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I'm reading Y. Manin's http://books.google.co.il/books?id=...resnum=1&ved=0CCkQ6AEwAA#v=onepage&q&f=false" and I've been having some difficulties. So I'm asking for help.

1. On page 64 of the book, (prooving Mostowski proposition). We have a subset N of the Von neumann universe, and [tex]N_\alpha \subset N[/tex] for all cardinals [tex]\alpha [/tex]. Now we want to prove that [tex] \cup N_\alpha= N [/tex], his first step is, assume otherwise and assume there is [tex]X\in N\setminus \cup(N_\alpha) [/tex] such that [tex]X \cap (N\setminus \cup N_\alpha) = \emptyset [/tex] then [tex]X\subset \cup N_\alpha [/tex] thus there is some [tex]\alpha_0 [/tex] s.t [tex]X\subset N_\alpha_0 [/tex]

Now, that last part I don't understand. why the fact that a set is a subset of a union of sets, it must be a subset of one of them? (which is clearly not true as a general argument, but why is it true here?)

2. On page 52, he presents Boolean algebras with axioms

[tex](A^')^' = A [/tex]

[tex] \vee \wedge [/tex] are associative commutative and distributive

[tex](a \vee b )^' = a \wedge b [/tex] [tex](a \wedge b)^' = a \vee b [/tex]

[tex] a \wedge a = a \vee a = a [/tex]

[tex] 1 \wedge a = a [/tex] [tex]0 \wedge a = a [/tex]

later he claims that for any map from a set of formulas to the boolean algebra, the value on the simple tautologies must be 1

but I don't see why it is so. isn't there something missing? for example, I can't see how to prove that [tex]a^' \vee a = 1 [/tex].

Thanks

[Sorry for all these Latex oddities, does anyone know how to exit the "uppercase" mode?]

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# Undergraduate mathematical logic questions

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