# [FOM] Arithmetic Axiom Systems for annotations

zhangyinsheng zhangyinsheng at istic.ac.cn
Wed May 15 10:09:44 EDT 2013

```Dear Sir.,

The following list of Arithmetic Axiom Systems is proposed by FOM without annotations. I tried to give some annotations yet 9 QUESTIONS left for your answers (and if any annotation wrong you are welcome to correct it)!

Godel Hierarchy
(Arithmetic Axiom Systems,whose consistensy is stronger in top-down)

(1) PFA:Proper Forcing Axiom.
(2) EFA: Elementary Functional Arithmetic,which removes induction from PA,so exponentiation
must be added as "Ax (x ^ y' )=x ^y multi x".[1]
(3) SEFA:Superexponential Function Arithmetic.
(4) PRA:Primitive Recursive Arithmetic.
(5) RCA_0:Recursive Comprehension Axiom,subscript "0" indicates restrict induction[2]
"AxEy phi(x,y) implies Ef Ax phi(x,phi(x))".[3]
(6) I sigma _2: Induction of sigma _2, sub "2" indicates one occurance of E-A in a Skolem prenex form of a formulas.
(7) I sigma _3.
(8) PA:Peano Arithmetic.
(9) ACA _0:Arithmetical Comprehension Axiom,the subscript 0 indicates that it includes only
a restricted portion of the full second-order induction scheme. [2,pp.6-9]
ACA=KL_0+ACA
KL_0: Koenig Lemma. [3, pp.114]
(10) ACA _0 + (An,X)(TJ(n,X)):ACA_ 0 + "for all n,X, the n-th Turing jump of X exists".     Turing jump TJ(X)is complete recuisively enumaerable set relative to X.
Turing jump TJ(n,X)is recuisively defined by TJ(0,X)=X,TJ(n+1,X)=TJ(TJ(n,X)).[2,pp.6-9]
(11) ACA:Arithmetical Comprehension Axiom.
(12) RCA _0 + TJ(omega): Omega is the set of natural numbers.
(13) ACA _0 + TJ(omega).
(14) ACA + TJ(omega).
(15) ACA _ 0 + (Ax)(TJ(omega,x)).
(16) ACA _ 0 + {(alpha,x)(TJ(alpha,x)):alpha < omega ^omega }.
(17) ACA _ 0 + {(alpha<omega^omega )(Ax) (TJ(alpha,x))}.
(18) RCA _ 0 + TJ(omega ^omega ).
(19) ACA _ 0 + TJ(omega ^omega ).
(20) ACA _ 0 + {(Ax)(TJ(omega ^omega,x))}.
(21) ACA _0 + {(Ax)(TJ(alpha,x)):alpha < OE_0}. (QUESTION 1: What does "OE _0" mean?)
(22) Delta _ 1:Bounded Arithmetics that comprehension scheme consists of the formulas of the form An(phi(n) double-direction implies psi(n)) implies EX An(n?X double-direction
implies phi(n))),where phi(n) is the form of An phi(n) and psi(n) is of En phi (n) [2,pp.25]
(23) Sub 1-CA:  (QUESTION 2: What does "sub 1" mean ?)
(24) RCA _ 0 + TJ(OE _0 ).
(25) ACA _ 0 + TJ(OE _0 ).
(26) ACA + TJ(OE _ 0 ).
(27) ACA _0 + (Ax)(TJ(OE_0 ,x)).
(28) {ATI(alpha):alpha < Gamma _0 }:Gamma _0 is the smallest impredicative  ordinal.(Bill Tait in FOM)
(29) ATR_0.Arithmetical transfinite recursion,"sub 0" indicates restricted induction.
(30) ATI( < gamma _0 ).
(31) ATR:Arithmetical Transfinite Recursion.
(32) Pi _ 1.
(33) sub-2-TI_0 : (QUESTION 3:What do "sub 2" and "sub 0" mean ? )
(32) Pi _1.
(34) sub 2-TI.
(35) TI: Transfinite Induction
(36) ID _2 :
ID_1: theory of inductive definitions (first level). This has the same strength as the Kleene-Vesley system FIM.
This is what Wim Veldman used for his proof of the Kruskal theorem. ID2, ID3, etc.:
All of these count as predicative in type theory.(Bas in FOM)
(37) ID _< omega.
(32) Pi _1.
(38)sub 1-CA _0:
(32) Pi _1.
(39) sub 1-CA
(32) Pi _1.
(40)sub 1-CA+TI
(32) Pi _1.
(41) ) sub 1-TR sub 0 : (QUESTION 4:What is _1-TR_0 ?)
(32) Pi _1.
(42) sub 1-TR.
