What is the difference between sets and classes in set theory?

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Discussion Overview

The discussion centers around the differences between sets and classes in set theory, exploring concepts related to membership, subsets, and the implications of these definitions. Participants engage with specific questions related to set membership and the nature of the empty set, as well as the implications of naive set theory versus class theory.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant questions how a set can be an element of another set, suggesting it might be clearer to describe the relationship as a subset instead.
  • Another participant confirms that a set can indeed be an element of another set, providing examples to illustrate this point, including the concept of sets as "packages."
  • There is a reiteration of the idea that while a set can be an element of another set, the individual elements of that set do not become members of the containing set.
  • Discussion includes an analogy comparing sets to items in a shopping bag to clarify the relationship between sets and their elements.
  • Participants explore the concept of the empty set and its relation to other sets, with one expressing confusion about how the empty set fits into a specific example involving nested sets.
  • A participant introduces the distinction between naive set theory and class theory, mentioning Russell's Paradox and the hierarchical structure of classes where sets are the lowest tier.

Areas of Agreement / Disagreement

Participants generally agree that a set can be an element of another set, but there is uncertainty regarding the implications of this relationship, particularly concerning the empty set and the distinctions between naive set theory and class theory. The discussion remains unresolved on some points, particularly regarding the understanding of the empty set in relation to nested sets.

Contextual Notes

Limitations include the potential confusion surrounding the definitions of sets and classes, the implications of Russell's Paradox, and the specific relationships between elements and sets that may depend on the definitions used.

Venomily
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Question 2 (a)

how is it possible? B is a set (since A is a set), how can a set be an element of another set?

Rather than saying: B is an element of C

I thought it would be better to say: B is a subset of C.



Also, can someone explain question 2 (d) to me? thanks
 

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Yes, a set can be an element of another set. There may be a set X={a,b,c}, where X is an element of Y.
But neither a, nor b, nor c becomes a member of Y; only the sets are members.

But you can have a set Z = {a, X} ... so X contains a,b,c, but neither b nor c is a member of Z.

Sets are "packages", and the set Z has the package X as an element - like having an apple and a package of sausages in your shopping bag. You can reach in and pull out the apple, but you cannot pull out a sausage - only a package of sausages.

But you could reach into that package of sausages and pull out a sausage!
 
2(d) is the null set. Try it with the package concept!
 
UltrafastPED said:
Yes, a set can be an element of another set. There may be a set X={a,b,c}, where X is an element of Y.
But neither a, nor b, nor c becomes a member of Y; only the sets are members.

But you can have a set Z = {a, X} ... so X contains a,b,c, but neither b nor c is a member of Z.

Sets are "packages", and the set Z has the package X as an element - like having an apple and a package of sausages in your shopping bag. You can reach in and pull out the apple, but you cannot pull out a sausage - only a package of sausages.

But you could reach into that package of sausages and pull out a sausage!

Great explanation, I understand now, thanks.

But how would I go about 2 (d)? we have the empty set, {}:

{A} = { { 0, {}, {{}} } }

Judging by what you said, I don't see how {} can be related to the above set? it is neither an element nor a set.
 
{} = sausages
{{}} = packet of sausages
0 = Orange

{ 0, {}, {{}} } = Shopping bag of (Orange + sausages + Packet of sausages).

{ { 0, {}, {{}} } } = car boot of Shopping bag.

We want {{}} i.e. the packet of sausages. It is not related to the Car boot because it is neither a SET nor a MEMBER of the Car boot, it is too deep inside.
 
Last edited:
By the way, this is true in "Naive set theory" which suffers from "Russel's Paradox". In "class theory" we do NOT allow "sets" to be contained in sets, but have a hierarchy of "classes" in which classes in one tier can be contained in classes of a higher tier. "Sets" are the lowest tier of classes.
 

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