R {0} st ε R + ( x < ε). (20) st fin F R + x R {0} ε F ( x < ε). (21) F x := 1 min F 2 0 R IST R 2.10 (2) f : R R x 0 R S- t R (t x 0 = f(t) f(x 0 )). (22) t x 0 f(t) f(x 0 ) st(f) st(x 0 ) f x 0 S-. (23) t R (t x 0 = f(t) f(x 0 )). (24) t R ( st δ R + t x 0 < δ ) = ( st ε R + f(t) f(x 0 ) < ε ). (25) *1 st(x) st y (y = x) [1] 9

10 t R st ε R + st δ R + ( t x 0 < δ = f(t) f(x 0 ) < ε). (26) t ε st ε R + t R st δ R + ( ). (27) st ε R + st fin F R + t R δ F ( ). (28) ε R + fin F R + t R δ F ( ). (29) ε R + δ R + t R ( t x 0 < δ = f(t) f(x 0 ) < ε). (30) [ ] δ := min F [ ] F := {δ } 3 IST book keeping [1] *2 IST IST *2!? (10) IST ( ) Twitter DM 10

11 IST t x 0 f(t) f(x 0 ) f x 0 *3 f ε > 0 δ > 0 t x 0 < δ f(t) f(x 0 ) < ε IST t x 0 f(t) f(x 0 ) IST ZFC IST ZFC [2] IST ZFC IST IST IST [14][16] IST [1] E. Nelson Internal Set Theory : A New Approach to Nonstandard Analysis, Bull.Amer.Math.Soc. 83 (1977),pp [2] E. Nelson The Syntax of Nonstandard Analysis, Ann.Pure.Appl.Logic. 38(1988), pp *3 f x 0 S- f x 0 11

12 (open archive). [3] E. Nelson Internal Set Theory, pdf [4] A. Robinson Non-standard Analysis, North-Holland (1974). Princeton University Press [5] ( ) (1991). H. -D. Ebbinghaus, et al. (Eds) Zahlen, Springer-Verlag Berlin Heiderberg (1983, 1988). [6] ( ) (2001) [7] =Gödel = (1995). [8] (2002). [9] (2010) [10] (2012). [11] M (1982). M. Davis Applied Nonstandard Analysis, John Wiley & Sons, Inc. (1977). Dover. [12] (1976,1987). [13] (1998,2017). [14] N. Vakil Real Analysis through Modern Infinitesimals, Cambridge(2011). [15] R. Lutz & M. Goze Nonstandard Analysis : A Practical Guide with Applications, Springer(1980). [16] A. Robert Nonstandard Analysis, John Wiley & Sons (1988). ( 1985) Dover [17] V. Kanovei & M. Reeken Nonstandard Analysis, Axiomatically, Springer(2004). [18] N. J. Cutland, et al. (Eds) Nonstandard Methods and Applications in Mathematics, Association for Symbolic Logic, Lecture Notes in Logic, 25 (2006). [19] K. Hrbacek, et al. Analysis with Ultrasmall Numbers, CRC Press(2015). [20] (2016). K. Kunen The Foundations of Mathematics, College Publications(2009). 2017/12/09 12

13 (25) (27) 2017/12/ /12/20 FV () 13

1 θ i (1) A B θ ( ) A = B = sin 3θ = sin θ (A B sin 2 θ) ( ) 1 2 π 3 < = θ < = 2 π 3 Ax Bx3 = 1 2 θ = π sin θ (2) a b c θ sin 5θ = sin θ f(sin 2 θ) 2

1 θ i (1) A B θ ( ) A = B = sin 3θ = sin θ (A B sin 2 θ) ( ) 1 2 π 3 < = θ < = 2 π 3 Ax Bx3 = 1 2 θ = π sin θ (2) a b c θ sin 5θ = sin θ f(sin 2 θ) 2 θ i ) AB θ ) A = B = sin θ = sin θ A B sin θ) ) < = θ < = Ax Bx = θ = sin θ ) abc θ sin 5θ = sin θ fsin θ) fx) = ax bx c ) cos 5 i sin 5 ) 5 ) αβ α iβ) 5 α 4 β α β β 5 ) a = b = c = ) fx) = 0 x x = x =