(32) Pi_1.
(43) sub 2-CA_0.
(32) Pi_1.
(44) sub 2-CA+TI.
(45) Z_ 2: Second-Order Arithmetic.
(46) Z _3 (QUESTION 5:What does "_ 3" mean ?)
(47) Type Theory.
(48) Weak Zermelo.
(49) ZC:Zermelo+Choice Axiom
(50) ZC + (A alpha < omega _ 1)(V(alpha)) :omega _ 1 is the first uncountable ordinal.  alpha is any ordinals.[5. pp.5]
V is the set-theoretic universe which is viewed as the cumulative hierarchy, open ended and under-determined by the set-theoritic axioms,and inviting further postulations based on reflection and generalization. [6.pp.XVII]
(51) KP(#): Kripke Platek set theory.The theory KP is an elementary first order theory in the vocabulary {epsilon}.
It is a weakening of ZF set theory where the power set axiom is removed, and the seperation and collection axiom acheme are restricted to "delta _ 0" formulas.
The "delta _ 0" formulas are the mmbers of the smallest class of formulas that contains the atomic formulas in the vocabulary {epsilon} and is closed under finite conjunction and
disjunction,bound quantifiers (Ex epsilon u) and (Ax epsilon u),and negation.[7]
(QUESTION 6:What does the Sharp symbol mean ?)
(52) ZFC: Zermelo–Fraenkel +Choice Axiom.
(53) ZFC + strongly inaccesible.
(54) ZFC + strongly Mahlo:there exists an Mahlo cardinal.
(55) ZFC + {strongly n-Mahlo: n < omega }:there exists an n-Mahlo cardinal,n < omega.
(56) ZFC + (An < omega )(strongly n-Mahlo).
(57) ZFC + (weakly compact).
(58) ZFC + (indescribable).
(59) ZFC + (subtle):ZFC + "there exists a subtle cardinal."
(60) ZFC + (almost ineffable):ZFC + "there exists an almost ineffable cardinal".
(61) ZFC + (ineffable).
(62) ZFC + {n-subtle: n < omega}
(63) ZFC + (An < omega)(n-subtle).
(64) ZFC + k emptyset omega :(QUESTION 7:What does  "k emptyset omega" mean ?)
(65) ZFC + (A alpha < omega_1)(k emptyset alpha ).
(66) ZFC + 0 # ."ZFC + 0 sharp "is a subset of omega satisfying Gaifman Theorem [8,pp.99]
(67) ZFC + (Ax pi omega)(x #):ZFC + "for all x containing omega, x sharp exists."(QUESTION 8:What does # mean ?)
(68) ZFC + k emptyset omega _1.(Question 8:What does "+ k emptyset omega_1" mean?)
(69) ZFC + Ramsey.
(70) ZFC + Measurable:ZFC + "there exists a measurable cardinal".
(71) ZFC + Concentrating Measurable.
(72) ZFC + Strong.
(73) ZFC + Woodin.
(74) ZFC + Superstrong.
(75) ZFC + Supercompact.
(76) ZFC + Extendible.
(77) ZFC + Vopenka.
(78) ZFC + Almost Huge.
(79) ZFC + Huge.
(80) ZFC + Superhuge.
(81) ZFC + (An < omega)(n-huge).
(82) ZFC + Rank into Itself.
(83) ZFC + Rank + 1 into Itself.
(84) VB + V into V. (Question 9:What does it mean?)

Reference:
[1]http://planetmath.org/ElementaryFunctionalArithmetic
[2] Stephen G.Simpson. "Subsystems of Second Order Arithmetic",https://www.math.psu.edu/simpson/sosoa/chapter1.pdf .
[3]H.Jerome Keisler.Nonstandard arithmetic and reverse mathematics.The Bulletine of Symbolic,Volume 12,Number 1,2006.
[4]Stephen G.Simpson.Friedman's research on subsystems of second order arithmetic, Harvey Friedman's research on the Foundations of Mathematics, Elsevier Science Publishers B.V.1985pp.137-160
[5] H.Jerome Keisler and Julia Knight.Barwise:infinitary logic and admissible sets,The Bulletine of Symbolic,Volume 10,Number 1,2004.
[6] Akihiro Kanamori.The Higher Infinite,Large Cardinals in Set Theory from Their Beginnings,Second Edition,`1994,Springer-Verlag.
[7]H.Jerome Keisler and Julia F.Knight.Barwise:infinitary logic and admissible sets,Bulletin of Symbolic Volume 10.Number 1,March 2004.pp.14
[8] Gaifman. Measurable cardinals and constructible sets (abstract).NAMS 11(1964),771.XX.99

Zhang Yinsheng,Ph.D

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