Can you answer that question?
>>17033811a purse
A miserable little pile of elements.
>>17033811Do all sets have to include the empty set, even the set that doesn't include the empty set?
>>17033837No. Just because a set may be the empty set doesn't mean it's membership to other sets is implied.It is true, though, that the empty set is a subset of all sets (including the empty set) ((proper subset is a mostly useless distinction)) (((according to them)))
>>17033811A set is a collection of elements. >so the empty set is not a set since it doesn’t have elements?Shut the fuck and trust the math.
>>17033837>>17033842a purse is a purse, whether or not your cop boyfriend emptied it all down your sink motor including your dog to save his own job.
>>17033842>collection of elements.a collection? elements? what are those?
A type A is a set if for all x, y : A and all p, q : x = y, we have p = q.
>>17033811A collection of things... Let's say you have three apples. Then you have another pile of five apples. That's {3,5}
>>17033815>empty set aggressively enters the chat
It is a container for elements. A container may be empty. A container that contains an empty container is, itself, not empty.A set is a container.
>>17033811a set is a collection of setsi have an erdos number of 2, so feel free to quote me
It's a biological ability of our brain to distinguish things from each other and also to see similarities. Plus, the ability to create abstractions to communicate those similarities: red apples, small apples, mine apples. Among all abstractions, the most general one is called a set.
>>17033811Smallest quantity of energy possible that keeps everything in motion
>>17033876sets are defined in terms of a membership functionfunctions are defined in terms of relationsrelations are defined in terms of sets
>>17033811This is a brilliant question because it exposes the ignorance of transfinitist mathematicians. Most people don't know this, but in mathematics sets are not defined at all! Instead mathematicians have a list of properties that these supposed sets are supposed to satisfy, with no indication whatsoever for what those objects that are supposed to satisfy those properties actually are.And actually this is not a trivial point, because some of the most basic questions about sets like the continuum hypothesis turn out to be unsolvable based on those axioms, and to resolve them you would need to specify more what you mean by a set, and mathematicians don't agree on that, and don't even know what an answer would look like. Because they've never had a real conception of a what a set is, they only pay lip service to it.
Wouldn't it be ironic if a actual set was dropped later today?
One can reasonably reply to the question that no notions have ultimate definitions. For any concept you can keep asking for the definitions of the constituent terms and never hit the end. So what's the point of asking what is a set? A more precise question, is when is asking for the definition of something warranted? I'd argue it's warranted when something demonstrates the intuitive conception we have of something is murky and doesn't answer the questions we want. And so it's not enough to ask, what is a set, one must also present reasons for why the intuitive conception that many people hold is not good enough.But of course there are by now many such reasons. First of all you had the whole Russel's paradox. Overly permissive notions of set don't work, because they allow you to form a set of all sets that don't contain themselves, whose existence implies a contradiction. A second reason to question the concept are objective vs subjective question. Does there exist a universe of sets before we start investigating them or do we make them up as we go along? Because if they don't exist before we talk about them, that is in conflict with impredicative definitions, which are at the basis of analysis, which explicitly define sets by quantifying on the whole universe of sets that were never specified previously. Similarly you have the question of axiom of choice. Do you accept it as true? On what basis? If you accept there are some sets that you can't specify or construct, in what sense do they exist? Especially sets which essentially contain infinite amount of information and can't even in principle be written down on a finite piece of paper.
>>17034000Third is the pervasive phenomenon of independence that was discovered in the second half of the 20th century. Most of the difficult questions of set theory that were initially important considered open problems turned out to be independent of our axioms. To this day people don't agree on a program even in principle on how to resolve them. They've tried with large cardinals and largely failed. They realized large cardinals do not resolve most of them (even though they do have many interesting consequences). A compelling way to view this pervasive independence phenomena, which is adopted by people such as Feferman, is by saying it demonstrates our conception of a set is simply not well defined. We don't know what we mean by a set, different conceptions of sets (simulated using forcing) give different answers to this question.
>>17033876Could be a stamp collection
>>17033992Why are mathematicians so gay?
>>17033992it's literally just a collection of thingies though
>>17033811Set S is defined by a predicate belongs(S, e) - if true then e is element of S.>but predicate is a function and functions are defined in terms of setsYes, and it was a mistake. You have to start with functions first.
A set is a ratio comparison observation between categories
>>17033811The mental act of grouping objects of thought.
>>17034175No it's not. Being *just* a collection of thingies means it's nothing more than a collection of thingies, which means whenever i give you a collection of thingies, it's a set. But that's patently not true, for the sets that mathematicians mean. I can give you a box of shoes, it's a collection of thingies, but no mathematician would say it's a valid set in ZFC for which whether 3 belongs to it or not is a meaningful thing to ask.
>>17034203>observation between categoriesobservation? category? what is that?
>>17034223perhaps not in ZFC but certainly in ZFC with atoms
>>17034223>>17034245btw I should add, a set isn't a BOX per se. A box of shoes is different from a bag of those same shoes, which is different from those shoes sitting on a table or stacked in a pyramid. The *set* of those shoes is the conceptual entity that remains the same in all of these situations. Kinda like how 3 isn't actually the same thing as 3 apples
https://doi.org/10.5281/zenodo.21866826
>>17033992>most basic questions about sets like the continuum hypothesis turn out to be unsolvable based on those axiomsit's not unsolvable, it's independent, like euclid's 5th axiom
>>17033811It’s like a tuple, doesn’t use 0 indexing
>>17034001>A compelling way to view this pervasive independence phenomena, which is adopted by people such as Feferman, is by saying it demonstrates our conception of a set is simply not well defined. We don't know what we mean by a set, different conceptions of sets (simulated using forcing) give different answers to this question.by what i wrote here >>17034404 according to feferman we don't even have a proper definition of a line, so bust your balls on that one
>>17033811Why are /sci/ schizos concerned about sets when they could start asking simpler yet more relevant questions first?E.g. "what is a group?" A model of the algebraic theory of groups.So, "what is a set theory?" A model of the first-order theory called ZF(C).
>>17033811A set is defined by the set of all things we call a set and by the set of all things we don't call a set
>>17033876>a collection? elements? what are those?whatever you want (almost)
>>17034854>A set is defined by the set
>>17033905what is a container
>>17035310That which contains.
>>17035315that would be?
>>17035320A container, of course.
