Number Systems Based on Logical Calculus

Transcription

1 International Mathematical Forum, Vol 8, 2013, no 34, HIKARI Ltd, wwwm-hikaricom Number Systems Based on Logical Calculus Pith Xie Axiom Studio, PO Box #3620 Jiangdongmen Postoffice, Gulou District Nanjing, , PR China pith Copyright c 2013 Pith Xie This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Abstract The reference [4] denote number systems with a logical calculus, but the form of natural numbers are not consistently in these number systems So we rewrite number systems to correct the defect Finally, we furtherly name the new operators according to the definitions of number systems Mathematics Subject Classification: 11A99, 03B10, 11B85, 03B80 Keywords: number, operator, number system, logical calculus 1 Introduction The refenences [1, 2, 3] construct surreal number system to extend real number system Surreal number system apply well to game theory and has a special merit In nature, surreal number system introduces limits of infinite dyadic rational sequences to denote irrational numbers According to [4, 32], surreal number system belongs to logical notations and lack intuition So it fails to join in algebraical operations In [4], the logical calculus for number systems can produce new real numbers with new operators However, the form of natural numbers are not consistently in these number systems So we rewrite number systems to correct this defect The number systems in this paper can show typical features as follows:

2 1664 Pith Xie 1 The logical calculus provides a uniform frame for arithmetic axioms Based on the same logical calculus, small number system can import new axioms to produce big number systems 2 Consistent binary relation are the nature of number systems The order relation and equivalence relation of each number system is consistent, so all numbers of each number system are layed in fixed positions of number line 3 The number systems unify the numbers and operators as a whole, and the only number 1 and various operators compose all numbers So operator distinguishes different number systems and is the foremost component of number system 4 The logical calculus for real number system produces new real numbers with new operators While producing new real numbers in arithmetic, the new operators certainly produce new equations and inequalities in algebra So the logical calculus not only extends real number system, but also extends equations and inequalities The paper is organized as follows In Section 2, we construct a general logical calculus for number systems In Section 3, we construct a logical calculus to denote natural number system In Section 4, we construct a logical calculus to denote integral number system In Section 5, we construct a logical calculus to denote rational number system In Section 6, we construct a logical calculus to denote real number system In Section 7, we furtherly name the new operators according to the definitions of number systems 2 Logical Calculus In mathematical logic, logical calculus is a formal system to abstract and analyze the induction and deduction apart from specific meanings In this section, however, we construct a logical calculus by virtue of formal language and deduce numbers to intuitively and logically denote number systems The logical calculus not only denotes real numbers, but also allows them to join in algebraical operations The introduction of formal language aims to use computer fast execute real number operations For clarity, we will explain the logical calculus with natural language In [5], the producer substitutes the right permutations for the left permutations to produce new permutations In [6], the connectives,,, and stand for not, and, or, implies and if and only if respectively Here, the producer is considered as a predicate symbol and embedded into the logical calculus Table 1 translates the syntax of the logical calculus to natural language

3 Number systems based on logical calculus 1665 Table 1: Translation From Syntax To Natural Language Syntax Natural Language { punctuation } punctuation, punctuation ( punctuation ) punctuation emptiness omission Φ{ } Denote Φ as a set of notations and particular axioms between { and } Different logical calculus correspond to different notations and particular axioms V { } Denote V as a set of variables between { and } C{ } Denote C as a set of constants between { and } P { } Denote P as a set of predicate symbols between { and } V C{ } Denote V C as a set of concatenations between V and C C C{ } Denote C C as a set of concatenations between C and C V C P { } Denote V C P as a set of concatenations among V, C and P ( ) Define the binary predicate symbol ( (V C)) Define a variable ranging over V C ( (V C P )) Define a variable ranging over V C P ( ) Define the binary predicate symbol Ψ{ } Denote Ψ as a set of general axioms between { and } Different logical calculus correspond to the same general axioms ( ) ( Define the binary predicate symbol ) ( ) Define the binary predicate symbol ( < ) Define the binary predicate symbol < ( = ) Define the binary predicate symbol = ( = ) Define the binary predicate symbol = ( ) Define the binary predicate symbol ( ) Define the binary predicate symbol

4 1666 Pith Xie Definition 21 {Φ, Ψ} is a logical calculus such that: (21) Φ{ (22) V {,a,b }, (23) C{, 1, + }, (24) P {,,,,, =,<,, }, (25) V C{,a,b, 1, +, aa, ab,a1,a+, ba, bb,b1, b +, aaa, aab,aa1,aa+, baa, bab,ba1,ba+ }, (26) C C{, 1, +, 11, 1+, 111, 11 + }, (27) V C P {,a,b, 1, +,,, aa, ab,a1,a+,a, a, ba, bb,b1,b+,b,b, aaa, aab,aa1, aa +,aa,aa, baa, bab,ba1,ba+,ba,ba }, (28) (â V ) ((â ) (â a) (â b) ), (29) (â C) ((â ) (â 1) (â +) ), (210) (â (V C)) ((â ) (â a) (â b) (â 1) (â aa) (â ab) (â a1) (â aaa) (â aab) (â aa1) ), (211) (â (C C)) ((â ) (â 1) (â +) (â 11) (â 1+) (â 111) (â 11+) ), (212) (â (V C P )) ((â ) (â a) (â b) (â ) (â aa) (â ab) (â a ) (â aaa) (â aab) (â aa ) ), (213) (ā (V C)) ( b (V C)) ( c (V C)) ( d (V C)) (ē (V C)) ( f (V C)) (ḡ (V C)) ( h (V C)) (ī (V C)) ( j (V C)) (ā (V C P )) ( b (V C P )) ( c (V C P )), (214) ((ā { b, c}) ((ā b) (ā c))) ((ā { b, c, d}) ((ā b) (ā c) (ā d))) ((ā { b, c, d, ē}) ((ā b) (ā c) (ā d) (ā ē))) ((ā { b, c, d, ē, f}) ((ā b) (ā c) (ā d) (ā ē) (ā f))) ((ā { b, c, d, ē, f,ḡ}) ((ā b) (ā c) (ā d) (ā ē) (ā f) (ā ḡ))) ((ā { b, c, d, ē, f,ḡ, h}) ((ā b) (ā c) (ā d) (ā ē) (ā f) (ā ḡ) (ā h))) ((ā { b, c, d, ē, f,ḡ, h, ī}) ((ā b) (ā c) (ā d) (ā ē)

