S. G. Simpson. Subsystems of Second-Order Arithmetic. Springer-Verlag, 1999.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Marsden Fund of New Zealand. The third author s research was partially supported by NSF grant DMS 9732526. We assume the reader is familiar with the basics of reverse mathematics and computability theory, including the method of organizing priority constructions on a tree. Standard references are [2] and [3] respectively. Although the motivation for our main result comes from reverse mathematics, its proof is completely computability theoretic. We begin with a few de nitions. De nition. X] fY X j jY j = 2g. A 2 coloring of [N ] is a function from [N ] into f0; 1g. ....

S. G. Simpson, Subsystems of Second Order Arithmetic, Perspect. Math. Logic (Springer{Verlag, Berlin, 1999).

....defection to Brouwer, stood H. Weyl who was inspired by the predicativism of H. Poincar e who accepted PA as true. Weyl in his seminal work Das Kontinuum [19, 4] was a forerunner of a ourishing and aesthetically pleasing school in the modern foundations of mathematics known as Reverse mathematics [15]. Weyl s formal system can be identi ed with the basic theory of Reverse mathematics, a subsystem of second order arithmetic ACA 0 which is conservative over PA. As it happens, HOL is based on a polymorphic (i.e. admitting variable types) extension of the system of typed lambda calculus of ....

....functions are called parameterized modules in computer programming. For reasons of modularity we have decided that the next version of CL will use the theory of nite sets FSI (see Sect. 3) which is a formal theory isomorphic to ACA 0 . We have contemplated a weaker second order system RCA 0 (see [15]) whose comprehension axioms allow to form at most 1 classes (graphs of all recursive functions are such) RCA 0 is conservative over PRA. At the end we have decided that a system like ACA 0 is easier to work with because of its more liberal comprehension axioms. ACA 0 is a vastly powerful ....

S. Simpson, Subsystems of Second Order Arithmetic, Springer, 1999.

....it remains a matter of investigation what partial realizations of Hilbert s program may still be vindicated. This line of research was forcefully articulated by Stephen Simpson in [Sim88] and can be viewed as a sub program of Simpson and Harvey Friedman s wider program of Reverse Mathematics (see [Sim99]) It is against this background that it is important to study formal systems of arithmetic that are (finitistically) conservative over primitive recursive arithmetic. Plainly, the parts of mathematics able to be carried out in these systems constitute partial realizations of Hilbert s original ....

....van Dalen [vDT88] Their presentation is of a piece with the original presentations of Skolem and Hilbert Bernays, in that it is framed in a quantifier free calculus. Nevertheless, we opt for a framework based on a first order language with equality, as expounded in section IX.3 of Simpson s book [Sim99]. In the sequel, PRA is such system: It is a first order universal theory (i.e. axiomatized by purely universal formulas) with a function symbol for each (description of a) primitive recursive function, and in which the principle of induction for quantifier free formulas holds. By Herbrand s ....

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Stephen Simpson. Subsystems of Second-Order Arithmetic. Perspectives in Mathematical Logic. Springer-Verlag, 1999.

....fails because there may be no algorithm for F which decides if a polynomial is irreducible. On the other hand, the answer to the second question is no, and in fact every infinite independent set in V can be as complex as the halting problem. See Metakides and Nerode [115, 116, 117] and Simpson [167]. Rabin s Theorem is interesting since it demonstrates that algorithmic considerations can generate new structure theory needed to perform algorithmic tasks. Combinatorial group was generated by algorithmic considerations such as the word problem of Dehn [30] and yet these considerations have ....

Simpson, S. Subsystems of Second Order Arithmetic, in preparation.

.... # 1 formula A there exists an arithmetic formula BA which contains the free variables of A plus two fresh variables w 0 and w 1 so that: A( # U , # U , # F , #v, G(S(x) G(x) Proof The proof of this theorem is more or less basic and can be found at many places (for example in Simpson [14]) # We are now ready for the preparation of our next embedding. Note that we can use the translation # from definition 3.2.1 and definition 3.2.2. In addition, the translation A N for every 2 formula A is the same as in definition 3.2.3. Lemma 3.3.2 (Characteristic term II) Let n be a ....

Simpson, S. G. Subsystems of Second Order Arithmetic. Perspectives in Mathematical Logic. Springer-Verlag, 1998.

....of ordinary mathematics. In this article, we consider Hahn s Theorem, one of the central results of ordered abelian group theory. This article is self contained with respect to the material on ordered groups (see Section 2) but the reader who is unfamiliar with reverse mathematics is referred to [8] or [3] for more background in this area. The work in this paper continues a line of inquiry into the computational and proof theoretic properties of ordered abelian groups started in [2] and continued in [5] 9] and [11] Downey and Kurtz began this study in [2] by showing that the e ective ....

....many of the arguments presented here. ACA 0 consists of RCA 0 plus the arithmetical comprehension scheme. Every model of ACA 0 is closed under the Turing jump, and the arithmetic sets form the minimum model of ACA 0 . When proving reversals, we will use the following well known result (see [8], Lemma III.1.3) Theorem 1.3. RCA 0 ) The following are equivalent. 1. ACA 0 . 2. The range of every one to one function exists. Given the characterizations of the models of RCA 0 and ACA 0 in terms of Turing degrees, it is not surprising that equivalences in reverse mathematics have ....

