thoughts from a restless mind.

ardentsonata:

ursineknight:

shaburdies:

sogreatandpowerful:

when does something belong? when is something included? say we have a set g containing three elements. these three elements are said to belong to g; they have been counted by the set — i.e., counted, presented, structured, situated. it is this situation to which they owe their *consistency*. one wonders: what do we owe to the situation that counts us? to that which we belong? to that which maintains our *consistency*? or, what does that situation owe to us? what if the name of this situation is, as it is in our case, “pony”?

a set’s subsets are not said to belong to that set. instead, we say they are included. for our set g of three elements, eight subsets are possible. (you might check our pictorial rigor above.) the empty or null set, { }, is trivially included in every set, so we count it as a subset of g. the set which actually does the counting of the subsets of g is called the power set, p(g).

so then, why make such a fuss about this rather simple distinction between belonging and inclusion? let’s first cast the issue in slightly different, yet familiar, terms: together, this pair draws out the fundamental distinction between what is mere presentation and the (immeasurable) excess of representation — where presentation stands for the originary count of a set’s elements (a count of belonging), and representation stands for the power set’s count of a set’s subsets (a count of inclusion).

the power set — the set of representations (read: re-presentation) — is always larger than the set it counts the subsets of. and, in the case of a large enough set — and all actual situations in the world are extremely, extremely large — the representations of the power set become so great and powerful as to render their domination over what is presented utterly immeasurable. this is why, if we continue to refer to a set as “the situation”, we may now refer to a power set as “the state of the situation”. now, since it will eventually become our explicit focus here, let’s take the pony community as an example. just how does this excess of subsets over elements, of inclusion over belonging, of p(g) over g, of representation over presentation, of the state of the situation over the situation apply to us? { }

basically me

Brendan, I couldn’t even finish this, but I figured you’d love it.

Ohhhh set theory.

I just had a test on you. .__________.

@ursineknight: thank you for showing me this xD

@ardentsonata: correct me on anything pl0x

any type of element may possess membership, may belong, but only a set may be said to be included; no element which is not a set may possess inclusion. that is, if we are elements, some set we are part of may be included, but we, as individuals alone, may never be included.

there are a few interesting points in set theory that, in its initial formulation, led to paradoxes and inconsistencies. a set is defined as loosely as possible to be a collection of objects, or elements. and the cool part is that a set is also an object! so a set can be an element of another set. and we have all these cool things we can do with power sets.

but then someone is curious and wants to find the upper limit, the largest set: the set of all possible sets. seems legitimate enough. say you get such a set, call it j. now define another set k = P(j), the power set of j. since k is a set, j must contain it. but how can a set contain it’s own power set? from above: ”the power set is always larger than the set it counts the subsets of.”

it’s actually worth checking the corner case to make sure this holds at the lower end. the null set contains zero elements. the power set of the null set contains the null set so it has at least one (and actually only one) element.

“the representations of the power set become so great and powerful as to render their domination over what is presented utterly immeasurable.” i’m not quite sure how to parse that semantically, but i have a bad feeling about “immeasurable”. part of why i enjoy mathematics is for the quantification and the ability to measure. even the infinities (yes, plural. there are infinitely many of them [the number of infinities is equivalent to the smallest infinity though, so don’t be too scared]) are well-ordered. much thanks to georg cantor here.

but infinities are confusing. there’s another one of my favorite paradoxes for the linguists out there. the word “autological” is an adjective which means “describes itself”. for example: “noun”, “word”, and “pentasyllabic” are all autological. then “heterological” means “does not describe itself”. for example “verb”, “banana”, and “monosyllabic” are heterological. by the law of the excluded middle, all words either do or do not describe themselves and so must be either autological or heterological. which type is “heterological”?

the relevant paradox in set theory is to define a sort of heterological set. that is, a set which contains: all sets which do not contain themselves. does this set contain itself?

set theory is cool. i need to look into taking a more formal class on it..

also math with ponies; yay ^__^