Nov
10th
2022
Oh dear. · 2:30am Nov 10th, 2022
I wrote a set-theoretic mini-textbook in the comments of one of Shakes's stories.
I need a better hobby. 
That said, I'm making some limited progress with stories. It's been very, very hard, but I'm trying to take more control of my life. I can't even look at my feed here anymore, it's just too much stuff. I should probably watch fewer accounts, but even trimming accounts is too difficult for me at this stage. I'll work on it this week and we'll see where it gets me.
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Trick Question · 215 views
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#set theory
#ordinal
#ordinal collapsing function
#shakespearicles
#geek
#progress
#disability
#feed
Interesting discussion. I've sometimes pondered how two lines of different lengths can each contain an infinite number of points. Would that fit into your description? Please do elaborate if you have time.
Countdown to the introduction of a character named Collapsing Function: ω, ω, ω, ω, ω, ω, ω…
5697129
5697129 The beginning of her long comment addresses that. It's hard to summarize. I would say that infinity is not a number in the common-sense understanding of numbers. Mathematicians, including Trixie, would say it is a number, and is in fact both a cardinal and an ordinal; but they get that not by counting, but by redefining what "equality" means. That seems like cheating to me.
Set theorists don't mean the same thing by "the same number" as you do. The way they talk about "the number of natural numbers" requires constructing a different ontology of the universe, one which considers the process by which sets are constructed, and therefore which makes sense only in a temporal universe. Most people have an ontology of a static, Platonist universe, which can't accommodate those sorts of constructions.
Um. I think.
5697201
This is not accurate. Infinity is not a number at all, it's just a concept (one we use quite frequently, but never as a number). Omega is an ordinal number which has a precise definition. Aleph-null is a cardinal number which has a precise definition.
The problem you're having is that you're trying to map the "essences" of words like "number" to things in your everyday experience to see if something qualifies for the label "number", but that label (at least, as you're using it) is wholly arbitrary. In math, as with all sciences, we don't believe that words have specific "essences" that determine truth. Truth is instead determined by using operational definitions, which are merely tools used so that there is no miscommunication.
A hundred years ago biologists used to debate what constituted "life". You hear this sort of silliness on both sides of the abortion argument today. But scientists no longer care whether viruses are "alive" or not, because we recognize the word "life" isn't holy or magical. It's just a word, and how we choose to define it has no effect on what truth is. Truth is determined by measurement and valid logical argument, not by the specific words we use to describe it. Thus operationism has trumped essentialism... except in literary discourse, where mindless masturbation over the meanings of words continues ad nauseum.
5697211
Yes, that's what I meant, but I was too lazy to write it out. Thanks for keeping me (a little bit) honest. I shouldn't have implied that the first infinite ordinal and the first infinite cardinal are defined the same way.
I'm not having that problem; I rant every day about other people having that problem. You'd know this if you'd read The Worst of Bad Horse!
But it's just of you to charge me, since I did say "that seems like cheating". I meant it seems like cheating to tell someone who is probably using a concept of "number" based on counting with fingers that eg Omega is an ordinal number. The way that Omega is operationalized covertly demands (I think) a different understanding of the nature of the universe and of what "being" or "is" means, one which uses operationalization (temporal) instead of essences (timeless, Platonic), so it's misleading to say "Omega is an ordinal" without mentioning "oh, and by the way you have to reconceptualize the universe, being, and how words work".
5697215
I appreciate your inquisitive mind and I know you always discuss in good faith (unlike roughly 100% of people on Twitter). You've also homed in on a very important but misleading concept which I believe I can further illuminate, and it will likely interest you.
It will take me some time to explain why some of what you say above is not a correct view of what math is. The short version is that math is not asserting anything about existence at all; you're confusing the application of math, which we call an interpretation (as you may guess, this has a very specific definition), with the actual process of proving things and building theories in math, which involves manipulation of symbols that have no intrinsic meaning whatsoever, by a very precise and systematic set of rules to ensure we only "prove" things that are true. The latter part is what math actually is, and omega "lives" there, in a sense. The former is (with a few exceptions) the realm of philosophy and applied mathematics, which is still important as it is what motivates the math, but it isn't actual math.
This is a very, very common mistaken impression, so much that I'm tempted to tattoo a response to it on my breasts. There are even professional mathematicians who don't understand the idea, and some of them mistakenly believe that set theory postulates that certain sets "exist" in an existential sense, or that performing the math requires an aspect of faith. Math is an enormous field, and said mathematicians tend to have limited knowledge of mathematical foundations and especially formalism (which is a perspective that allows us to "prove" that the proofs we've made for centuries in math are actually well-founded and true).
The primary reason for this confusion is that we use simple language to describe things in shorthoof. When a set theorist postulates that "this thing exists", it doesn't mean what it sounds like—there is no true existential meaning being asserted. It's a short way of saying something that is far more complicated and abstract, and does not involve perceptions of reality, because that assertion still lies in the realm of math where we aren't assigning meaning to the symbols. What it does mean is that we can logically demonstrate that a certain finite string of symbols can be explicitly constructed from strings we already know exist within a theory (a theory is a set of these strings closed under logical implication, meaning everything you can prove from a theory is also in the theory) we've previously shown is consistent (meaning we can prove there is no way to construct certain other strings that would ruin the entire theory).
Why bother doing math at all, if it has no intrinsic meaning? The usefulness of math comes from the fact that if we can prove something is true in the mathematical realm, and if we can also prove that it has all of the features that explicitly describe something we care about in the real world, that proof will carry over into the real world and can be trusted as true. Sometimes math gets very, very abstract and it can be difficult to imagine how some of the math we do could ever be useful, but you'd be amazed how seemingly unrelated abstractions can apply to the real world.
I'll reply to this in greater detail when I'm feeling better, which may be in a few minutes or a few months. Probably closer to the first.
"I can't even look at my feed here anymore, it's just too much stuff."
I believe that an appropriate piece of modern slang here is "mood".
You may notice how long it took me to get to this blog post, and... yeeeeah, this is very, very far from being one of the last ones on my current list.
Good luck with, well, life in general.