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Cake day: July 2nd, 2023

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  • i’m starting to wonder if maybe it’s better for me financially to not interact with the sales and then just occasionally eat the price of a full price game if i really want to play it. unless of course the sale is for a game that i’ve taken a sacred vow to play within the next year







  • i genuinely feel that if some of the “great” 17th to 19th century thinkers and philosophers wrote their works today they would be considered cringe. i will not be naming examples for my own safety, with one exception: nietzsche would be super cringe.

    don’t get me wrong though, lots of cool stuff was written before and after that time period. and not everything from that time period is bad, but a decent chunk of it is.






  • affiliate@lemmy.worldOPto196@lemmy.blahaj.zonemath rule
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    7 months ago

    in this context, the “universe” means “the collection of all sets”, or more specifically, the Von Neumann universe (which is just a method of iteratively constructing “all” the sets). and so the figure is providing a way to visualize the collection of all sets.

    this is done by assigning a so-called “rank” to each set. the notion of “rank” is kind of annoying to define in simple terms, but it’s basically used as a tool for proving things by induction. it does this by assigning an ordinal number to each set. (ordinal numbers are “basically” a “continuation” of the positive whole numbers, in the sense that for any ordinal number ɑ, you can define the successor ordinal ɑ + 1; so, it’s kind of like a way of formalizing the concept of ∞ + 1.)

    you can think of “rank” as analogous to “cardinality”, in the following way. the “cardinality” of a set is a cardinal number that basically says “how big” the set is. meanwhile, the “rank” of a set is an ordinal number that roughly says “”“how big”“” that set is. (notice that there are a few extra scare quotes this time.)

    lastly, the the set R(ɑ), where ɑ is an ordinal, is the set of all sets that have rank less than ɑ. i.e., the R(ɑ) is the set of all sets that have “”“size”“” smaller than ɑ.

    and this kind of explains the visualization using a cone: there are more sets of “size” < 4 than there are of sets “size” < 3, and there are more sets of “size” < 3 than there are sets of “size” < 2. so it kind of lets you see “the universe” as a cone in that way.