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Results tagged with set-theory
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user 1309249
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.
2
votes
2
answers
110
views
Formal definition of $\aleph_0$
Usually, I see $\aleph_0$ defined to be equal $|\omega|$. However, I don't really understand: in ZFC, everything should be a set. But $\aleph_0$ is not defined as a set, but rather as the image of a s …
- 1,677
2
votes
1
answer
61
views
Optimal definition of intersection of sets in ZFC
The definition I'm used to is
$$\bigcap x=\left\{y\in\bigcup x:\forall z(z\in x\implies y\in z)\right\}$$
However, this doesn't seem to work when $x=\varnothing$, as the usual convention is to define …
- 1,677
0
votes
1
answer
76
views
Definition of the Cartesian product in ZFC
The definition I know is
$$A\times B\triangleq\{x\in\mathscr P(\mathscr P(A\cup B)):\exists u\exists v(x=(u,v))\},$$
whose existence and uniqueness follow from the axioms of separation and extensional …
- 1,677
0
votes
1
answer
64
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Proof that the intersection of the inductive subsets of $a$ is independent of $a$
For brevity, I will write $\mathcal I(a)$ in place of "$a$ is an inductive set" from now on. I want to show that, for all $a$ and $b$, if $\mathcal I(a)\land\mathcal I(b)$, then
$$\bigcap\{e\subseteq …
- 1,677
1
vote
0
answers
51
views
Parameters in the axioms of separation and replacement
Usually, the axiom schemata of separation and replacement are stated for a given formula $\varphi(x,u_1,\dots,u_n)$. Would it be wrong if I just stated them for a formula $\varphi(x)$? Can't we just t …
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0
votes
0
answers
42
views
Are classes countable? [duplicate]
So, I'm not really asking whether classes (like the "set" of all classes) is countable, because it's obviously not the case, since all sets are classes and sets are clearly uncountable. At the same ti …
- 1,677
5
votes
1
answer
212
views
Is $\mathbb N$ formally not a subset of $\mathbb Z$? [duplicate]
I read this while browsing. Basically, the thesis is that $\mathbb N \subset \mathbb Z$ is false, because actually $\mathbb Z_{\ge0}$ is only isomorphic and not actually equal to $\mathbb N$, the same …
- 1,677
1
vote
1
answer
137
views
Models of $\mathsf{ZFC}$
I have three questions about this topic:
How do you denote a generic model of $\mathsf{ZFC}$? For example, a model of $\mathsf{PA}$ is usually denoted as a triple $(\mathbb N, 0, S)$. The first thing …
- 1,677
0
votes
1
answer
174
views
Does the fact that CH is independent of ZFC imply that it's true? [duplicate]
Assume that CH is false. Then, there exists a counterexample; that is, there exists some set $A$ such that there exist an injection $f\colon \Bbb N\to A$ and an injection $g\colon A\to\Bbb R$, but the …
- 1,677
2
votes
1
answer
81
views
Are these statements for the axioms of separation and replacement as formal?
I am writing an introductory text to set theory (it's just for me though, to see if I'm able to do that and practice mathematical writing). Would it be okay if I enunciated the axiom schema of separat …
- 1,677
1
vote
1
answer
75
views
Formal definition of function [duplicate]
In set theory, functions are usually defined as subsets of a Cartesian product. However, this way, the functions $f\colon\Bbb R\to\Bbb R$ and $g\colon\Bbb R\to[-1,1]$, defined as $f(x)=g(x)=\sin x$ fo …
- 1,677
1
vote
0
answers
77
views
Proof of the existence of an inductive set from the "apparently weaker" form of the axiom of...
In the Wikipedia page of the axiom of infinity, an "apparently weaker" of the axiom is stated. It is said that one can prove the existence of $\omega$ using this form as well together with the axioms …
- 1,677
3
votes
1
answer
354
views
Does the axiom of infinity state the existence of the empty set?
The axiom of infinity is usually presented as follows:
$$\exists x((\exists y(\forall z\,\neg(z\in y))\land y\in x)\land\forall s(s\in x\implies\bigcup\{s,\{s\}\}\in x))$$
Now, this does in fact state …
- 1,677
9
votes
4
answers
1k
views
Why do people speak about truth value of undecidable propositions?
I have heard people say things like "If the Goldbach conjecture was proven to be independent of PA, then it would follow it's true." The reasoning behind this is that, if it was false, we could explic …
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