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S.G. SIMPSON, Subsystems of second order arithmetic, Springer-Verlag, 1999.

....(cf. Sm1] page 864) However, such proofs are fomalizable within a subsystem of second order arithmetic WKL 0 ( KT] and it turns out that such proofs give proofs of the formalized second incompleteness theorem by a theorem of H. Friedman that any 5 2 theorem of WKL 0 is provable in PRA (cf. [Si]) 2. Modal logic. A realization of a sentence OE in a language of propositional modal logic with one modal operator 3 to a sentence OE in LA is obtained by replacing propositional variables by sentences in LA and the modal operator by a provability predicate. Solovay proved in 1976 that OE is ....

Simpson, S.G., Subsystems of Second-Order Arithmetic, forthcoming.

....there exists a countable c model of the theory ACA0 containing these parameters so that A is true in this model. In [1] it is also announced that the principle of c model reflection is equivalent to the principle of bar induction. A detailed proof of this important result is given in Simpson [4]. Friedman s result provides the equivalence of the full scheme of c model reflection and the full scheme of bar induction. It is of course a very natural question to ask how much c model reflection corresponds to how much of bar induction. Building upon work of Friedman and Howard, Simpson [3] ....

....for each natural number n 1. All these theories comprise the standard system ACA0 of second order arithmetic which includes comprehension for arithmetic formulas and induction on the natural numbers for sets. It is well known that ACA0 is finitely axiomatizable by a II 1 sentence, say FACA, cf. [4]. The schema of IIl bar induction comprises for each IIl formula A the statement (nl B0 (VX) WO(X) TI(X,A) The theory II1 BI0 extends ACAo by each instance of (IIi BI) Observe that induction on the natural numbers is available in ACA0 q (IIi BI) at least) for IIl (and hence also for 1) ....

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SIMPSON, S. G. Subsystems of Second Order Arithmetic. Perspectives in Mathematical Logic. Springer-Verlag, 1998.

....strong but proof theoretically weak subsystems of analysis. In particular the fragment (WKL 0 ) of second order arithmetic which is based on recursive comprehension (with set parameters) 1 induction (with set parameters) and WKL occurs prominently in the context of reverse mathematics (see [16]) Although (WKL 0 ) allows to carry out a great deal of classical mathematics, it 2 conservative over primitive recursive arithmetic PRA as was shown rst by H. Friedman using a model theoretic argument. In [15] a proof theoretic argument is given for a variant of (WKL 0 ) which uses function ....

Simpson, S.G., Subsystems of Second Order Arithmetic. Perspectives in Mathematical Logic. Springer-Verlag. xiv+445 pp. 1999.

.... In a subsequent paper we will illustrate the greater mathematical strength of these principles (compared to WKL) 1 Introduction The so called weak K onig s lemma WKL is of crucial signi cance in the study of on the other hand have a low proof theoretic and computational strength (see e.g. [15], 10] The pre x weak has a twofold meaning: the full statement of K onig s lemma is restricted in the formulation of WKL in two ways 1) instead of allowing arbitrary nitely branching trees we only have binary trees in WKL (note however that it wouldn t make a di erence if we would allow ....

.... has a twofold meaning: the full statement of K onig s lemma is restricted in the formulation of WKL in two ways 1) instead of allowing arbitrary nitely branching trees we only have binary trees in WKL (note however that it wouldn t make a di erence if we would allow bounded trees in the sense of [15]) 2) the tree is represented by a function f and consequently the tree predicate f(n) 0 expressing that n is the code of a nite branch in the tree represented by f is quanti er free. In view of 2) WKL could be denoted by QF WKL, where QF refers to quanti erfree . It is known that the ....

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Simpson, S.G., Subsystems of Second Order Arithmetic. Perspectives in Mathematical Logic. Springer-Verlag. xiv+445 pp. 1999.

....using the reverse direction explicitly) that substantial parts of mathematics (and in particular analysis) can be carried out in systems T which are conservative Basic Research in Computer Science, Centre of the Danish National Research Foundation. over primitive recursive arithmetic PRA (see [25] for a systematic account) This is of interest for mainly two reasons 1) If a 2 sentence A is provable in T and the conservation of T over PRA has been established proof theoretically, then one can extract a primitive recursive program which realizes A from a given proof. Typically the ....

.... what has been called nitistic reasoning (see e.g. 26] If the conservation of T over PRA has been established nitistically (which is possible for mathematically strong systems T (see [22] 8] then all the mathematics which can be carried out in T has a nitistic justi cation (see [24] [25] for a discussion of this) In this paper we exhibit a sharp boundary between nistically reducible parts of analysis and extensions which provably go beyond the strength of PRA. More precisely we study the (proof theoretical and numerical) strength of function parameterfree schematic forms of ....

Simpson, S.G., Subsystems of Second Order Arithmetic. Perspectives in Mathematical Logic, Springer-Verlag. To appear.

