#Zermelo

[omegaphilosophia]

The Philosophy of Set Theory

The philosophy of set theory explores the foundational aspects of set theory, a branch of mathematical logic that deals with the concept of a "set," which is essentially a collection of distinct objects, considered as an object in its own right. Set theory forms the basis for much of modern mathematics and has significant implications for logic, philosophy, and the foundations of mathematics.

Key Concepts in the Philosophy of Set Theory:

Definition of Set Theory:

Basic Concepts: Set theory studies sets, which are collections of objects, called elements or members. These objects can be anything—numbers, symbols, other sets, etc. A set is usually denoted by curly brackets, such as {a, b, c}, where "a," "b," and "c" are elements of the set.

Types of Sets: Sets can be finite, with a limited number of elements, or infinite. They can also be empty (the empty set, denoted by ∅), or they can contain other sets as elements (e.g., {{a}, {b, c}}).

Philosophical Foundations:

Naive vs. Axiomatic Set Theory:

Naive Set Theory: In its original form, set theory was developed naively, where sets were treated intuitively without strict formalization. However, this led to paradoxes, such as Russell's paradox, where the set of all sets that do not contain themselves both must and must not contain itself.

Axiomatic Set Theory: In response to these paradoxes, mathematicians developed axiomatic set theory, notably the Zermelo-Fraenkel set theory (ZF) and Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). These formal systems use a set of axioms to avoid paradoxes and provide a rigorous foundation for set theory.

Set Theory and the Foundations of Mathematics:

Role in Mathematics: Set theory serves as the foundational framework for nearly all of modern mathematics. Concepts like numbers, functions, and spaces are all defined in terms of sets, making set theory the language in which most of mathematics is expressed.

Mathematical Platonism: The philosophy of set theory often intersects with debates in mathematical Platonism, which posits that mathematical objects, including sets, exist independently of human thought. Set theory, from this perspective, uncovers truths about a realm of abstract entities.

Philosophical Issues and Paradoxes:

Russell's Paradox: This paradox highlights the problems of naive set theory by considering the set of all sets that do not contain themselves. If such a set exists, it both must and must not contain itself, leading to a contradiction. This paradox motivated the development of axiomatic systems.

Continuum Hypothesis: One of the most famous problems in set theory is the Continuum Hypothesis, which concerns the possible sizes of infinite sets, particularly whether there is a set size between that of the integers and the real numbers. The hypothesis is independent of the ZFC axioms, meaning it can neither be proven nor disproven within this system.

Axioms of Set Theory:

Zermelo-Fraenkel Axioms (ZF): These axioms form the basis of modern set theory, providing a formal foundation that avoids the paradoxes of naive set theory. The axioms include principles like the Axiom of Extensionality (two sets are equal if they have the same elements) and the Axiom of Regularity (no set is a member of itself).

Axiom of Choice (AC): This controversial axiom asserts that for any set of non-empty sets, there exists a function (a choice function) that selects exactly one element from each set. While widely accepted, it has led to some counterintuitive results, like the Banach-Tarski Paradox, which shows that a sphere can be divided and reassembled into two identical spheres.

Infinity in Set Theory:

Finite vs. Infinite Sets: Set theory formally distinguishes between finite and infinite sets. The concept of infinity in set theory is rich and multifaceted, involving various sizes or "cardinalities" of infinite sets.

Cantor’s Theorem: Georg Cantor, the founder of set theory, demonstrated that not all infinities are equal. For example, the set of real numbers (the continuum) has a greater cardinality than the set of natural numbers, even though both are infinite.

Philosophical Debates:

Set-Theoretic Pluralism: Some philosophers advocate for pluralism in set theory, where multiple, possibly conflicting, set theories are considered valid. This contrasts with the traditional view that there is a single, correct set theory.

Constructivism vs. Platonism: In the philosophy of mathematics, constructivists argue that mathematical objects, including sets, only exist insofar as they can be explicitly constructed, while Platonists hold that sets exist independently of our knowledge or constructions.

