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proof of properties of universe
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(Proof)
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- This is the special case of axiom 2 with $x=y$ since $\{x,x\} = \{x\}$ . (In other words, in set theory, we do not count duplicate entries twice.)
- By definition of power set, if $x \subset y$ , then $x \in \mathcal{P} (y)$ . By axiom 3, $\mathcal{P} (y) \in \mathbf{U}$ . By axiom 1, it follows that $x \in \mathbf{U}$ .
- By axiom 2, $\{x,y\} \in \mathbf{U}$ . By axiom 2 again, it follows that $\{\{x,y\},x\} \in \mathbf{U}$ .
- By axiom 2, $\{x,y\} \in \mathbf{U}$ . If we set $z_x = x$ and $z_y = y$ , then $x \cup y = \bigcup_{i \in \{x,y\}} z_i$ , hence, by axiom 4, $x \cup y \in \mathbf{U}$ . -- If $x \in U$ and $y \in U$ then, by axiom 1, $a \in \mathbf{U}$ for all $a \in x$ and $b \in \mathbf{U}$ for all $b \in y$ . By property 3, if $a \in \mathbf{U}$ and $b \in \mathbf{U}$ , then $(a,b) \in \mathbf{U}$ ; further, by property 1, $\{(a,b)\} \in \mathbf{U}$ . Hence, by axiom 4, $\{(a,b) \mid b \in
y \} = \bigcup_{b \in y} \{(a,b)\} \in \mathbf{U}$ for all $a \in x$ . Using axiom 4 again, we conclude that $x \times y = \{(a,b) \mid a \in x \wedge b \in y \} = \bigcup_{a \in y} \{(a,b) \mid b \in y \} \in \mathbf{U}$
- By axiom 4, $\bigcup_{i \in I} x_i \in \mathbf{U}$ . By property 4, $I \times \bigcup_{i \in I} x_i \in \mathbf{U}$ . Now, every function from $I$ to $\bigcup_{i \in I} x_i \in \mathbf{U}$ is a subset of $I \times \bigcup_{i \in I} x_i \in \mathbf{U}$ . Since $\prod_{i \in I} x_i$ is a set of functions from $I$ to $\bigcup_{i \in I} x_i \in \mathbf{U}$ , we have, by defintion of power set, $\prod_{i \in I} x_i \subset \mathcal(P) (I \times \bigcup_{i \in I} x_i)$ . Hence, by axiom 3 and property 2, we conclude that $\prod_{i \in I} x_i \in
\mathbf{U}$ .
- Assume the contrary, namely that $x \in \mathbf{U}$ and $\#x \ge \#\mathbf{U}$ . By axiom 3, $\mathcal{P} (x) \in \mathbf{U}$ but $\#(\mathcal{P}(x)) = 2^{\#x} \ge 2^{\#\mathbf{U}}$ . Since, by axiom 1, every element of $\mathcal{P} (x)$ belongs to $\mathbf{U}$ , this would mean that we would have at least $2^{\#\mathbf{U}}$ elements of $\mathbf{U}$ , which contradicts the fact that $\#U < 2^{\#\mathbf{U}}$ . (This argument is a variation on Cantor's paradox.)
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"proof of properties of universe" is owned by rspuzio.
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Cross-references: Cantor's paradox, variation, argument, mean, subset, function, property, power set, set theory, axiom
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This is version 5 of proof of properties of universe, born on 2005-12-29, modified 2005-12-29.
Object id is 7544, canonical name is ProofOfPropertiesOfUniverse.
Accessed 1087 times total.
Classification:
| AMS MSC: | 03E30 (Mathematical logic and foundations :: Set theory :: Axiomatics of classical set theory and its fragments) | | | 18A15 (Category theory; homological algebra :: General theory of categories and functors :: Foundations, relations to logic and deductive systems) |
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Pending Errata and Addenda
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