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I would like to show that if {$A_{i}$: i$\in$I} $\subseteq$ $A$ $\in$ $\mathbb{U}$, then $\bigcup_{i \in I}$$A_{i}$ $\in$ $\mathbb{U}$, where $\mathbb{U}$ is a universe and the capital $A's$ are all sets (p. 162-163, Lectures on the Hyperreals, William Goldberg).

A universe is a strongly transitive set $\mathbb{U}$ such that:

1) if $a,b\in\mathbb{U}$, then ${a,b}\in\mathbb{U}$

2) if there are sets $A,B\in\mathbb{U}$, then $A\cup{B}\in\mathbb{U}$.

3) if there is a set $A\in\mathbb{U}$, then $\mathcal{P}(A)\in\mathbb{U}$

Strong Transitivity: for any set $A\in\mathbb{U}$, there exists a transitive set $B\in\mathbb{U}$ with $A\subseteq{B}\subseteq{\mathbb{U}}$. Transitivity of $B$ means that $c\in{C}\in{B}\implies{c\in{B}}$

theperennialstudent
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  • Could you clarify what "universe" is for you? – Wojowu Jul 01 '15 at 08:30
  • I'll quote the author: "In... analyzing a particular structure, access may be needed to a wide range of entities: sets, members of sets, sequences, relations, functions, etc. We will posit the existence of a "universe" U that contains all such entities that might be required." Furthermore, such a U will be a called "universe over X" if X is a set that is an element of U and the members of X are regarded as individuals that are not sets and have no members themselves. U will contain at least one set and also the positive integers 1,2,... to ensure that m-tuple formation can be carried out. – theperennialstudent Jul 01 '15 at 09:17

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