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5 votes

Why is my answer incorrect? (Indexed Sets)

The notation $\bigcup_{X \in \mathcal{P}(\mathbb{N})} X$ does not mean $\{X \mid X \in \mathcal{P}(\mathbb{N})\}$. Rather, it means the union of all $X$ in $\mathcal{P}(\mathbb{N})$. For example, $$\...
Sambo's user avatar
  • 6,713
4 votes
Accepted

Logically equivalent statements about Set Union

First step: forget for the moment the quantifiers and consider Exportatrion: $(P∧Q)→R$ is equivalent to $P→(Q→R)$. Now we have to take care of the quantifier, using the so-caleld Prenex rules: $ϕ → ∀...
Mauro ALLEGRANZA's user avatar
3 votes

Why is my answer incorrect? (Indexed Sets)

Recall that $X\in\mathcal{P}(\mathbb{N})\iff X\subseteq \mathbb{N}$. So the stuff becomes $$\bigcup_{X\subseteq\mathbb{N}}X$$ which is the union of all subsets of $\mathbb{N}$. This clearly gives $\...
Angae MT's user avatar
  • 1,335
2 votes

Countable Choice from Finite Sets

The correct order of implications is $$(1)\to(3)\to(2)\to(4).$$ And all of these implications are strict. To note that (3) implies (2), note that every finite set is Dedekind-finite, but not ...
Asaf Karagila's user avatar
  • 397k
1 vote

Cantor's paradox and the universal set

In an edition of the text this appears to be from, “Iterative Conceptions of Set,", this definition appears (emphasis mine): Definition 13. The Naive Comprehension Schema asserts that for every ...
Steve Kass's user avatar
1 vote
Accepted

Does $\forall x, \exists x$ also needs to be quantified over some sets in zfc subset axiom?

You wrote: "I understand $x \in \cup\cup F$ and $z \in \cup\cup G$, so $\langle x, z\rangle$ exists." You don't need to know $x \in \cup\cup F$ and $z \in \cup\cup G$ to conclude that $\...
Dan Velleman's user avatar
  • 2,901

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