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#set theory
snail-and-snail · 2 years
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countability
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m---a---x · 5 months
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So you people like convoluted diagrams, too?
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Here ya go, filthy
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STE(A)M Meeting
Engineer: What if we added Art to STEM, so it says STEAM? Like a STEAM ENGINE?
Biologist: but i like stems…
Physicist: Sorry, but STEAM’s got my vote. I approve of all 7(ish?) phases of water. I think.
Computer Scientist: I vote for STEAM too, #PC gaming master race
Set Theorist: I will also vote in favor of increasingly large collections of seemingly unrelated things.
Education Professor: That's all very... dumb. F-. But, I have a pretty good idea on how to use the A so I'll vote for it too!
Biologist: :(
Artist: wtf, why am I here? What kinda nerdy sausage party is this?
Education Professor: (on hands and knees) PLEASE PLEASE PLEASE MAKE OUR CURRICULUMS LESS BORING I DON'T KNOW HOW TO STOP ALL MY STUDENTS FALLING ASLEEP IF THEY CAN'T BE CREATIVE PLEASE
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prokopetz · 1 year
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BRB, starting a sideblog which reblogs from all those – and only those – blogs which do not reblog themselves.
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boycritter · 9 months
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spectrallysequenced · 3 months
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Just learned that "|A|>|B| implies 2^|A|>2^|B|" is independent of ZFC. Shit's fucked up.
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model-theory · 4 months
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Of course all of my notes and exposition are completely professionally written.
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art-of-mathematics · 7 months
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The task I wanted to do:
- task 1: Visualize the fourth dedekind number
What I needed to do in order to do task 1:
- task 1.1: Draw a hypercube and
- task 1.2: then give each of the 16 vertices a name - which is the 16 different "cells" (I dunno which word to use, sorry) of a venn diagram with 4 overlapping sets plus a circle around that 4-venn diagram.
- - -- ---
//additional info: For the 3rd dedekind number visualization I already posted I needed to use 20 cubes. Each of the 8 vertices of a cube was related to 3 sets - A, B and C - and the intersections of these sets. (AB, AC, BC, ABC)
You remember this post:
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... and these details:
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...and now back to the 4th dedekind number, which follows the same principle, - but in 4 dimensions.
- So we need a hypercube - and a 4th set/letter:
Now we have 4 letters - alias 4 sets
- and 11 combinations of these sets:
AB, AC, AD, BC, BD, CD, (6)
ABC, ABD, ACD, BCD, (4)
ABCD (1)
=> 11 different intersections + 4 "pure" sets. (sorry for ignorant wording)
So, we have 15 of these now. What is with the 16th (as the hypercube has 16 vertices)? Yeah, that one is the circle you imagine around the 4-set venn diagram.
- - -- ---
This is a 4-set venn diagramm btw:
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So, back to me attempting to do the 4th dedekind-number visualization:
So I started to draw that venn diagram using ellipses, but I slightly altered it, because I wanted to use my isometric grid paper. (The angles of my drawing are different than in the picture depicted above.)
I started to draw the 4 ellipses, and I somehow started to see two intertwined/interlocked tubes due to the additional helplines I used for drawing.
Then I started to use my thicker black pen to make this effect of these two interlocked tubes more visible:
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Now I plan to write the sets/letters at the backside of that piece of paper - in mirrored, so I can hold this piece of paper (with the tube drawing in front) against the light - and see the letters of the set names shining through.
i might also add some details to the tubes.
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garadinervi · 23 days
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Plato, Phaedrus, 265D[-E], in Christopher Alexander, Notes on the Synthesis of Form, Harvard University Press, Cambridge, MA, 1964
«First, the taking in of scattered particulars under one Idea, so that everyone understands what is being talked about… Second, the separation of the Idea into parts, by dividing it at the joints, as nature directs, not breaking any limb in half as a bad carver might.» – Plato, Phaedrus, 265D[-E]
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lesbianslovebts · 6 months
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My mom wrote "potatos - real" on the list as opposed to "potatos - instant" but I, the jokester that I am, added this:
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extinctavialae · 5 days
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bubbloquacious · 8 months
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m---a---x · 5 months
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Oooooh, i fucking love convoluted, indescipherable diagrams, really gotta show restraint not putting them all over my paper
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But maybe, sometimes, formula variable notation is better :(
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real-real-numbers · 9 months
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the chad category theorist vs the virgin set theorist
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greetings-inferiors · 5 months
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I absolutely adore that the set of rationals are countable infinite but the set of irrationals are uncountable infinite, so there are many more irrationals than rationals, but between any two irrationals there are infinitely many rationals. It’s one of those facts that are so absurd that you laugh because you don’t know any other way to process it
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flameshadowconjuring · 7 months
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I am really tired of all the propaganda which says that the axiom of choice is this necessary evil, because it is really useful in proofs but the choice functions are not constructive. Because yes, the axiom itself gives you no scheme for constructing an explicit choice function. That much is true. But I would argue that the axiom's negation is actually way more non-constructive!
The axiom of constructibility (also known as V=L) implies the axiom of choice, so the negation of the axiom of choice implies the existence of sets which are not ordinal definable, and thus cannot be constructed by exclusively appealing to the axioms of ZF (which, with the exception of extensionality and foundation, are all axioms which exclusively describe closure properties of the set-theoretic universe).
In fact, the proof of V=L implying choice is constructive! If V=L holds, we can explicitly define a well-ordering of the entire set-theoretic universe! Not only that, but the existence of a transitive model of ZF implies a transitive model of ZF+V=L which is point-definable, i.e. every set in that model is uniquely characterized by a first-order formula of set theory. Every set being definable strikes me as very constructive.
Sure, things like Banach-Tarski are pretty weird results that require the axiom of choice or a similar axiom. But the only real issue here is that we ascribe our geometric intuition to a partition of a sphere that is impossible to do in real life! Not only can we not divide real-life objects infinitely often, but even if we could, we physically could not select points the way that Banach-Tarski requires.
I think it is interesting to think about set theories where choice does not hold or may not hold. Certainly, a set theory where all subsets of R are measurable is appealing, if only for convenience. And it is legitimately fun to prove equivalences between the axiom of choice and various other propositions, such as the total ordering of the cardinals or Tychonov's theorem, or to reason about the relative strengths of different weaker variants like dependent choice, or my personal favorite, the ultrafilter lemma. It gives you more insight into the beauty of choice. But, in as much as you can 'believe' in an axiom, I believe that the axiom of choice is true in the platonic ideal of mathematics, and I will not stand for slander against it.
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