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A note on ordinal numbers and rings of formal power series. (English) Zbl 0808.03042

In “Ordinal numbers and the Hilbert basis theorem” [J. Symb. Log. 53, No. 3, 961-974 (1988; Zbl 0661.03046)], S. G. Simpson has shown that over \(\text{RCA}_ 0\), for any or all countable fields \(K\), a formal version of Hilbert basis theorem is equivalent to the assertion that the ordinal number \(\omega^ \omega\) is well ordered. It is well known that there is a basis theorem for rings of formal power series whose statement is: “Let \(R\) be a commutative ring all of whose ideals are finitely generated. Then, all ideals of the commutative ring of formal power series with coefficients from \(R\) are also finitely generated.” In this paper we establish that \(\omega^ \omega\) also “measures” the “intrinsic logical strength” of a version of this assertion formalized in second-order arithmetic and in which the ring of coefficients can be any countable field.

MSC:

03F35 Second- and higher-order arithmetic and fragments
03B30 Foundations of classical theories (including reverse mathematics)

Citations:

Zbl 0661.03046
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References:

[1] Brown, D., Simpson, S.G.: Which set existence axioms are needed to prove the separable Hahn-Banach theorem? Ann. Pure Appl. Logic,31, 123-144 (1986) · Zbl 0615.03044 · doi:10.1016/0168-0072(86)90066-7
[2] Chevalley, C.: On the theory of local rings. Ann. Math.44, 690-708 (1943) · Zbl 0060.06908 · doi:10.2307/1969105
[3] Simpson, S.G.: Ordinal numbers and the Hilbert basis theorem. J. Symb. Logic53, 961-974 (1988) · Zbl 0661.03046 · doi:10.2307/2274585
[4] Simpson, S.G.: Which set existence axioms are needed to prove the Cauchy/Peano theorem for ordinary differential equations? J. Symb. Logic49, 783-802 (1984) · Zbl 0584.03039 · doi:10.2307/2274131
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