On the value group of a model of Peano arithmetic. (English) Zbl 1375.03039
In this paper, the authors investigate real closed fields (that is, ordered fields which are elementarily equivalent to the real numbers), looking in particular at the existence of total exponential functions on the integer part of the field. An integer part of a real closed field \(K\) is a discretely ordered subring \(Z\) such that for each \(r\in K\), there exists some \(z\in Z\) with \(z \leq r \leq z+1\). Moreover, there is a natural valuation on \(K\), in which the valuation ring \(R_v\) is the convex hull of \(\mathbb Z\) (the usual integers), and \(R_v\cap Z = \mathbb Z\).
If the field \(K\) is viewed as a structure in the language \((+,-,\cdot,0,1,<)\) of ordered rings, then \(Z\) is a substructure of \(K\), and one can ask about the theory of \(Z\). The main focus of the paper, as the title suggests, is on the case where \(Z\) is a model of Peano arithmetic (PA), although the authors do consider several other theories. In all of the considered theories, there is a formula \(E(x,y)\) which stands for \(2^x = y\). This formula satisfies many of properties one would expect from an exponential function, including “\(2^{a+b} = 2^a\cdot 2^b\)” and “for all \(x\in Z\), there exists \(y\) such that \(2^y \leq x < 2^{y+1}\).”
The paper uses these properties about exponentiation on \(Z\) to determine when a real closed field satisfies related properties about exponentiation. Specifically, let \(K\) be a non-Archimidean real closed field whose integer part \(Z\) is a model of PA (hereafter referred to as “admitting an IPA”). The main theorem concludes that:
The paper is very well written, and the content contained in the paper is accessible for a reader with a small amount of familiarity with either real closed fields or valued fields. The only background required about models of arithmetic is an intuitive understanding of the structure of the natural numbers. Some of the proofs are referenced to other papers: most notably, part (2) of the main theorem follows from part (1) and two results from [S. Kuhlmann, Ordered exponential fields. Providence, RI: American Mathematical Society (AMS) (2000; Zbl 0989.12003)]. The final construction uses a small amount of computability theory, but the main results can be understood without a strong background in computability.
If the field \(K\) is viewed as a structure in the language \((+,-,\cdot,0,1,<)\) of ordered rings, then \(Z\) is a substructure of \(K\), and one can ask about the theory of \(Z\). The main focus of the paper, as the title suggests, is on the case where \(Z\) is a model of Peano arithmetic (PA), although the authors do consider several other theories. In all of the considered theories, there is a formula \(E(x,y)\) which stands for \(2^x = y\). This formula satisfies many of properties one would expect from an exponential function, including “\(2^{a+b} = 2^a\cdot 2^b\)” and “for all \(x\in Z\), there exists \(y\) such that \(2^y \leq x < 2^{y+1}\).”
The paper uses these properties about exponentiation on \(Z\) to determine when a real closed field satisfies related properties about exponentiation. Specifically, let \(K\) be a non-Archimidean real closed field whose integer part \(Z\) is a model of PA (hereafter referred to as “admitting an IPA”). The main theorem concludes that:
- 1.
- \(K\) admits left exponentiation: if \(A\) is a vector space complement of \(R_v\) in \((K,+)\) and \(B\) is a vector space complement of \(\{a\in K^{>0} : v(a)=0\}\) in \((K,\cdot)\) then there is an order-preserving group isomorphim from \(A\) onto \(B\).
- 2.
- The value group of \(K\) is an exponential group in the additive group of the residue field: if \((G, +, -, 0, <)\) is the value group and \((\overline K, +, -, 0, <)\) is the additive group of the residue field, then the rank of \(G\) is isomorphic to the negative cone \(G^{<0}\) and the Archimidean components of \(G\) are all isomorphic (as ordered groups) to \(\overline K\).
The paper is very well written, and the content contained in the paper is accessible for a reader with a small amount of familiarity with either real closed fields or valued fields. The only background required about models of arithmetic is an intuitive understanding of the structure of the natural numbers. Some of the proofs are referenced to other papers: most notably, part (2) of the main theorem follows from part (1) and two results from [S. Kuhlmann, Ordered exponential fields. Providence, RI: American Mathematical Society (AMS) (2000; Zbl 0989.12003)]. The final construction uses a small amount of computability theory, but the main results can be understood without a strong background in computability.
Reviewer: Peter Sinclair (Hamilton)
MSC:
03C60 | Model-theoretic algebra |
03C64 | Model theory of ordered structures; o-minimality |
06A05 | Total orders |
12J10 | Valued fields |
12J15 | Ordered fields |
12L12 | Model theory of fields |
13A18 | Valuations and their generalizations for commutative rings |
12F05 | Algebraic field extensions |
12F10 | Separable extensions, Galois theory |
12F20 | Transcendental field extensions |
Keywords:
left exponentiation; natural valuation; value group; residue field; valuation rank; power series fields; maximally valued fields; ordered fields; real closed fields; integer parts; Peano arithmetic; recursive saturationCitations:
Zbl 0989.12003References:
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