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Mathematics Stack Exchange

Questions tagged [set-theory]

This tag is for set theory topics typically studied at the advanced undergraduate or graduate level. These include cofinality, axioms of ZFC, axiom of choice, forcing, set-theoretic independence, large cardinals, models of set theory, ultrafilters, ultrapowers, constructible universe, inner model theory, definability, infinite combinatorics, transfinite hierarchies; etc. More elementary questions should use the "elementary-set-theory" tag instead.

This tag is for set theory topics typically studied at the advanced undergraduate or graduate level. More elementary questions should use the tag instead.

Topics relevant to this tag include cofinality, axioms of ZFC and of close variants (such as ZFA, KP, NBG, MK), axiom of choice, forcing, set-theoretic independence, large cardinals, models of set theory, ultrafilters, ultrapowers, constructible universe, inner model theory, definability, infinite combinatorics, transfinite hierarchies; etc.

For questions about alternative set theories, use instead the tag .

10895 questions
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Are there any objects which aren't sets?

What is an example of a mathematical object which isn't a set? The only object which is composed of zero objects is the empty set, which is a set by the ZFC axioms. Therefore all such objects are sets. Objects composed of many objects are obviously…
user325165
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56
votes
3 answers

Set Theoretic Definition of Numbers

I am reading the book by Goldrei on Classic Set Theory. My question is more of a clarification. It is on if we are overloading symbols in some cases. For instance, when we define $2$ as a natural number, we define $$2_{\mathbb{N}} =…
user17762
38
votes
3 answers

What was the definition of "set" that resulted in Russell's paradox?

Russell's paradox, the set of all sets not containing themselves can be broken down to two statements: A thing that contains all sets that don't contain themselves. This thing/one such thing would necessarily qualify as a set. Now, what was the…
user2268997
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28
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7 answers

Please Explain Kuratowski Definition of Ordered Pairs

I've seen this Kuratowski definition for ordered pairs, but can't fathom why it implies an order to $x$ and $y$ $(x,y):=\{\{x\}, \{x,y\}\}$ As I understand sets, $\{\{x\}, \{x,y\}\}$ is also $\{\{x,y\}, \{x\}\}$. Only when I think about the Axiom of…
147pm
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25
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2 answers

Bijection between $2^{\mathbb R}$ and $\mathbb{ R ^ R}$

I'm well aware of the standard proof based on cardinality arithmetic to show that these two sets have the same cardinality and the question of defining a bijection between the two sets came up. I worked on it a little while and was wondering if…
JSchlather
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24
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1 answer

Do sets, whose power sets have the same cardinality, have the same cardinality?

Is it generally true that if $|P(A)|=|P(B)|$ then $|A|=|B|$? Why? Thanks.
benjamin
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23
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9 answers

What is it about modern set theory that prevents us from defining the set of all sets which are not members of themselves?

We can clearly define a set of sets. I feel intuitively like we ought to define sets which do contain themselves; the set of all sets which contain sets as elements, for instance. Does that set produce a contradiction? I do not have a very firm…
crf
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21
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2 answers

Are there any areas of mathematics that are known to be impossible to formalise in terms of set theory?

Are there any areas of mathematics that are known to be impossible to formalise in terms of set theory?
user7485
20
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2 answers

Difference between ZFC & NBG

Can someone tell me what's the advantages and disadvantages of using NBG rather than ZFC and what's the advantages and disadvantages of using ZFC rather than NBG?
Katlus
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17
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2 answers

$f:P(X)\to X$ property

For any nonempty set $X$ and any function $f:P(X)\to X$, there exist different $A,B\subseteq X$ such that $A\subseteq B$ and $f(A)=f(B)$. One attempt is trying to find a chain A s.t. $|X|<|A|\leq |P(X)|$ from here Thank you.
TKM
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17
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3 answers

Counter example of Zorn's lemma when we only take countable chains

I was learning Zorn's lemma yesterday and I couldn't find any example, where Zorn's lemma fails when we only require that every countable chain in P has a maximal element in P. Does anyone know an example?
Long
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16
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4 answers

Sets and classes

I don't quite understand the difference between sets and classes. A class consists of sets. Why is not a class a set? A group itself is a set. A isomorphic class of group is not a set, right? Could you explain how to distinguish a set from a class.
user65175
16
votes
3 answers

Why do we use von Neumann ordinals and not Zermelo ordinals?

Why do we use von Neumann ordinals, $$ 0 = \emptyset $$ $$ n+1 = n \cup \{n\} $$ and not Zermelo ordinals? $$ 0 = \emptyset $$ $$ n+1 = \{ n \} $$
Maicake
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do all uncountable sets have same cardinality as real numbers?

Countable sets have the same cardinality as the natural numbers (or a subset of the natural numbers, depending on who you ask). Can one make the same claim about uncountable sets and real numbers? In other words, do all uncountable sets have same…
user105966
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2 answers

Does $2^X \cong 2^Y$ imply $X \cong Y$ without assuming the axiom of choice?

A friend of mine told me that $X \cong Y \Rightarrow 2^X \cong 2^Y$ ($X$ and $Y$ being sets), which is very easy to prove, but he was wondering about the converse in ZF, i.e., can one take logarithms? Since the (apparently) simpler question of…
Abel
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