WIP video about The Axiom of Choice - Urgently Requesting Feedback
Added 2025-03-26 07:57:02 +0000 UTCWe've got an exciting video coming up in the next few days!
A video about the axiom of choice and it's implications to mathematics.
We’d love to hear your feedback and suggestions as we wrap up this video! Animations are still in progress and placeholders are used for Derek talking to the camera.
Producer Casper has also recorded some temporary lines of voiceover, so you'll hear his voice mixed in where Derek's will eventually be.
Is there anything wrong? Any other questions? Anything more you'd like to see? Explanations you think don't do a good enough job or lack depth? Let us know!
Thanks everyone!
Comments
Just recently caught this video and enjoyed the exploration into not just the Axiom itself but the supporting ideas and socio-historical context. I went back and cracked open some Wittgenstein after viewing.
Ebony Belhumeur
2025-04-04 22:37:36 +0000 UTCNice presentation. Reminds me of a joke that went around the department back in the day: The Axiom of choice seems obviously true, the well ordering principle seems obviously false, and nobody has any idea about Zorn's lemma. IIRC if you built up math w/o the Axiom of choice, you can get a measure theory (on the reals), in which all sets are measurable. For me this, was mind blowing - a set feels fundamental, but Vitali's sets literally don't exist in that world. BTW, if you move Vitali's argument to $S^1$ the end of the argument a little simpler, but I understand why you'd focus on the argument as presented.
Ben Galehouse
2025-03-30 00:10:43 +0000 UTCThus it follows from Cohen's work that axiom of choice is independent of the other axioms of mathematics. Thus is some sense one can say that "axiom of choice" is a matter of choice!
Soren Riis
2025-03-27 00:21:09 +0000 UTC13:30 After selecting points x1,x2, to form a well-ordered set S⊂R, if S≠R we can extend S by choosing another point from R∖S. Although this might seem like a countable process, it must proceed through transfinite stages to eventually cover all of R. The point is that if we consider a chain of well-ordered subsets (ordered by extension), their union is also well-ordered; if this union were not all of R, it could be further extended, contradicting maximality—a fact typically formalized using Zorn’s lemma. It may not be necessary to reference Zorn's lemma here, as the core idea is clear: any well-ordered subset of R can be extended unless it already equals R.
Soren Riis
2025-03-27 00:12:57 +0000 UTC4:17 Technically speaking, Cantor first proved that the reals are uncountable by a different argument, but later he came up with his famous diagonal proof. In his initial proof, he observed that a countable set must have isolated points, whereas a perfect set—one that is closed and has no isolated points—must be uncountable. This line of reasoning provided an indirect proof that the reals are uncountable.
Soren Riis
2025-03-26 23:49:27 +0000 UTC26.50: Kurt Godel grew up in a German-speaking environment in what is now Austria. As a result, he is usually referred to as an Austrian (not Czech ) logician and mathematician.
Soren Riis
2025-03-26 23:40:13 +0000 UTCCorrect but my point is that zero is a mathematical construct.
Keith Arnold
2025-03-26 20:19:20 +0000 UTCMore direct feedback on the video: 1) math does not require ordering, partial ordering is also important, 2) total ordering is optional, you can force order in a very limited scope (e.g. in a sort function), 3) programmers should think of total ordering as a property modeling a type. If the reality being modeled is not always order-able maybe your define greater than and less than functions should either return a value of undefined or if too confusing they should not be defined.
Gregory Lawson
2025-03-26 17:59:55 +0000 UTCI think Computer Science takes a more practical approach to this by defining Total Ordering and Partial Ordering. The axiom of Choice seems to apply to Total ordering which is very useful for sorting. Partial Ordering happens when some ordering is sensible but some is not. Two common examples of partial ordering are measurements and regular expressions. Measurement noise means some measurements that are close cannot be reasonably ordered. Regular expressions define sets (in CS jargon Languages) of characters or bits. You might want to define greater than to mean a language is a superset of another, so that you can build a tree of generalizations. You can often build very useful trees out of partital orders. But what if you want to print out a list, you need a total order. Then you create a total order by a forced choice for that purpose.
Gregory Lawson
2025-03-26 17:19:29 +0000 UTC0 is real. In fact, it is REQUIRED in any practical number system. For instance, our number system is based ten but ten is written "10" and symbols used are 0 1 2 3 4 5 6 7 8 9. In binary or base two, the symbols are 0 1 and two is 10. 11 cheers for 0!
