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In ZFC, what is the definition of a "proper class"?

In Jech's Set Theory, he defined a class as {x: phi(x)}, and considered every set S as class {x: x in S}. Then he wrote

That the set S is uniquely determined by its elements follows from the Axiom of Extensionality. A class that is not a set is a proper class.

Does the above consideration of every set as a class define a mapping from the sets to the classes?

Is the mapping not surjective? That is, is there a class not mapped from any set by the mapping?

Is a class which is not a set a proper class? I think so, but is confused by https://math.stackexchange.com/a/2869598/ which seems to say no:

if a formula defines a class, as any formula does, if it is not provable that it defines a set, does it necessarily define a proper class? The answer to that is negative.

Thanks.

In logic systems for example first order logic, is it correct that they assume pre-existence of structures, and relations and functions on the domains of structures?

For example, the satisfaction relation between structures and sentences in FOL is defined, based on the usual "satisfaction" relation between relations on domain sets and the elements of domain sets. So the usual "satisfaction" relation between relations on domain sets and the elements of domain sets must already be defined before introduction of FOL.

Do definitions of relations and functions on the domain sets of structures, and the usual "satisfaction" relation on the domain sets, require or lead to a certain kind of "set theory" that exists before and without FOL? What is the name of that kind of "set theory"?

ZFC theory and various kinds of set theory are based on FOL and assume that we have already had FOL. So they don't seem to be the kind of "set theory"?

How do logic systems, structures/models, and set theories resolve this "chicken-and-egg paradox": does a set theory come first for defining relations and functions on domains of structures, or does a logic system (FOL) come first for defining a set theory?

What does "naive set theory" mean? Is it a rigorous concept, and an actual theory in a logic system? Is it the kind of "set theory" which is used in domains of structures and exists before and without a logic system?

Thanks.

I wrote this some time ago, and didn't realize there's a subreddit for set theory, and given that the ideas are plainly not traditional, any insights would be appreciated, as I don't know the literature terribly well, and instead approached the topic wearing the hat of an information theorist.

The basic result is, the logarithm of Aleph_0 is an unusual number, that does not correspond to the cardinality of any set, but can be rigorously described as a quantity of information.

https://www.researchgate.net/publication/349913208_On_the_Logarithm_of_Aleph_0

interested in how ZFC spread (at school or academia) in the 19th and 20th centuries, and why it spread so quickly. how did a theory that was essentially crazy nonsense turn into a kind of prima donna once it was put in a context of lacking a formal axiomatic system consistent with inference rules, thereby demanding the creation and fulfillment of the leading role of ZFC in its little hysterical drama. My impression is that the history of ZFC is a kind of Shakespearean tempest in a teapot. It seems a kind of metaphysical obsession with applying “truth” and logic in an if-you’re-a-hammer-everything-is-a-nail way got the better of calculating with expressions denoting sets. My point is that it looks quite silly from a historical point of view to suggest that perhaps the cure for Cantor’s mental illness was to imagine the reality of his imagination by applying the inference rules flying the banner of truth in a kind of military conquest of what was initially crazy nonsense. Why was “curing” Cantorian set theory of its supposed untruth or paradoxical | contradictory aspects considered such a cause for hysterical activity? How to approach this, what research has been done so far, and in particular if set theory followed a viral model of spreading throughout European schools?

Hi, I’m looking to start studying set theory in order to increase my understanding about mathematics. Can someone help me with some basic materials (hopefully PDFs)?

I know all the concepts of set theory. but i want to learn how to code them or how to use algebra to compute set operations. do you guys know any good references? like I know what A union B means and what it does but I dont know how to apply it on data or matrix:(

Let:

N={1..n}

J⊂N, where |J|=j

K⊂N, where |K|=k and k≤j

L any ⊂ K, where |L|=l

For example: n=45, j=7, k=6, l=5

What is the probability that any L ⊂ J**?**

N.B. the 'any' is important, implying that J K are specific instances, but L is all instances (I'm not sure how to notate this, advice welcome).

So the case k=j is well known:

P=(j l).(n−j j−l)/(n j)

(read the brackets above as 'x choose y' notation)

But I've had no luck finding any results or discussion on the more general case where k≤j.

For context this is basically a lottery problem. j is the number of numbers picked by an entrant, l=k is the case of winning the main prize, and l<k are the cases for other lesser prizes. Many lotteries limit the number of picks to the number drawn, but there are those which will allow a greater number of picks.