Applications Beyond Mathematics:

Set Theory in Logic: Set theory is foundational not only to mathematics but also to formal logic, where it provides a framework for understanding and manipulating logical structures.

Philosophy of Language: In philosophy of language, set theory underlies the formal semantics of natural languages, helping to model meaning and reference in precise terms.

The philosophy of set theory is a rich field that explores the foundational principles underlying modern mathematics and logic. It engages with deep philosophical questions about the nature of mathematical objects, the concept of infinity, and the limits of formal systems. Through its rigorous structure, set theory not only provides the bedrock for much of mathematics but also offers insights into the nature of abstraction, existence, and truth in the mathematical realm.

[emmy-the-witch-of-eternity]

-why do you give me your silliest battles?

-because you are my silliest soldier

[tagitables]

Ernst Zermelo (1871 - 1953)

[heirtocragflame]

Is mouse robot girl not a reference to mice?

"In set theory, a mouse is a small model of (a fragment of) Zermelo–Fraenkel set theory with desirable properties. The exact definition depends on the context. In most cases, there is a technical definition of "premouse" and an added condition of iterability (referring to the existence of wellfounded iterated ultrapowers): a mouse is then an iterable premouse."

Mouse robotgirl is a reference to an animal on planet earth called mice.

[kaiasky]

Hi KaiaGPT, can you explain what set, category, and group theory, are in a way that the philosophically - but not mathematically - inclined, will understand(or even just one of them) sincerely yrs, hopeless wordcel

I can certainly help explain what set theory, category theory, and group theory are in an approachable way! The branches of math you're talking about have a lot in common with philosophy.

Each of these theories is named after a core mathematical object that they study. Set theory studies sets, group theory studies groups, and category theory studies categories. These terms refer to types of axiomatically-defined mathematical objects. So group theory isn't related to real-world "groups" like a bunch of grapes or a group of friends, but to anything that obey the axioms that define a group.

We could choose any axioms and study the resulting behavior, but mathematicians tend to focus their effort on axioms that lead to interesting and complicated behavior.

Set theory studies sets, which are, roughly speaking, unordered collections of things. We could have the set of {1, 2,3}, and say that "2 is in the set {1, 2, 3}". We could also have the set "the set of all even numbers".

A natural thing to do with sets is ask about their size. We can intuitively say that {1, 5, 9} and {1000, 2000, 3000} are the same size, because they each have 3 elements. The way mathematicians formalize this is by saying sets are the same cardinality (size) if you can make a 1-to-1 mapping from one to another. If one set has items left over after you pair all the items in the other set, the one with leftover items is bigger. That leads to surprising conclusions, like that "the set of all numbers divisible by 100" is just as big as "the set of all numbers divisible by 1". (You can line up 1 and 100, 2 and 200, and so on.) On the other hand, it turns out that no matter how hard you try, you can't make a mapping between the natural numbers and the real numbers--there are real numbers left over at the end. That means set theory is a very powerful way to talk about various kinds of infinities!

Another part of set theory is that it can be used to construct and define other parts of mathematics. For instance, we've been talking about sets that contain numbers, but it's possible to define numbers using set theory, starting with only the empty set. We start by saying that that "0" is the fancy name we give for empty set {}, and "1" is the fancy name we give for {{}}, the set containing 0. And every number after 1 is simply defined as "the set of all numbers before it". Because it is capable of defining many other types of mathematical objects, a list of axioms (ZFC, "Zermelo-Fraekel set theory + axiom of choice") is often taken as the foundations of mathematics.

Group theory studies groups. Groups are sets, along with a binary operation that is 1) associative, 2) has an identity element, 3) and has an inverse element for each element. That definition isn't too helpful, so let's talk about some examples.

Imagine I have 4 cards on a table, labeled 1, 2, 3, 4.

I ask you to turn them face-down, and then rearrange them in any order you like. Then, I get somebody else to rearrange the cards however they like, not seeing the original numbers and not knowing how you rearranged them. Combined, those two shuffles would create a new ordering, and how you rearranged the cards would affect how their shuffle rearranged them.

