What is a set (the term commonly used by mathematicains), mathematically speaking?

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# What is a set (the term commonly used by mathematicains), mathematically speaking?

What is a set (the term commonly used by mathematicains), mathematically speaking?

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{math speech}

object in the category of sets

How is the category of sets defined?

Why are people here obsessed with defining a set? It's simply any nonnegative number of unique elements whose order doesn't matter. You should be more concerned with defining a unique element.

https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory

>1. Axiom of extensionality

>Two sets are equal (are the same set) if they have the same elements.

>2. Axiom of regularity (also called the axiom of foundation)

>Every non-empty set x contains a member y such that x and y are disjoint sets.

>3. Axiom schema of specification (or of separation, or of restricted comprehension)

>Any definable subclass of a set is a set.

>4. Axiom of pairing

>If x and y are sets, then there exists a set which contains x and y as elements.

>5. Axiom of union

>The union over the elements of a set exists.

>6. Axiom schema of replacement

>The image of a set under any definable function will also fall inside a set.

>7. Axiom of infinity

>There exists at least one infinite set, namely the set of natural numbers.

>8. Axiom of power set

>For any set x, there is a set y that contains every subset of x.

>9. Axiom of well-ordering (choice)

>Given any collection of sets, each containing at least one element, it is possible to construct a new set by choosing one element from each set, even if the collection is infinite.

you're welcome

further, why is a set a primitive notion?

The “element” relation is undefined, and the class of sets is defined by a set of axioms. Sets are tuned to behave well by setting axioms, but the membership relation doesn’t need to contain any information

An abstract object obeying the axioms of a particular set theory, typically ZFC if not otherwise specified.

Because something must be, or it's impossible to define anything. Sets are often chosen for this purpose because they are intuitive, powerful enough to define all of mathematics, yet philosophically fairly minimal and therefore unlikely to be inconsistent. However, set theory is not the only option as there are other systems such as type theories which can do the same.

You are a low IQ schizo. Take your med please

>Sets are often chosen for this purpose because they are intuitive

>Intuitive

what is intuitive about the axiom of infinity?

The fact that you can look out into the sky and see that it doesn't just end, but seems to go on indefinitely.

>The fact that you can look out into the sky and see that it doesn't just end, but seems to go on indefinitely.

sure, and how does that imply that an "infinite set" exists?

Are you really asking how the set of infinite points in the sky that never seems to ends which you can intuitively see with your own eyes implies the existence of an infinite set?

>infinite set implies existence of an infinite set

nice circular reasoning

Not him but you asked what is intuitive about it which is necessarily gonna sound circular when you then replace the tenor with the vehicle in the intuitive metaphor.

It's not actually circular though since noone claims the axiom of infinity implies the infinite sky.

It isn't reasoning it is a concrete example.

Also, circular reason is the best of the three possible reasoning methods because assertion has no logical basis and regressive reasoning has no conclusive end.

It's not an example. I don't think there's infinitely many "points" in the sky.

Then how many are there and where exactly does it stop?

something something planck length

not that I think that's an intuitive way to look at the sky but it's a possible answer

>what is intuitive about the axiom of infinity?

What is unintuitive about it? Infinity is everywhere in mathematics. The axiom of infinity simply asserts the existence of the natural numbers, which is one of the simplest and most useful infinite sets.

mathgays trying to invent another language instead of using english and numbers

its caused by boredom, considering just how boring maths are, it requires mathematicans take various forms of psychodelics to cope, sometimes they forget they took drugs and write down some delusional concepts like "sets" or "imaginary numbers" thinking its smart

>or "imaginary numbers" thinking its smart

tell me you are a brainlet without saying that you are a brainlet

an array of numbers

It's a collection of objects distinct from each other, with no particular order.

It's like the commonsense notion of a list (like a shopping list), but specifically with the idea that the order doesn't matter, and with an additional rule that if an item is on there twice, you just regard it as if it's on there once. There's not considered to be a difference, for example, between {2, 9, 9, 1} and {2, 9, 1}.

A set is an abstract collection of possibly abstract objects.

By an “aggregate” (Menge) we are to understand any collection into a whole (Zusammenfassung zu einem Ganzen) [math]M[/math] of definite and separate objects [math]m[/math] of our intuition or our thought. These objects are called the “elements” of [math]M[/math].

a box

From a completely formal, logicist amd foundational point of view, a set is that whose existence is guaranteed by the axiom of existence.

Now, this is how set theory builds upon itself, creating the sets we're all familiar with in mathematics. Name one of such existing sets as [math]A[/math]. Adding specification axioms to your theory lets you construct the set [math]{x in A : x neq x}[/math] (called the empty set, since [math] x in varnothing[/math] is a false sentence for every [math] x [/math]).

The rest of the axioms in any theory of sets will lead to construct the sets used in mathematics.

On the other hand, mathematically speaking, sets are not objects of mathematics per se. They're a metamathematical object. Mathematical objects include numbers, functions, rings, triangles and martingales.

One last thing. The question of to what extent can topics studied in set theory be considered mathematical or metamathematical investigations has no single answer and is a topic of debate in itself.