5 Number systems based on logical calculus 1667 (ā f) (ā ḡ) (ā h) (ā ī))) ((ā { b, c, d, ē, f,ḡ, h, ī, j}) ((ā b) (ā c) (ā d) (ā ē) (ā f) (ā ḡ) (ā h) (ā ī) (ā j))) }, (215) Ψ{ (216) (ā b) ( b = cā d), (217) (ā b c d) ( c ē) (ā bē d), (218) (ā b c) ((ā b) (ā c)), (219) (ā b c) ((ā c) ( b c)), (220) (ā < b) ( b <ā), (221) (ā < b) (ā = b), (222) (ā < b) ( b < c) (ā < c), (223) (ā < b) (ā (C C)) ( b (C C)) (ā b), (224) (ā < b c d) ( c =ē) (ā < bē d), (225) (ā b c < d) ( b =ē) (āē c < d), (226) (ā < b c d) ( c ē) ( c {ā, b, d}) (ā < bē d), (227) (ā < b c d cē) ( c f) ( c {ā, b, d, ē}) (ā < b f d fē), (228) (ā b c < d) ( b ē) ( b {ā, c, d}) (āē c < d), (229) (ā b c < d bē) ( b f) ( b {ā, c, d, ē}) (ā f c < d fē), (230) (ā b c < d bē b f) ( b ḡ) ( b {ā, c, d, ē, f}) (āḡ c < dḡēḡ f), (231) (ā b c b d <ē) ( b f) ( b {ā, c, d, ē}) (ā f c f d <ē), (232) (ā b c b d <ē b f) ( b ḡ) ( b {ā, c, d, ē, f}) (āḡ cḡ d <ēḡ f), (233) (ā b c b d <ē b f bḡ) ( b h) ( b {ā, c, d, ē, f,ḡ}) (ā h c h d <ē h f hḡ), (234) ā =ā, (235) (ā = b) ( b =ā), (236) (ā = b) (ā < b), (237) (ā = b c d) ( c =ē) (ā = bē d), (238) (ā b c) ( b = d) (ā b c =ā d c), (239) (ā = b c d) ( c ē) ( c {ā, b, d}) (ā = bē d), (240) (ā = b c d cē) ( c f) ( c {ā, b, d, ē}) (ā = b f d fē), (241) (ā b c = d bē) ( b f) ( b {ā, c, d, ē}) (ā f c = d fē), (242) (ā b c = d bē b f) ( b ḡ) ( b {ā, c, d, ē, f}) (āḡ c = dḡēḡ f), (243) (ā b c b d =ē b f bḡ) ( b h) ( b {ā, c, d, ē, f,ḡ}) (ā h c h d =ē h f hḡ),

6 1668 Pith Xie (244) (ā b) ((ā < b) (ā = b)), (245) (ā b) ( b c) (ā c) } Then we define number with above logical calculus Definition 22 In a logical calculus {Φ, Ψ}, ifā true, then ā is a number In the following sections, we will prove that the logical calculus {Φ, Ψ} can deduce common number systems 3 Natural Number System Theorem 31 If Φ{ (31) V {,a,b}, (32) C{, 1, +, [, ]}, (33) P {,,,,, =,<, }, (34) V C{,a,b, 1, +, aa, ab,a1,a+, ba, bb,b1, b +, aaa, aab,aa1,aa+, baa, bab,ba1,ba+ }, (35) C C{, 1, +, 11, 1+, 111, 11 + }, (36) V C P {,a,b, 1, +,,, aa, ab,a1,a+,a, a, ba, bb,b1,b+,b,b, aaa, aab,aa1, aa +,aa,aa, baa, bab,ba1,ba+,ba,ba }, (37) (â V ) ((â ) (â a) (â b) ), (38) (â C) ((â ) (â 1) (â +) ), (39) (â (V C)) ((â ) (â a) (â b) (â 1) (â aa) (â ab) (â a1) (â aaa) (â aab) (â aa1) ), (310) (â (C C)) ((â ) (â 1) (â +) (â 11) (â 1+) (â 111) (â 11+) ), (311) (â (V C P )) ((â ) (â a) (â b) (â ) (â aa) (â ab) (â a ) (â aaa) (â aab) (â aa ) ), (312) (ā (V C)) ( b (V C)) ( c (V C)) ( d (V C)) (ē (V C)) ( f (V C)) (ḡ (V C)) ( h (V C))

7 Number systems based on logical calculus 1669 (ī (V C)) ( j (V C)) (ā (V C P )) ( b (V C P )) ( c (V C P )), (313) ((ā { b, c}) ((ā b) (ā c))) ((ā { b, c, d}) ((ā b) (ā c) (ā d))) ((ā { b, c, d, ē}) ((ā b) (ā c) (ā d) (ā ē))) ((ā { b, c, d, ē, f}) ((ā b) (ā c) (ā d) (ā ē) (ā f))) ((ā { b, c, d, ē, f,ḡ}) ((ā b) (ā c) (ā d) (ā ē) (ā f) (ā ḡ))) ((ā { b, c, d, ē, f,ḡ, h}) ((ā b) (ā c) (ā d) (ā ē) (ā f) (ā ḡ) (ā h))) ((ā { b, c, d, ē, f,ḡ, h, ī}) ((ā b) (ā c) (ā d) (ā ē) (ā f) (ā ḡ) (ā h) (ā ī))) ((ā { b, c, d, ē, f,ḡ, h, ī, j}) ((ā b) (ā c) (ā d) (ā ē) (ā f) (ā ḡ) (ā h) (ā ī) (ā j))), (314) a 1 [a + a], (315) a<[1 + a], (316) ā b ([ā + b] =[ b +ā]), (317) ā b c ([ā +[ b + c]] = [[ā + b]+ c]) }, Proof then N{Φ, Ψ} denotes natural number system (A1) (a 1 [a + a]) (a 1) by(314), (218) (A2) (a [a + a]) by(218) (A3) (a [1 + a]) by(a2), (A1), (217) (A4) (a [1 + 1]) by(a3), (A1), (217) (A5) (a <[1 + a]) (1 < [1 + 1]) by(315), (A1), (229) (A6) 1 by(223) (A7) [1+1] by(a5), (223) (A8) ([1 + 1] < [1 + [1 + 1]]) by(315), (A4), (229) (A9) [1+1] by(223) (A10) [1+[1+1]] by(a8), (223) (A11) (a [1+[a + a]]) by(a3), (A2), (217) (A12) (a [1 + [1 + a]]) by(a11), (A1), (217)