This is a group! We have:

A set, which is "all possible orders for the 4 cards",

An operation, "rearranging the cards". This is a "binary" operation because it takes two inputs (the order of the cards at the start, and the shuffle applied to it). We might write down "switching the first and second cards" as <2 1 3 4>, and then we could write <2 1 3 4> ∘ <1 3 2 4> to denote a situation where the first person swapped the left two cards, and the second person swapped the middle two cards. If you try it, you get <2 1 3 4> ∘ <1 3 2 4> = <2 3 1 4>!

An identity element, <1 2 3 4>, which is just "leaving the cards where they are".

An inverse element for every element. If you're the second shuffler in our game, and you knew what the first person did, you could pick your shuffle so the cards end up back in <1 2 3 4> order.

The tricky one, associativity. That means it doesn't matter what order we do our ∘ operation in: (A ∘ B) ∘ C = A ∘ (B ∘ C). If you draw a diagram like this, where each horizontal set of arrows represents a rearrangement, you can see that it doesn't matter if you trace the arrows from top to bottom or bottom to top, you end up with the same correspondence in the end.

But because these properties are relatively common, it turns out they describe a lot of things. For instance:

Addition on the whole numbers (x + 0 = x)

Adding times (11:30 pm + 0:00 = 11:30 pm)

Multiplication on the nonzero fractions (x * 1 = x)

Rotating 90 degrees and flipping an image in photoshop

Rubicks cubes

Symmetries of shapes and crystals

That means that any result or proof that applies to groups applies to any of these, along with many other types of groups!

Category theory deals with categories. The formal definition here is even less helpful than the formal definition of groups, so we'll skip it. We might talk about the "category of sets", or the "category of groups". This lets us talk about the relation between different types of mathematical objects or their similarities. So like, being able to say that certain categories are identical to one another, or in some sense the mirror image of one another. It also demonstrates how certain operations on different mathematical structures can be thought of as the same underlying operation.

For instance, one way we combine sets is the Cartesian product, and it creates the set of all combinations from set 1 and set 2. E.g. {blue, red} x {shirt, shoes} gives {(blue, shirt), (blue, shoes), (red, shirt), (red, shoes)}. There ends up being similar ideas all over the place, including in group theory.(Imagine putting a rubicks cube next to the cards from the group theory example, and mixing up both of them separately.) And one result of category theory is that you can point to all of these and say "all of these different types of operations are the same underlying pattern, just expressed in different categories."

I hope this was helpful in understanding these parts of mathematics and the mathematical structures they study!

----

i am not a mathematician, many of my mutuals know a lot more math than I do, so i'm sure they'll correct me if i got things wrong or left things out. and if u have questions im happy to answer or they will have better answers than i do. i dont know shit about category theory especially, but it's not in the nature of a gpt chat assistant to admit it doesn't know things. i thiiink the thing about products is approximately a correct summary of universal properties but idk. (also <> is weird notation for the permutations but (1234) etc would look like it was cyclic notation which is another topic.)

[keycrash]

semi-daily math post since people asked—

you may have heard about historical arguments in mathematics— irrational numbers, imaginary numbers, even quaternions— but one of the more modern divides is over something called the axiom of choice. an axiom is one of the base assumptions of a system of logic— things that we presume to be true so that we can rely on their logic to create new conclusions. our common system of logic is called zermelo-fraenkel set theory. (if you choose to accept the axiom of choice, it’s abbreviated ZFC to include that.) set theory is extremely foundational and has to do with how we group collections of abstract mathematical objects; one axiom in ZFC, for example, is ‘if we have two sets, there exists a union of the sets.’ for example, the union of {x,y} and {y,z} is {x,y,z}.