.... order arithmetic) of the system from [5] which is formulated in the language of functionals of all nite types) Via appropriate representations and codings of higher objects (like continuous functions between Polish spaces) a great deal of mathematics can be developed already in ACA 0 (see [34] for a comprehensive treatment) Feferman s system, however, allows a more direct treatment of such objects and their mathematics and also contains a strong uniform ( explicit ) version of arithmetical comprehension via a non constructive operator. These features hold in an even stronger form ....

....H. Friedman observed that large parts of the mathematics that can be carried out in ACA 0 are already formalizable in a subsystem WKL 0 which instead of the schema of arithmetical comprehension is based on the binary K onig s lemma (for quanti er free trees) and 1 induction only (see again [34] for a comprehensive treatment of ordinary mathematics in WKL 0 ) This fact is of foundational relevance since WKL 0 can be proof theoretically reduced to and is PRA (H. Friedman (1976, unpublished) and [32] for a historical discussion which in particular points out various errors in the ....

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Simpson, S.G., Subsystems of Second Order Arithmetic. Perspectives in Mathematical Logic. Springer-Verlag. xiv+445 pp. 1999. 33

.... is highly non constructive relying both on classical logic and non computational analytical principles which correspond in logical terminology to the so called binary ( weak ) Konig s lemma, a principle which has received considerable attention in various parts of logic in recent years (see [25]) In this paper we carry out a complete logical analysis of Cheney s proof and show how the explicit modulus mentioned above can be extracted from this seemingly hopelessly non constructive proof. Consequently, our result, like Cheney s proof, does not require any measure theory. The main result ....

....analytical terms correspond to applications of Heine Borel compactness of e.g. 0, 1] d . In logical terms, these principles correspond to the so called binary ( weak ) Konig s lemma WKL which su#ces to derive a substantial amount of mathematics relative to weak fragments of arithmetic (see [25]) 6 In this paper the only genuine analytical tool # (which goes beyond E PA # QF AC) is the attainment of the minimum of f # C[0, 1] #) #f # C[0, 1]#x # [0, 1] # f(x) inf y#[0,1] f(y) # . 4 Note that this notion used also in constructive mathematics and computable and ....

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S.G. Simpson. Subsystems of Second Order Arithmetic. Perspectives in Mathematical Logic. Springer-Verlag, 1999.

....(CH) in V , Cohen s proof of the independence of CH from ZFC, with all the information about models of set theory that came with the proof, is a huge step toward understanding why CH has not yet been settled in V . Likewise, the study of Henkin models of second order artihmetic (see e.g. [Sim99]) isolate reasons why some results of number theory or analysis are hard to prove. 2 Preliminary example Mathematicians argue exactly but informally. This has worked well for centuries. However, if we want to understand the way mathematicians argue, it is necessary to formalize basic concepts ....

Stephen G. Simpson. Subsystems of second order arithmetic. Springer-Verlag, Berlin, 1999.

....that the Bolzano Weierstrass theorem is logically equivalent to the axioms of ACA 0 , the equivalence being proved in the weaker system RCA 0 . For a survey of subsystems of second order arithmetic and their role in foundational studies, see my article [16] A fuller treatment will appear in [17]. For additional results and open problems concerning logical and foundational aspects of combinatorics, see the articles in Logic and Combinatorics [8] especially [3] Aharoni, Magidor and Shore [2] made a major contribution to the foundational program of [16] They obtained two important ....

....background material concerning ATR 0 and related systems. We present little more than what is needed for our main result, the provability of CKDT in ATR 0 . For a broad survey of subsystems of second order arithmetic, see [16] For detailed information on ATR 0 , see [6] 14] 15] 16] and [17]. All of the systems which we shall consider are first order theories in the language of second order arithmetic. This is a first order language with two sorts of variables: number variables i, j, k, m, n, and set variables U , V , W , X, Y , Z, Number variables are intended to ....

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S. G. Simpson, Subsystems of Second Order Arithmetic, in preparation.

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S. G. Simpson. Subsystems of Second-Order Arithmetic. Springer-Verlag, 1999.

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S. G. Simpson. Subsystems of Second Order Arithmetic. Perspectives in Mathematical Logic. Springer, Berlin, 1999.

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Stephen G. Simpson, Subsystems of second order arithmetic, Berlin 1999 [Perspectives in Mathematical Logic]

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Stephen Simpson. Subsystems of Second-Order Arithmetic. Springer, Berlin, 1998.

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Stephen G. Simpson, Subsystems of Second Order Arithmetic, Springer-Verlag, Berlin, 1999.

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Stephen G. Simpson, Subsystems of second order arithmetic, Springer-Verlag, 1998. 1, 1, 5, 5 30

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S. G. Simpson. Subsystems of Second Order Arithmetic. Perspectives in Mathematical Logic. Springer-Verlag, 1999.

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S. Simpson, Subsystems of Second Order Arithmetic, Springer, 1999.

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S. G. Simpson, Subsystems of second order arithmetic, Perspectives in Mathematical Logic, Springer, Berlin, 1999.

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S.G. Simpson (1999). Subsystems of Second Order Arithmetic. Springer.

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