Fluxes of galactic cosmic rays (GCR) observed at 1 AU are modulated inside the heliosphere at different time scales. Here we study the properties of the power spectral density (PSD) of galactic cosmic ray variability using hourly data from 31 neutron monitors (NM) from 1953 to 2016. We pay particular attention to the reliability of the used datasets and methods. We present the overall PSD and discuss different parts of the spectrum and the related periodicities. We find significant spectral peaks at the periods of 11 years, 1.75 years, 155 days, 27 days and 24 hours and the harmonics of the latter two peaks. We calculate a power law slope of −1.79 ± 0.13 for the period range between 50 and 130 hours and a slope of −1.34 ± 0.17 for the period range between 40 days and 3.4 years (1000 − 30000 h).

Suppose λ is a singular cardinal of uncountable cofinality κ. For a model of cardinality λ, let No() denote the number of isomorphism types of models of cardinality λ which are L∞λ-equivalent to . In [7] Shelah considered inverse κ-systems of abelian groups and their certain kind of quotient limits Gr()/ Fact(). In particular Shelah proved in [7, Fact 3.10] that for every cardinal Μ there exists an inverse κ-system such that consists of abelian groups having cardinality at most Μκ and card(Gr()/ Fact()) = Μ. Later in [8, Theorem 3.3] Shelah showed a strict connection between inverse κ-systems and possible values of No (under the assumption that θκ < λ for every θ < λ): if is an inverse κ-system of abelian groups having cardinality < λ, then there is a model such that card() = λ and No() = card(Gr()/ Fact()). The following was an immediate consequence (when θκ < λ for every θ < λ): for every nonzero Μ < λ or Μ = λκ there is a model , of cardinality λ with No() = Μ. In this paper we show: for every nonzero Μ ≤ λκ there is an inverse κ-system of abelian groups having cardinality < λ such that card(Gr()/ Fact()) = Μ (under the assumptions 2κ < λ and θ<κ < λ for all θ < λ when Μ > λ), with the obvious new consequence concerning the possible value of No. Specifically, the case No() = λ is possible when θκ > λ for every λ < λ.