the axiom of choice essentially states that given an infinite collection of sets, you can make a new set by choosing one element from each of those sets. kinda abstract. kinda not as abstract as you’d think, too? but once you start thinking about choosing from infinite sets without a ‘rule’ to follow— infinite arbitrary choices— it can get dicey. it was originally controversial because some of its conclusions were kind of counterintuitive; for example, the banach-tarski paradox, which lets you divide an ideal sphere (so, infinitely divisible) into complex parts such that you can manipulate those parts into two identical spheres of the same volume as the original. there’s even a common math joke about it by jerry bona— “the axiom of choice is obviously true, the well-ordering principle obviously false, and who can tell about zorn’s lemma?” poking fun at the fact that… those three things are all equivalent to the same thing, the axiom of choice, just presented in different ways that make them seem either very intuitive or very counterintuitive!

these days the axiom of choice is widely used. i wouldn’t say ‘widely accepted,’ exactly, because axioms aren’t exactly ‘true’ or ‘false’; they’re a basis of logic we either decide to use or decide not to use based on whether it’s useful for us. (people study other systems of logic too! look up peano arithmetic). that being said, apparently it’s useful enough to have justified its existence to most mathematicians :-)

[youzicha]

People were talking about the Löwenheim-Skolem theorem, which reminded me of Skolem's 1922 paper Some remarks on axiomatized set theory, surely one of the greatest papers ever. I don't think it's directly available online, but there is a translation in the book From Frege to Gödel, which in turn can be found on the usual pirate sites.

In terms of content, it is packed with foundational results, including

First description of how to write the Axiom of Separation in first-order logic.

First complete proof of the Löwenheim-Skolem theorem, and a discussion of what is now known as Skolem's paradox.

Independent invention of the Axiom of Replacement (you know, what turned Zermelo set theory into Zermelo-Fraenkel set theory).

Conjecture that the continuum hypothesis is unprovable.

But the most memorable part is the Conclusion:

The most important result above is that set-theoretic notions are relative. I had already communicated it orally to F. Bernstein in Gottingen in the winter of 1915-16. There are two reasons why I have not published anything about it until now: first, I have in the meantime been occupied with other problems; second, I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come to publish a critique.

"I didn't work out the axiomatic foundations because I had better things to do, but since you guys still aren't getting it..."

[coraniaid]

(Answering @badwolfwho1's questions for this character ask game; four of four.)

Willow

5 What's the first song that comes to mind when you think about them

(Honestly, this was a surprisingly hard question to answer and I don't really know if I think this song fits Willow at all or I just subconciously gave up and picked a song I liked. It's a good song though?)

12 What's a headcanon you have for this character?

Not exactly a fully developed headcanon (so much as it is a stubborn refusal to accept a fictional character I like is not actually Just Like Me) but ... I'm rewatching early Season 4 now and it is honestly so hard to persuade myself that Willow would be content to sit in this (objectively not very good!) pysch class (where they are teaching Jung and Freud as fact!) when the show had previously established her as somebody who was hacking into government computers for fun before she ever met Buffy, and who talks about liking math, and who regularly competed in her school's science fair, and started work on trying to rebuild Ted the robot when she was a teenager, and who was headhunted by a company that was not quite explicitly mid-1990s Microsoft, and who taught her high school computer class while a high school student. Let Willow be computer science student you wrote her as, you cowards!

Yes, later in the season the show will use Willow "not being as interested in computers any more" as a(n honestly not very coherent) metaphor for her coming out as a lesbian, but we aren't at that point yet! (And besides, why would getting into magic make Willow more interested in outdated pop pysch? As opposed to, say, quantum mechanics or category theory or anything else that more closely resembles the show's take on magic?) We haven't even met Tara! It feels very obvious to me that the writers just want Willow to go to classes with Buffy and don't particularly care that the character they created in the first three seasons wouldn't want to go to those classes.

At least, Willow wouldn't want to go to those classes unless her best friend was also going. (And, actually, why is Buffy apparently majoring in pysch now, anyway? What happened to her previously established love of English literature? I know the writers bring that up again next season; it feels a bit pointless they ignore it now.)