8 1670 Pith Xie (A13) (a [1 + [1 + 1]]) by(a12), (A1), (217) (A14) ([1 + [1 + 1]] < [1 + [1 + [1 + 1]]]) by(315), (A13), (229) (A15) [1 + [1 + [1 + 1]]] by(223) Then according to Definition 22, we can deduce from N{Φ, Ψ} the numbers as follows: {1, [1 + 1], [1+[1+1]], [1 + [1 + [1 + 1]]] } (B1) [1+1]=[1+1] by(a6), (316) (B2) [1+[1+1]]=[[1+1]+1] by(a6), (A7), (316) (B3) [1 + [1 + [1 + 1]]] = [[1 + [1 + 1]] + 1] by(a6), (A10), (316) (B4) [1 + [1 + [1 + 1]]] = [[1 + 1] + [1 + 1]] by(a6), (A7), (317) Then we can deduce from N{Φ, Ψ} the equalities as follows: [1+1] = [1+1], [1+[1+1]] = [[1+1]+1], [1 + [1 + [1 + 1]]] = [[1 + [1 + 1]] + 1], [1 + [1 + [1 + 1]]] = [[1 + 1] + [1 + 1]], The deduced numbers correspond to natural numbers as follows: 1 1, [1+1] 2, [1+[1+1]] 3, [1 + [1 + [1 + 1]]] 4, The equalities on deduced numbers correspond to the addition in natural number system So the claim follows

9 Number systems based on logical calculus Integral Number System Theorem 41 If Φ{ (41) V {,a,b,c}, (42) C{, 1, +, [, ], }, (43) P {,,,,, =,<, }, (44) V C{,a,b, 1, +, aa, ab,a1,a+, ba, bb,b1, b +, aaa, aab,aa1,aa+, baa, bab,ba1,ba+ }, (45) C C{, 1, +, 11, 1+, 111, 11 + }, (46) V C P {,a,b, 1, +,,, aa, ab,a1,a+,a, a, ba, bb,b1,b+,b,b, aaa, aab,aa1, aa +,aa,aa, baa, bab,ba1,ba+,ba,ba }, (47) (â V ) ((â ) (â a) (â b) ), (48) (â C) ((â ) (â 1) (â +) ), (49) (â (V C)) ((â ) (â a) (â b) (â 1) (â aa) (â ab) (â a1) (â aaa) (â aab) (â aa1) ), (410) (â (C C)) ((â ) (â 1) (â +) (â 11) (â 1+) (â 111) (â 11+) ), (411) (â (V C P )) ((â ) (â a) (â b) (â ) (â aa) (â ab) (â a ) (â aaa) (â aab) (â aa ) ), (412) (ā (V C)) ( b (V C)) ( c (V C)) ( d (V C)) (ē (V C)) ( f (V C)) (ḡ (V C)) ( h (V C)) (ī (V C)) ( j (V C)) (ā (V C P )) ( b (V C P )) ( c (V C P )), (413) ((ā { b, c}) ((ā b) (ā c))) ((ā { b, c, d}) ((ā b) (ā c) (ā d))) ((ā { b, c, d, ē}) ((ā b) (ā c) (ā d) (ā ē))) ((ā { b, c, d, ē, f}) ((ā b) (ā c) (ā d) (ā ē) (ā f))) ((ā { b, c, d, ē, f,ḡ}) ((ā b) (ā c) (ā d) (ā ē) (ā f) (ā ḡ))) ((ā { b, c, d, ē, f,ḡ, h}) ((ā b) (ā c) (ā d) (ā ē)

10 1672 Pith Xie (ā f) (ā ḡ) (ā h))) ((ā { b, c, d, ē, f,ḡ, h, ī}) ((ā b) (ā c) (ā d) (ā ē) (ā f) (ā ḡ) (ā h) (ā ī))) ((ā { b, c, d, ē, f,ḡ, h, ī, j}) ((ā b) (ā c) (ā d) (ā ē) (ā f) (ā ḡ) (ā h) (ā ī) (ā j))), (414) a 1 [aba], (415) b c +, (416) a<[1 + a], (417) ā ([ā ā] =[1 1]), (418) ā b ([ā + b] =[ b +ā]), (419) ā b ([[ā b]+ b] =ā), (420) ā b c ([[ā b]+ c] =[ā +[ c b]]), (421) ā b c ([ā +[ b + c]] = [[ā + b]+ c]), (422) ā b c ([ā +[ b c]] = [[ā + b] c]), (423) ā b c ([ā [ b + c]] = [[ā b] c]), (424) ā b c ([ā [ b c]] = [[ā b]+ c]) }, Proof then Z{Φ, Ψ} denotes integral number system (A1) (a 1 [aba]) (a 1) by(414), (218) (A2) (a [aba]) by(414), (218) (A3) (a [1ba]) by(a2), (A1), (217) (A4) (a [1b1]) by(a3), (A1), (217) (A5) (b c + ) (b c ) by(415), (218) (A6) (b ) by(a5), (219) (A7) (a [1 1]) by(a4), (A6), (217) (A8) (a <[1 + a]) (1 < [1 + 1]) by(416), (A1), (229) (A9) 1 by(223) (A10) [1+1] by(a8), (223) (A11) (a <[1 + a]) ([1 1] < [1+[1 1]]) by(416), (A7), (229) (A12) ([1 1] < [[1 1] + 1]) by(a11), (418), (224) (A13) ([1 1] < 1) by(a12), (419), (224) (A14) [1 1] by(223)

