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7

No (assuming there are standard models at all). $\operatorname{Con}\sf(ZF)$ is always true in standard models, and $\sf ZF$ proves that $\operatorname{Con}\sf(ZF)$ is always true in standard models. But of course that $\operatorname{Con}\sf(ZF)$ is not provable from $\sf ZF$ itself. The reason is that standard models agree with the universe about the ...


4

Assume that $\square_{\omega_1}$ holds and let $(C_\alpha \mid \alpha \in \operatorname{Lim}(\omega_2))$ be a witnessing sequence, i.e. $C_\alpha \subseteq \alpha$ is club, $C_\alpha$ is countable, whenever $\operatorname{cf}(\alpha) = \omega$, $C_\beta = C_\alpha \cap \beta$, whenever $\beta$ is a limit point of $C_\alpha$. Note that each $C_\alpha$ has ...


3

A few examples where the difference is important: The Baire Category Theorem: $\mathbb R$ (or any complete metric space) is not the union of countably many closed sets with empty interiors. The union of countably many sets of Lebesgue measure zero has Lebesgue measure zero. Given countably many sequences of real numbers, there is a single sequence ...


3

Yes. Every infinite structure has a strongly $\omega_1$-homogeneous elementary extension. So you can start with the rationals and find an $\omega_1$-homogeneous elementary extension $(L, <)$ which is countably transitive being $\omega_1$-homogeneous. You can find a construction here. If you also assume CH, then there is a saturated DLO without end points ...


2

One possible way to build sets iteratively from "the universe" instead of the empty set (and the process if fairly symmetrical to the usual one) is to replace axioms with duals. Instead of the axiom of extensionality $$X = Y \equiv (t \in X \equiv t \in Y)$$ use the axiom of "intensionality", according to which, sets belonging to the same sets are equal: ...


1

In ordinal arithmetic $w(w+1)=w^2+w>w+1$ . Also $(w_1)*(w_1+w)=(w_1)^2 + w_1w>w_1+w$.


1

I think the following argument works to show that generic conditions do extend to totally generic ones. As mentioned in the comments, a poset $Q$ is totally proper iff it is proper and countably distributive. I will only need the forward direction and this isn't difficult to see: given a condition $p$ and a name $\dot{f}$ for a countable sequence take a ...


1

First of all, it is not the case that undergraduates are expected to have research experience when applying to grad school. Certainly I didn't. What are super important are letters of recommendation. As to your question: it's great that you want to do research in set theory! However, this is very hard, especially if you have no professor at your school to ...



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