So my current headcanon is that Willow is going to a bunch of computing and math classes this semester (or at least she will be until gets distracted by magic/Tara), on top of Maggie Walsh's pysch classes, she just pretends she isn't because she doesn't want Buffy to think she's showing off by taking such a high course load. Whenever there's a college scene with Buffy present and not Willow, I assume Willow is somewhere off-screen learning about the axioms of Zermelo Fraenkel set theory or about assembly language or about crystal oscillators or ... you know, something I would she would actually care about.

13 What's an emoji, an emoticon and/or any symbol that reminds you of this character or you think the character would use a lot?

Willow started using the internet in the mid- to late 90s, which I think would have had a big impact on the sorts of symbols she’d use.  That’s a bit too early for emoji, I think: I don’t see Willow using them. I can see Willow using the old classic of :/ a lot though (especially if she also introduced Xander to IRC at some point)

[bubbloquacious]

7, 23, 45, 71 👀

From Real's Math Ask Meme.

7. What do you like about math?

The creativity :) I can make up whatever I want and play around with it, and no matter what way I try to solve a problem, to unkink a tangle in a structure, I know it will give the same answer.

1↋. Will P=NP? Why or why not?

It can't be true right? Like that would be crazy. One joke that people make occasionally is like what if the equivalence between P and NP problems is independent of the Zermelo-Frænkel axioms, like the continuum hypothesis, wouldn't that be wild. I'm not so sure this could happen! So there's this idea that the Collatz conjecture is not independent of ZFC, because if it is false, then it is provably false; if there is a number that ends up in a loop that does not contain zero, then trying every number in sequence will eventually get you a disproof.

The same thing doesn't quite work for P = NP, but it's similar. If it is true, then you can try every Turing machine in sequence, and see if it solves an NP-complete problem in polynomial time, so it is provably true. It's not exactly right, because you need to show that the machine solves the problem for any input size, which is not necessarily computable. But it seems like a proof like that might work, which is why I feel like P = NP is not independent of ZFC.

39. Are you a Formalist, Logicist, or Platonist?

Answered here.

5↋. 👀

👀

[lonewolflink]

me: i should write my dissertation

my brain: what if in ch. 8 of wsc you actually have lia and ryujin argue about whether or not zermelo-fraenkel set theory should include the axiom of choice

[tournevole]

Gilbert Zermelo Cram

[max1461]

Not to rag on Zermelo and Fraenkel, but they punch somewhat above their weight in name recognition

[emmy-the-witch-of-eternity]

New Alix portrait!! This time with coherent hair!

I also added a color guide to my page to know which exact pencils i used and not have the issue i had of "oh lol that was the wrong pink"

[perdvivly]

Whoops! Sorry I didn’t see that. I do want to point out (I’m sure you know) that in set theory a bijection *is* a set. So I disagree that there’s meaningful “opinion” to be had on which of these claims is “more ontological”

No need to apologise ^_^! I mean, the context here is that we're living in a post forcing world and we're still talking about the continuum hypothesis so clearly *something* a little funky is going on. We're not *just* talking about axiomatic set theory as laid down by the great prophet zermelo. And while I don't think anyone would question whether or not you could *represent* a bijection as a set of ordered pairs, I do think there's a legitimate question as to whether or not that's what a bijection in some deep down way *is*.

But as I mentioned in the tag here (x) I do think that option is untenable. Like, you're right. We agree. I just don't think it's an utterly ridiculous position to hold. I don't know who you are b/c nonny, but I think it's really understandable that you didn't see that I'd said any of this if you aren't following me or something.

[furry-er]

Not me dreaming about

telling the class about thicc math teaching angels whilst teaching Zermelo-Fraenkel set theory, and emphasizing how 'thicc' 🍑😫 they are 😉 (*idk ZFC i just watched a short video on the topic the night before)

[mathhombre]

When Math Sounds Like Magic

Mathober Day 3: Internal

Looking around I found "Internal Set Theory" which is formally an extension of Zermelo Fraenkel Set Theory but spends time talking about the predicate and somehow means you can treat infinitesimals as numbers...?