11 Number systems based on logical calculus 1673 Then according to Definition 22, we can deduce from Z{Φ, Ψ} the numbers as follows: {1, [1 + 1], [1 1], [1+[1+1]], [1 [1 + 1]] } (B1) [1+[1+1]]=[[1+1]+1] by(a9), (A10), (418) (B2) [1 + [1 1]] = [[1 1]+1] by(a9), (A14), (418) (B3) [1 + [1 1]] = [[1 + 1] 1] by(a9), (422) (B4) [[1 + 1] + [1 1]] = [[1 1]+[1+1]] by(a10), (A14), (418) (B5) [[1 + 1] + [1 1]]=[1+[1+[1 1]]] by(a10), (A14), (421) (B6) [[1 + 1] + [1 1]] = [[[1 + 1] + 1] 1] by(a9), (A10), (422) Then we can deduce from Z{Φ, Ψ} the equalities as follows: [1+[1+1]] = [[1+1]+1], [1+[1 1]] = [[1 1]+1], [1+[1 1]] = [[1 + 1] 1], [[1+1]+[1 1]] = [[1 1]+[1+1]], [[1+1]+[1 1]] = [1+[1+[1 1]]], [[1+1]+[1 1]] = [[[1 + 1] + 1] 1], The deduced numbers correspond to integral numbers as follows:, [1 [1 + [1 + 1]]] 2, [1 [1 + 1]] 1, [1 1] 0, 1 1, [1+1] 2, [1+[1+1]] 3, [1 + [1 + [1 + 1]]] 4, The equalities on deduced numbers correspond to the addition and subtraction in integral number system So the claim follows

12 1674 Pith Xie 5 Rational Number System Theorem 51 If Φ{ (51) V {,a,b,c,d}, (52) C{, 1, +, [, ], }, (53) P {,,,,, =,<, }, (54) V C{,a,b, 1, +, aa, ab,a1,a+, ba, bb,b1, b +, aaa, aab,aa1,aa+, baa, bab,ba1,ba+ }, (55) C C{, 1, +, 11, 1+, 111, 11 + }, (56) V C P {,a,b, 1, +,,, aa, ab,a1,a+,a, a, ba, bb,b1,b+,b,b, aaa, aab,aa1, aa +,aa,aa, baa, bab,ba1,ba+,ba,ba }, (57) (â V ) ((â ) (â a) (â b) ), (58) (â C) ((â ) (â 1) (â +) ), (59) (â (V C)) ((â ) (â a) (â b) (â 1) (â aa) (â ab) (â a1) (â aaa) (â aab) (â aa1) ), (510) (â (C C)) ((â ) (â 1) (â +) (â 11) (â 1+) (â 111) (â 11+) ), (511) (â (V C P )) ((â ) (â a) (â b) (â ) (â aa) (â ab) (â a ) (â aaa) (â aab) (â aa ) ), (512) (ā (V C)) ( b (V C)) ( c (V C)) ( d (V C)) (ē (V C)) ( f (V C)) (ḡ (V C)) ( h (V C)) (ī (V C)) ( j (V C)) (ā (V C P )) ( b (V C P )) ( c (V C P )), (513) ((ā { b, c}) ((ā b) (ā c))) ((ā { b, c, d}) ((ā b) (ā c) (ā d))) ((ā { b, c, d, ē}) ((ā b) (ā c) (ā d) (ā ē))) ((ā { b, c, d, ē, f}) ((ā b) (ā c) (ā d) (ā ē) (ā f))) ((ā { b, c, d, ē, f,ḡ}) ((ā b) (ā c) (ā d) (ā ē) (ā f) (ā ḡ))) ((ā { b, c, d, ē, f,ḡ, h}) ((ā b) (ā c) (ā d) (ā ē)

13 Number systems based on logical calculus 1675 (ā f) (ā ḡ) (ā h))) ((ā { b, c, d, ē, f,ḡ, h, ī}) ((ā b) (ā c) (ā d) (ā ē) (ā f) (ā ḡ) (ā h) (ā ī))) ((ā { b, c, d, ē, f,ḡ, h, ī, j}) ((ā b) (ā c) (ā d) (ā ē) (ā f) (ā ḡ) (ā h) (ā ī) (ā j))), (514) a 1 [aba], (515) b +, (516) c d b ++, (517) a<[1 + a], (518) (ā < b) c ([ā + c] < [ b + c]), (519) (ā < b) c ([ c b] < [ c ā]), (520) ([1 1] < ā) (ā < b) ([1 1] < c) ([ā c] < [ b c]), (521) ā ([ā ā] =[1 1]), (522) ā b ([ā + b] =[ b +ā]), (523) ā b ([[ā b]+ b] =ā), (524) ā b c ([[ā b]+ c] = [[ā + c] b]), (525) ā b c ([ā +[ b + c]] = [[ā + b]+ c]), (526) ā b c ([ā +[ b c]] = [[ā + b] c]), (527) ā b c ([ā [ b + c]] = [[ā b] c]), (528) ā b c ([ā [ b c]] = [[ā b]+ c]), (529) ā ([ā + +1] = ā) ([ā 1] = ā), (530) (ā =[1 1]) ([ā ā] =1), (531) ā b ([ā ++ b] =[ b ++ā]), (532) ā b c ([ā ++[ b ++ c]] = [[ā ++ b]++ c]), (533) ā b c ([ā ++[ b + c]] = [[ā ++ b]+[ā ++ c]]), (534) ā b c ([ā ++[ b c]] = [[ā ++ b] [ā ++ c]]), (535) ā ( b =[1 1]) ([[ā b]++ b] =ā), (536) ā ( b =[1 1]) c (([[ā b]++ c] = [[ā ++ c] b]) ([[ā + c] b] = [[ā b]+[ c b]]) ([[ā c] b] = [[ā b] [ c b]])), (537) ā ( b =[1 1]) ( c =[1 1]) (([ā ++[ b c]] = [[ā ++ b] c]) ([ā [ b ++ c]] = [[ā b] c]) ([ā [ b c]] = [[ā b]++ c])) },

14 1676 Pith Xie then Q{Φ, Ψ} denotes rational number system Proof (A1) (a 1 [aba]) (a 1) by(514), (218) (A2) (a [aba]) by(514), (218) (A3) (a [1ba]) by(a2), (A1), (217) (A4) (a [1b1]) by(a3), (A1), (217) (A5) (b + ) (b ) by(515), (218) (A6) (a [1 1]) by(a4), (A5), (217) (A7) (a <[1 + a]) (1 < [1 + 1]) by(516), (A1), (229) (A8) 1 by(223) (A9) [1+1] by(a7), (223) (A10) (a <[1 + a]) ([1 1] < [1+[1 1]]) by(516), (A6), (229) (A11) ([1 1] < [[1 1] + 1]) by(a10), (518), (224) (A12) ([1 1] < 1) by(a11), (519), (224) (A13) [1 1] by(223) (A14) (b + ) (b +) by(515), (218) (A15) (a [1 + 1]) by(a4), (A14), (217) (A16) (a <[1 + a]) ([1 + 1] < [1 + [1 + 1]]) by(516), (A15), (229) (A17) ([1 1] < [1 + 1]) by(a12), (A7), (222) (A18) ([1 1] < [1 + [1 + 1]]) by(a17), (A16), (222) (A19) ([1 [1 + [1 + 1]]] < [[1 + 1] [1 + [1 + 1]]]) by(a12), (A7), (A18), (520) (A20) [1 [1 + [1 + 1]]] by(223) (A21) [[1 + 1] [1 + [1 + 1]]] by(a19), (223) (A22) ([1 + 1] = [1 1]) by(a17), (221) Then according to Definition 22, we can deduce from Q{Φ, Ψ} the numbers as follows: {1, [1+1], [1 1], [1+[1+1]], [1 [1 + 1]], [1 [1 + [1 + 1]]] }