Edwin Pole II
2025-03-26 14:02:31 +0000 UTC@14:30 What does "well ordered" mean in light of the axiom of choice? Is it an axiom to say that x1, x2, x3, ..., w, w+1 represents the "true order" of numbers (for lack of a better term)? Could we declare x42, x5, x37, ..., w2, w7 as well ordered?
chromicacid
2025-03-26 13:56:51 +0000 UTC@1:20 reminded me of something I learned in a book call The Mathematics of History: A Critical Measurement. It's a tough read, but I remember reading that there was once a lot of resistance in scholarly circles to even considering a number to be disconnected from reality, hence some fierce debates about 0 as a number (a quantity of nothing), infinity, etc. Maybe this isn't the video for it, but could you explain why "real", "rational", and "irrational" are called that? Those sound like loaded terms from a philosophy fight.
chromicacid
2025-03-26 13:55:09 +0000 UTC@0:50 Unrelated to topic, but interesting note about computers: What does memory address 0 mean? Somewhere, someone had to decide what 0 meant. When first receiving power, the mainboard bios starts pulling instructions from memory address 0, which the circuitry routes to the bios flash memory as the first instruction for the bios boot sequence. In practice, the memory is protected from most access, and any programmers attempting to access memory address 0 will result in a null reference exception, even though memory address 0 technically exists.
chromicacid
2025-03-26 13:54:06 +0000 UTCThe reality of the universe adheres to a set of rules - physics. In reality, there is no such thing as infinite or zero these things are mathematical creations of our own mind. Tying systematic processes or rules to unrealistic numbers does not make them real. I must agree with Edwin that this seems like mentalbation.
Keith Arnold
2025-03-26 13:31:10 +0000 UTCOh good so it’s not just me thinking that. 😂
HydrochloRick
2025-03-26 12:44:52 +0000 UTCI'd like to try to provide my non-mathematician's perspective to maybe help us make this teachable to people who, like myself, are very much not mathematicians lol. Honestly, the problem that's making this so hard for me to intuit is just my own executive functioning and attention, but acknowledging that provides an opportunity. What it boils down to is that I'm struggling to conceive any pragmatic relevance for any of this (the Axiom of Choice and whatnot). Sorry to sound so typical and foolish, but my brain can't help but think: "What's the point?". If I can actually map these ideas onto something that matters (or something that *is* matter 😂), then it becomes infinitely easier for me to understand. While it remains abstract—spheres of uncountably infinite this's and that's—my brain adamantly refuses to attribute any salience to the information. I can't remember it no matter how hard I try. How can this apply to something "real"? Even if that "real" thing is literally just theoretical physics lol. As it stands, this is among the few videos you've done recently from which I can't seem to make myself learn anything. But it REALLY feels like I'm only missing a piece or two of relevance in order for the things you're saying in the video to start making sense to my mind. It does *feel* like something here could be relevant to some of the odd stuff I've been engaging lately trying to empirically quantify cognition. But I can't quite wrap my head around it yet. Still waiting for that FAQ about the Principle of Least Action! Do you still want some more questions for that? I've thought of some fun ones!
HydrochloRick
2025-03-26 12:41:26 +0000 UTCAny time you "divide by 0" you run into this kind of problem. This happens in quantum theory as well. 0 is an essential component of math but it is unique in that 1/0 is essentially meaningless. Trying to fit 1/0 into reality is pointless. I refer to doing that as mentalbation. Feel free to use the word but an occasional attribution would be appreciated especially if $ and worldwide fame are involved.
Edwin Pole II
2025-03-26 10:50:43 +0000 UTCI agree with Kyle that I don't understand the point of the Axiom video- "what will I take away?" On a more basic level I noted a mismatch between the voiceover and the words on the screen at ~1:15. V/O says "choose" but screen reads "select". Appreciate V/O is only temp but differences cause viewer to stall and so miss the next bit. BTW, loved the Mars 'copter video and by coincidence my 7-year old grandson's teacher has just covered the Wright Flyer- so cool that part of its fabric was on the Mars machine!
Martin Gee
2025-03-26 10:20:03 +0000 UTCThe explanation about parallel lines and choice would have been much more helpful at the beginning of the video. I'm still lost and don't understand the point of the Axiom of Choice. But I understood the part about parallel lines. If you could explain Aziom of Choice with a more relatable metaphor that would help.
Kyle Nishioka
2025-03-26 08:38:14 +0000 UTCMake the comma after x1 look more like a comma. Right now, looks like x11.
Burt Humburg
2025-03-26 08:33:17 +0000 UTCYou describe Zermelo's proof as perfect but you begin the video by saying it is the most attacked theorem in math or something like that.
Burt Humburg
2025-03-26 08:27:56 +0000 UTCDoesn't the diagonalizarion require the list to be ordered? Otherwise, how would you know if there isn't a copy of the diagonal number already in your list? Ordering it gives you that certainty. I'm not a mathematician. Please confirm before you change the animation.
Burt Humburg
2025-03-26 08:20:03 +0000 UTC
