About: Fundamental sequence

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A fundamental sequence (FS) is an important concept in the study of ordinal hierarchies. If \(\alpha\) is a countable limit ordinal, a fundamental sequence for \(\alpha\) is a monotonically increasing sequence of length \(\omega\) consisting of ordinals, supremum of which is equal to \(\alpha\). Due to poor standardization in set theory, definitions of valid FS's vary. Some authors use "least strict upper bound" instead of "supremum," some relax the monotonicity condition to only require nondecreasing sequences, and some even allow fundamental sequences for successor ordinals.

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rdfs:label	Fundamental sequence
rdfs:comment	A fundamental sequence (FS) is an important concept in the study of ordinal hierarchies. If \(\alpha\) is a countable limit ordinal, a fundamental sequence for \(\alpha\) is a monotonically increasing sequence of length \(\omega\) consisting of ordinals, supremum of which is equal to \(\alpha\). Due to poor standardization in set theory, definitions of valid FS's vary. Some authors use "least strict upper bound" instead of "supremum," some relax the monotonicity condition to only require nondecreasing sequences, and some even allow fundamental sequences for successor ordinals.
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abstract	A fundamental sequence (FS) is an important concept in the study of ordinal hierarchies. If \(\alpha\) is a countable limit ordinal, a fundamental sequence for \(\alpha\) is a monotonically increasing sequence of length \(\omega\) consisting of ordinals, supremum of which is equal to \(\alpha\). Due to poor standardization in set theory, definitions of valid FS's vary. Some authors use "least strict upper bound" instead of "supremum," some relax the monotonicity condition to only require nondecreasing sequences, and some even allow fundamental sequences for successor ordinals. Typically when we speak of FS's we refer to systems of FS's that generate these sequences. For a countable ordinal \(\mu\), a fundamental sequence \(S\) is a function over \(\mu \cap ext{Lim}\) where each \(S(\alpha)\) is a fundamental sequence for \(\alpha\). If \(S\) has been established as the fundamental sequence we are using, we use \(\alpha[n]\) as an abbreviation for \(S(\alpha)(n)\). Sequences are always zero-indexed, so \(\alpha[0]\) is the first member of the sequence. Some authors have used \(\alpha_n\), but most modern papers use the square-bracket notation. In ZF, it is impossible to show that there exists an FS system that works for all countable limit ordinals, although with the axiom of choice we can nonconstructively prove that such a system exists. Unfortunately, there is no such constructive proof. Indeed, axiom of choice is necessary for this. It is an open problem whether an ordinal hierarchy can be defined without using fundamental sequences. More explicitly, the problem concerns whether there is a model of ZF such that there exists an \(F : \omega_1 ightarrow (\mathbb{N} ightarrow \mathbb{N})\) where for all \(\alpha > \beta\), \(F(\alpha)\) eventually outgrows \(F(\beta)\), but there does not exist an \(S: \omega_1 \cap ext{Lim} ightarrow (\mathbb{N} ightarrow \omega_1)\) such that for all \(\alpha\), \(\sup(R) = \alpha\) where \(R\) is the range of \(S(\alpha)\).

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