15 Number systems based on logical calculus 1677 (B1) [1+[1+1]]=[[1+1]+1] by(a8), (A9), (522) (B2) [1 + [1 1]] = [[1 1]+1] by(a8), (A13), (522) (B3) [1 + [1 1]] = [[1 + 1] 1] by(a8), (526) (B4) [[1 + 1] + [1 1]] = [[1 1]+[1+1]] by(a9), (A13), (522) (B5) [[1 + 1] + [1 1]]=[1+[1+[1 1]]] by(a9), (A13), (525) (B6) [[1 + 1] + [1 1]] = [[[1 + 1] + 1] 1] by(a8), (A9), (526) (B7) [1 + [1 [1 + [1 + 1]]]] = [[1 [1 + [1 + 1]]] + 1] by(a8), (A20), (522) (B8) [[1 [1 + 1]] + +[1 + 1]] = 1 by(a8), (A22), (535) Then we can deduce from Q{Φ, Ψ} the equalities as follows: [1+[1+1]] = [[1+1]+1], [1+[1 1]] = [[1 1]+1], [1+[1 1]] = [[1 + 1] 1], [[1+1]+[1 1]] = [[1 1]+[1+1]], [[1+1]+[1 1]] = [1+[1+[1 1]]], [[1+1]+[1 1]] = [[[1 + 1] + 1] 1], [1+[1 [1 + [1 + 1]]]] = [[1 [1 + [1 + 1]]] + 1], [[1 [1 + 1]] + +[1 + 1]] = 1, The deduced numbers correspond to rational numbers as follows:, [1 [1 + 1]] 1,, [[1 1] [1 [1 + 1]]] 1 2,, [1 1] 0,, [1 [1 + 1]] 1 2,, 1 1,

16 1678 Pith Xie, [1+[1 [1 + 1]]] 3 2,, [1+1] 2, The equalities on deduced numbers correspond to the addition, subtraction, multiplication, division in rational number system So the claim follows 6 Real Number System Definition 61 Real number system is a logical calculus R{Φ, Ψ} such that: (61) (62) (63) (64) (65) (66) (67) (68) (69) Φ{ V {,a,b,c,d,e,f,g,h,i,j,k,l}, C{, 1, +, [, ],,/,,, }, P {,,,,, =,<,, }, V C{,a,b, 1, +, aa, ab,a1,a+, ba, bb,b1, b +, aaa, aab,aa1,aa+, baa, bab,ba1,ba+ }, C C{, 1, +, 11, 1+, 111, 11 + }, V C P {,a,b, 1, +,,, aa, ab,a1,a+,a, a, ba, bb,b1,b+,b,b, aaa, aab,aa1, aa +,aa,aa, baa, bab,ba1,ba+,ba,ba }, (â V ) ((â ) (â a) (â b) ), (â C) ((â ) (â 1) (â +) ), (â (V C)) ((â ) (â a) (â b) (â 1) (â aa) (â ab) (â a1) (â aaa) (â aab) (â aa1) ), (610) (â (C C)) ((â ) (â 1) (â +) (â 11) (â 1+) (â 111) (â 11+) ), (611) (â (V C P )) ((â ) (â a) (â b) (â ) (â aa) (â ab) (â a ) (â aaa) (â aab) (â aa ) ), (612) (ā (V C)) ( b (V C)) ( c (V C)) ( d (V C)) (ē (V C)) ( f (V C)) (ḡ (V C)) ( h (V C))

17 Number systems based on logical calculus 1679 (ī (V C)) ( j (V C)) ( k (V C)) ( l (V C)) (ā (V C P )) ( b (V C P )) ( c (V C P )), (613) ((ā { b, c}) ((ā b) (ā c))) ((ā { b, c, d}) ((ā b) (ā c) (ā d))) ((ā { b, c, d, ē}) ((ā b) (ā c) (ā d) (ā ē))) ((ā { b, c, d, ē, f}) ((ā b) (ā c) (ā d) (ā ē) (ā f))) ((ā { b, c, d, ē, f,ḡ}) ((ā b) (ā c) (ā d) (ā ē) (ā f) (ā ḡ))) ((ā { b, c, d, ē, f,ḡ, h}) ((ā b) (ā c) (ā d) (ā ē) (ā f) (ā ḡ) (ā h))) ((ā { b, c, d, ē, f,ḡ, h, ī}) ((ā b) (ā c) (ā d) (ā ē) (ā f) (ā ḡ) (ā h) (ā ī))) ((ā { b, c, d, ē, f,ḡ, h, ī, j}) ((ā b) (ā c) (ā d) (ā ē) (ā f) (ā ḡ) (ā h) (ā ī) (ā j))), ((ā { b, c, d, ē, f,ḡ, h, ī, j, k}) ((ā b) (ā c) (ā d) (ā ē) (ā f) (ā ḡ) (ā h) (ā ī) (ā j) (ā k))), ((ā { b, c, d, ē, f,ḡ, h, ī, j, k, l}) ((ā b) (ā c) (ā d) (ā ē) (ā f) (ā ḡ) (ā h) (ā ī) (ā j) (ā k) (ā l))), (614) (ā b c = d bē fḡ) ( b {ā, c, d, ē, f,ḡ}) ( f {ā, b, c, d, ē, ḡ}) (( b h) ( f ī)) (ā h c = d hēīḡ), (615) (ā b c dē = f) ( b {ā, c, d, ē, f}) ( d {ā, b, c, ē, f}) (( b ḡ) ( d h)) (āḡ c hē = f), (616) a 1 [aba], (617) b +, (618) c d e f g, (619) e + + e, (620) f f, (621) g / /g, (622) (h +) (i ), (623) (h +h) (i i), (624) (i ) (h +), (625) (i i) (h +h), (626) (h +) (j /),

