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ORDERINGS OF MONOMIAL IDEALS
, 2003
"... We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular ..."
Abstract

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We study the set of monomial ideals in a polynomial ring as an ordered set, with the ordering given by reverse inclusion. We give a short proof of the fact that every antichain of monomial ideals is finite. Then we investigate ordinal invariants for the complexity of this ordered set. In particular, we give an interpretation of the height function in terms of the HilbertSamuel polynomial, and we compute upper and lower bounds on the maximal order type.
History of Constructivism in the 20th Century
"... notions, such as `constructive proof', `arbitrary numbertheoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifierfree statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented by providi ..."
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notions, such as `constructive proof', `arbitrary numbertheoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifierfree statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented by providing an x which satisfies A. Establishing :8xAx finitistically means: providing a particular x such that Ax is false. In this century, T. Skolem 4 was the first to contribute substantially to finitist 4 Thoralf Skolem 18871963 History of constructivism in the 20th century 3 mathematics; he showed that a fair part of arithmetic could be developed in a calculus without bound variables, and with induction over quantifierfree expressions only. Introduction of functions by primitive recursion is freely allowed (Skolem 1923). Skolem does not present his results in a formal context, nor does he try to delimit precisely the extent of finitist reasoning. Since the idea of finitist reasoning ...