- Oct 2024
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Local file Local file
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Thus if P isthe set of all sets, we can apparently form the set Q = {Ae P| A ¢ A}, leading tothe contradictory Oe Q iff O¢€ Q. This is Russell’s paradox (see Exercise 1A)and can be avoided (in our naive discussion) by agreeing that no aggregate shallbe a set which would be an element of itself.
Russell's paradox (1901) in set theory can be stated as:
If $$P$$ is the set of all sets, one can form the set $$Q = {A \in P | A \notin A}$$ which can lead to the contradiction $$Q \in Q$$ iff $$Q \notin Q$$.
This can be done by dividing P into two non-empty subsets, $$P_1 = {A \in p | A \notin A}$$ and $$P_2={A \in P | A \in A}$$. We then have the contradiction $$P_1 \in P_1$$ iff $$P_1 \notin P_1$$.
The paradox happens when we allow as sets A for which $$A \in A$$. It can be remedied by agreeing that no collection can be a set which would be an element of itself.
Relation to Groucho Marx's quote (earliest 1949) about resigning membership of a club which would have him as a member: https://hypothes.is/a/3_zAfITjEe-H5-PlfOlK8A
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- Dec 2022
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genizalab.princeton.edu genizalab.princeton.edu
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But a 1901 article(link is external) by Solomon Schechter hinted at the cache’s potential for social history.
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- Apr 2021
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www.carbonbrief.org www.carbonbrief.org
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New research shows that lake "stratification periods" – a seasonal separation of water into layers – will last longer in a warmer climate. These longer periods of stratification could have "far-reaching implications" for lake ecosystems, the paper says, and can drive toxic algal blooms, fish die-offs and increased methane emissions.
Eine neue Studie ergibt, dass die globale Erhitzung die Ökosysteme von Seen weltweit schädigt. Die Periode, in der sich das Wasser in Schichten teilt, dauert länger. Dramatische Veränderungen mit Schäden für die Biodiversität lassen sich z.B. in den großen amerikanischen Seen beobachten. Kurzbericht: Climate Change Could Cause ‘Irreversible Impacts’ to Lakes - EcoWatch, Details: Climate change could cause ‘irreversible impacts’ to lake ecosystems | Carbon Brief, Studie: Phenological shifts in lake stratification under climate change | Nature Communications
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- Mar 2017
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www.jstor.org.ezproxy.umw.edu www.jstor.org.ezproxy.umw.edu
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Wed Mar 29 2017 09:21:31 GMT-0400 (Eastern Daylight Time)
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