18 1680 Pith Xie (627) (h +h) (j /j), (628) k [1+1] [1 + k], (629) l 1 [1 + l], (630) a<[1 + a], (631) (ā < b) c (([ā + c] < [ b + c]) ([ā c] < [ b c]) ([ c b] < [ c ā])), (632) ([1 1] ā) (ā < b) ([1 1] < c) ([ā c] < [ b c]), (633) ([1 1] < ā) (ā < b) ([1 1] < c) ([ c b] < [ c ā]), (634) (1 < ā) ([1 1] < b) (1 < [ā f b]), (635) (1 < ā) (1 < b) ([1 1] < [ā/g b]), (636) (1 < ā) (ā < b) (1 < [ b/gā]), (637) (1 ā) (ā < b) ([1 1] < c) ([āe c] < [ be c]), (638) (1 ā) (1 b) ([1 1] < c) ([āe c] < [ be c]) (ā < b), (639) (1 < ā) ([1 1] b) ( b < c) ([āe b] < [āe c]), (640) (1 < ā) ([1 1] b) ([1 1] c) ([āe b] < [āe c]) ( b < c), (641) ā b ([ā b] =[ā/ b]), (642) ā ( b =[1 1]) ([ā b] =[ā// b]), (643) ā ([ā ā] =[1 1]), (644) ā b ([ā + b] =[ b +ā]), (645) ā b ([[ā b]+ b] =ā), (646) ā b c ([[ā b]+ c] = [[ā + c] b]), (647) ā b c ([ā +[ b + c]] = [[ā + b]+ c]), (648) ā b c ([ā +[ b c]] = [[ā + b] c]), (649) ā b c ([ā [ b + c]] = [[ā b] c]), (650) ā b c ([ā [ b c]] = [[ā b]+ c]), (651) ā ([ā + +1] = ā) ([ā 1] = ā), (652) (ā =[1 1]) ([ā ā] =1), (653) ā b ([ā ++ b] =[ b ++ā]), (654) ā b c ([ā ++[ b ++ c]] = [[ā ++ b]++ c]), (655) ā b c ([ā ++[ b + c]] = [[ā ++ b]+[ā ++ c]]), (656) ā b c ([ā ++[ b c]] = [[ā ++ b] [ā ++ c]]), (657) ā ( b =[1 1]) ([[ā b]++ b] =ā), (658) ā ( b =[1 1]) c (([[ā b]++ c] = [[ā ++ c] b]) ([[ā + c] b] = [[ā b]+[ c b]]) ([[ā c] b] = [[ā b] [ c b]])),

19 Number systems based on logical calculus 1681 (659) ā ( b =[1 1]) ( c =[1 1]) (([ā ++[ b c]] = [[ā ++ b] c]) ([ā [ b ++ c]] = [[ā b] c]) ([ā [ b c]] = [[ā b]++ c])), (660) ā ([ā +++1]=ā) ([ā 1]=ā), (661) ā ([1+++ā] =1), (662) (ā =[1 1]) ([ā +++[1 1]] = 1), (663) ([1 1] < ā) ([[1 1]+++ā] =[1 1]), (664) ([1 1] < ā) ( b =[1 1]) c (([[ā b]+++ b] =ā) ([[ā b]+++ c] = [[ā +++ c] b]) ([ā +++[ c b]] = [[ā +++ c] b])), (665) ([1 1] < ā) ([1 1] < b) c (([ā +++[ b///ā]] = b) ([[ā +++ c]/// b] =[ c ++[ā/// b]]) ([[ā b]+++ c] = [[ā +++ c] [ b +++ c]])), (666) ([1 1] < ā) ([1 1] < b) ([1 1] < c) (([[ā/// c] [ b/// c]] = [ā/// b]) ([[ā ++ b]/// c] = [[ā/// c]+[ b/// c]]) ([[ā b]/// c] = [[ā/// c] [ b/// c]])), (667) (ā =[1 1]) ( b =[1 1]) ( c =[1 1]) (([[ā ++ b]+++ c] = [[ā +++ c]++[ b +++ c]]) ([ā +++[ b ++ c]] = [[ā +++ b]+++ c]) ([ā +++[ b + c]] = [[ā +++ b] + +[ā +++ c]]) ([ā +++[ b c]] = [[ā +++ b] [ā +++ c]])), (668) ([1 1] < ā) ( b =[1 1]) ( c =[1 1]) (([ā [ b ++ c]] = [[ā b] c]) ([ā [ b c]] = [[ā b]+++ c])), (669) ([1 1] < ā) ([1 1] < b) ( c =[1 1]) (([[ā ++ b] c] = [[ā c]++[ b c]]) ([[ā b] c] = [[ā c] [ b c]])), (670) (1 ā) ([ā + e1] = ā), (671) (1 ā) ([ā ++e[1 1]] = 1), (672) (1 ā) ([ā f1] = ā), (673) ([1 1] ā) ([1 + +eā] =1), (674) ([1 1] < ā) ([1 fā] =1), (675) (1 < ā) ([1//gā] =[1 1]),

20 1682 Pith Xie (676) (1 < ā) ([ā/gā] =1), (677) (1 ā) ([1 1] < b) ([[āi b]h b] =ā), (678) (1 ā) ([1 1] < b) ([[āh b]i b] =ā), (679) (1 ā) (1 < b) ([ bh[āj b]] = ā), (680) (1 ā) (1 < b) ([[ bhā]j b] =ā), (681) (1 ā) (1 b) ([ā + e b] =[āe[ā + e[ b 1]]]), (682) 1 1 =[1 1], (683) 1 [1+1] =1, (684) 1 1 =1, (685) 1 [1+1] =1, (686) k [[[1 + 1] + +l] 1] = [k 1] l, (687) k [[[1 + 1] + +l] 1] = [k 1] l, (688) k [[1+1]++l] =[ [k 1] l + [k 1] [l +1] ], (689) k [[1+1]++l] =[ [k 1] l + [k 1] [l +1] ], (690) (1 ā) ( b c ) ( b c ) ([ā + h[ b c b c ]] = [[ā + h b c ] i b c ]) } It should be noted that (682) (690) restrict [ b c b c ]tobe a Farey fraction In the following, we will deduce some numbers and equalities as examples (A1) (a 1 [aba]) (a 1) by(616), (218) (A2) (a [aba]) by(616), (218) (A3) (a [1ba]) by(a2), (A1), (217) (A4) (a [1b1]) by(a3), (A1), (217) (A5) (b + ) (b ) by(617), (218) (A6) (a [1 1]) by(a4), (A5), (217) (A7) (a <[1 + a]) (1 < [1 + 1]) by(630), (A1), (229) (A8) 1 by(223) (A9) [1+1] by(a7), (223) (A10) (a <[1 + a]) ([1 1] < [1+[1 1]]) by(630), (A6), (229) (A11) ([1 1] < [[1 1] + 1]) by(a10), (644), (224) (A12) ([1 1] < 1) by(a11), (645), (224) (A13) [1 1] by(223) (A14) (b + ) (b +) by(617), (218) (A15) (a [1 + 1]) by(a4), (A14), (217)

21 Number systems based on logical calculus 1683 (A16) (a <[1 + a]) ([1 + 1] < [1 + [1 + 1]]) by(630), (A15), (229) (A17) ([1 1] < [1 + 1]) by(a12), (A7), (222) (A18) ([1 1] < [1 + [1 + 1]]) by(a17), (A16), (222) (A19) ([1 [1 + [1 + 1]]] < [[1 + 1] [1 + [1 + 1]]]) by(a12), (A7), (A18), (632) (A20) [1 [1 + [1 + 1]]] by(223) (A21) 1 < [1+[1+1]] by(a7), (A16), (222) (A22) ([1 + 1] = [1 1]) by(a17), (221) (A23) (1 < [[1 + [1 + 1]] f[1 + 1]]) by(a21), (A17), (634) (A24) (f f) (f ) by(620), (218) (A25) (f f) by(620), (218) (A26) (f ) by(a25), (A24), (217) (A27) (1 < [[1 + [1 + 1]] [1 + 1]]) by(a23), (A26), (226) (A28) [[1 + [1 + 1]] [1 + 1]] by(223) (A29) ([[1 + 1] [1 + 1]] < [[1 + [1 + 1]] [1 + 1]]) by(a17), (A16), (632) (A30) (1 < [[1 + [1 + 1]] [1 + 1]]) by(a22), (652), (225) (A31) [[1 + [1 + 1]] [1 + 1]] by(223) Then according to Definition 22, we can deduce from R{Φ, Ψ} the numbers as follows: {1, [1+1],[1 1], [1 + [1 + 1]], [1 [1 + 1]], [1 [1 + [1 + 1]]], [[1 + [1 + 1]] [1 + 1]], [[1 + [1 + 1]] [1 + 1]] } (B1) ( k [[1+1]++l] = [ [k 1] l + [k 1] [l +1] ]) by(688) (B2) ( [1+1] [[1+1]++l] = [ [[1 + 1] 1] l + [[1 + 1] 1] [l +1] ]) by(628), (218), (242) (B3) ( [1+1] [[1 + 1] + +1] = [ [[1 + 1] 1] 1 + [[1 + 1] 1] [1+1] ]) by(629), (218), (242) (B4) ( [1+1] [[1 + 1] + +1] =

22 1684 Pith Xie [ [[1 1]+1] 1 + [[1 1]+1] [1+1] ]) by(646), (237) (B5) ( [1+1] [[1 + 1] + +1] = [ [1+1] ]) by(645), (237) (B6) ( [1+1] [1+1] = [ [1+1] ]) by(651), (235), (237) (B7) ( [1+1] [1+1] = [[1 1] + 1]) by(682), (683), (237) (B8) ( [1+1] [1+1] =1) by(645), (237) (B9) ([1 1] < [1+1] [1+1] ) by(a12), (224) (B10) [1+1] [1+1] by(223) (B11) ( k [[1+1]++l] = [ [k 1] l + [k 1] [l +1] ]) by(689) (B12) ( [1+1] [[1+1]++l] = [ [[1 + 1] 1] l + [[1 + 1] 1] [l +1] ]) by(628), (218), (242) (B13) ( [1+1] [[1 + 1] + +1] = [ [[1 + 1] 1] 1 + [[1 + 1] 1] [1+1] ]) by(629), (218), (242) (B14) ( [1+1] [[1 + 1] + +1] = [ [[1 1]+1] 1 + [[1 1]+1] [1+1] ]) by(646), (237) (B15) ( [1+1] [[1 + 1] + +1] = [ [1+1] ]) by(645), (237) (B16) ( [1+1] [1+1] = [ [1+1] ]) by(651), (235), (237) (B17) ( [1+1] [1+1] = [1 + 1]) by(684), (685), (237) (B18) ([1 1] < [1+1] [1+1] ) by(a17), (224) (B19) [1+1] [1+1] by(223) (B20) ([[1 + 1] + e[[1 + [1 + 1]] [1 + 1]]] = [[1 + 1]e[[1+1]+e[[[1 + [1 + 1]] [1 + 1]] 1]]]) by(a7), (A30), (681) (B21) ([[1 + 1] + +e[[1 + [1 + 1]] [1 + 1]]] = [[1+1]+e[[1+1]++e

23 Number systems based on logical calculus 1685 [[[1 + [1 + 1]] [1 + 1]] 1]]]) by(619), (242) (B22) ([[1 + 1]+++e[[1 + [1 + 1]] [1 + 1]]] = [[1+1]++e[[1+1]+++e [[[1 + [1 + 1]] [1 + 1]] 1]]]) by(619), (242) (B23) ([[1 + 1]++++[[1 + [1 + 1]] [1 + 1]]] = [[1+1]+++[[1+1]++++ [[[1 + [1 + 1]] [1 + 1]] 1]]]) by(619), (242) (B24) ([[1 + 1]++++[[1 + [1 + 1]] [1 + 1]]] = [[1+1]+++[[1+1]++++ [[[1 [1 + 1]] + [[1 + 1] [1 + 1]]] 1]]]) by(658), (237) (B25) ([[1 + 1]++++[[1 + [1 + 1]] [1 + 1]]] = [[1+1]+++[[1+1]++++ [[[1 [1 + 1]] + 1] 1]]]) by(a22), (652), (237) (B26) ([[1 + 1]++++[[1 + [1 + 1]] [1 + 1]]] = [[1+1]+++[[1+1]++++ [[[1 [1 + 1]] 1] + 1]]]) by(646), (237) (B27) ([[1 + 1]++++[[1 + [1 + 1]] [1 + 1]]] = [[1+1]+++[[1+1]++++[1 [1 + 1]]]]) by(645), (237) (B28) ([[1 + 1] + h[ [1+1] [1+1] [1+1] [1+1] ]] = [[[1 + 1] + h [1+1] [1+1] ] i [1+1] [1+1] ]) by(a7), (B10), (B19), (690) (B29) ([[1 + 1] + h[1 [1 + 1]]] = [[[1 + 1] + h1] i[1 + 1]]) by(b8), (B17), (235), (237) (B30) ([[1 + 1] + +h[1 [1 + 1]]] = [[[1 + 1] + +h1] i[1 + 1]]) by(623), (614) (B31) ([[1 + 1]+++h[1 [1 + 1]]] = [[[1 + 1]+++h1] i[1 + 1]]) by(623), (614) (B32) ([[1 + 1]++++[1 [1 + 1]]] = [[[1 + 1]++++1] [1 + 1]]) by(622), (614) (B33) ([[1 + 1] + e1] = [1 + 1]) by(a7), (670) (B34) ([[1 + 1] + +e1] = [1 + 1]) by(619), (235),

24 1686 Pith Xie (239) (B35) ([[1 + 1]+++e1] = [1 + 1]) by(619), (235), (239) (B36) ([[1 + 1]++++1]=[1+1]) by(619), (235), (239) (B37) ([[1 + 1]++++[1 [1 + 1]]] = [[1 + 1] [1 + 1]]) by(b32), (B36), (237) (B38) ([[1 + 1]++++[[1 + [1 + 1]] [1 + 1]]] = [[1+1]+++[[1+1] [1 + 1]]]) by(b27), (B37), (237) Then we can deduce from R{Φ, Ψ} the equalities as follows: [[1 + 1] + +[[1 + 1] [1 + 1]]] = [[[1 + 1] [1 + 1]] + +[1 + 1]], [[1+1]++++[[1 + [1 + 1]] [1 + 1]]] = [[1+1]+++[[1+1] [1 + 1]]], The deduced numbers correspond to real numbers as follows: [[1 1] [[1 + [1 + 1]] [1 + 1]]],, [[1 1] [[1 + 1] [1 + 1]]] 2 2,, [[1 1] [[1 + 1] [1 + [1 + 1]]]] 3 2,, [[1 1] 1] 1,, [[1 1] [1 [1 + 1]]] 1 2,, [1 1] 0,

25 Number systems based on logical calculus 1687, [1 [1 + 1]] 1 2,, [[1 + 1]///[1 + [1 + 1]]] log 3 2,, 1 1,, [[1 + 1] [1 + [1 + 1]]] 3 2,, [[1 + 1] [1 + 1]] 2 2,, [1+[1 [1 + 1]]] 3 2,, [[1 + [1 + 1]]///[1 + 1]] log 2 3, [[1 + [1 + 1]] [1 + 1]],, [1+1] 2,, [[1+1]+[1 [1 + 1]]] 5 2,, [1+[1+1]] 3, [[[1 + [1 + 1]] [1 + 1]] + +[1 + 1]], The equalities on deduced numbers correspond to addition, subtraction, multiplication, division, exponentiation operation, root-extraction operation, logarithm operation and more other operations in real number system These deduced numbers hold the consist order relation and equivalence relation, so the logical calculus R{Φ, Ψ} is a consist axiom Since Q{Φ, Ψ} is dense in R{Φ, Ψ}(with its standard topology), every de-

26 1688 Pith Xie duced number has rational numbers arbitrary close to it So every deduced number holds only one position in number line Note that in the correspondence above, some deduced numbers such as [[1 1] [[1 + [1 + 1]] [1 + 1]]], [[1 + [1 + 1]] [1 + 1]] and [[[1 + [1 + 1]] [1 + 1]] + +[1 + 1]] do not correspond to any known real numbers So the logical calculus R{Φ, Ψ} can deduce more real numbers than before In fact, the logical calculus R{Φ, Ψ} not only deduces more real numbers than before, but also makes its deduced numbers join in algebraical operations So the logical calculus R{Φ, Ψ} intuitively and logically denote real number system 7 Operator In conclusion, the number systems by logical calculus forms the new arithmetic axioms Since operators distinguish all number systems, we can furtherly define new operators according to the definitions of number systems, as is shown in Table 2 In each row, the column Number System stands for the name of number system, the column Operator stands for the name of operators, the column Form stands for the form of the operators Table 2: Definitions Of New Operators Number System Operator Form natural number system natural operator + integral number system integral operators +, rational number system rational operators +, ++,, real number system real operators +, ++, +++, ++++,,,,,,, /, //, ///, ////, To distinguish among real number systems in history, we name the real number system by logical calculus after Operator Axiom While producing new real numbers in arithmetic, the new operators in Operator Axiom certainly produce new equations and inequalities in algebra So Operator Axiom not only extends real number system, but also extends equations and inequalities References [1] JH Conway, On Numbers and Games, Second Edition, A K Peters, 2001 [2] DE Knuth, Surreal Numbers, Addison-Wesley, 1974

27 Number systems based on logical calculus 1689 [3] H Gonshor, An Introduction to the Theory of Surreal Numbers, Cambridge University Press, 1986 [4] P Xie, A logical calculus to intuitively and logically denote number systems, Progress in Applied Mathematics, Vol1, No2, (2011), [5] P Linz, An Introduction to Formal Languages and Automata, Third Edition, Jones and Bartlett Publishers, Inc, 2001 [6] A Nerode and RA Shore, Logic for Application, Second Edition, Springer- Verlag New York, Inc, 1997 [7] HB Enderton, A Mathematical Introduction to Logic, Second Edition, Elsevier, 2001 [8] H-D Ebbinghaus, J Flum and W Thomas, Mathematical Logic, Second Edition, Springer Science+Business Media, Inc, 1994 [9] C Haros, Tables pour evaluer une fraction ordinaire avec autand de decimals qu on voudra; et pour trover la fraction ordinaire la plus simple, et qui a approche sensiblement d une fraction decimale, J Ecole Polytechn 4 (1802), Received: September 1, 2013