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The Harmonic series is well known and its divergence was proven back in the middle ages. I've taken an introductory course in model theory so I know a bit about RCF and some properties of it. We did …

The answer to the second question is negative. Take, for example, Sacks forcing over $L$. The result is $L[r]$ where $r$ is a real number and the following property is true: $$\forall x(x\in L\lor r\i …

Note that it is consistent that $\sf ZFC$ doesn't have weak JFEP either. Suppose that there are exactly two different height of transitive models, that is some $\alpha<\beta$ such that $L_\alpha$ and …

If $M$ is supertransitive and satisfies $\sf ZFC$, then $\omega\in M$, and more importantly, $V_\omega\in M$. Now by recursion, if $\alpha$ is an ordinal in $M$, then $V_\alpha\in M$ as well. Therefor …

Yes, that is true. But note that in its nature statements like $\operatorname{Con}(T)$ are meta-theoretic statements. So when we say that $V$ is a model of $\sf ZF$, we mean that in the meta-theory it …

The usual set up of first-order logic is with an infinite reservoir of variables which we can use in formulas. This is one of the annoying reasons why we need to put $\aleph_0$ into the cardinal equat …

What $\sf ZF$ actually proves is that there is a class called $L$, and for every axiom of $\sf ZFC$, the relativization of that axiom holds in $L$. Therefore, the fact we are talking about non-transi …

Obviously not. Any model living inside a pointwise definable model is going to be "externally pointwise definable" for obvious reasons. Now take any generic, symmetric, or otherwise extension of the m …

Very clearly not. Take some countable elementary submodel $M_0$ of $L_{\omega_2}$, and take $M_1$ to be another one, but with $M_1$ a end extension of $M_0$. We can find such models by first finding t …

I don't have anything in mind which "tames" the cardinals structure. I also think that you don't fully understand the consequences of $\sf AD$ on the cardinal structure if you say that it tames the st …

If $T$ is a theory which proves "there is no extension of the model to a model of $\sf ZFC$ without adding ordinals", then there is no extension of models of $T$ by a class forcing to a pointwise defi …

Take a countable elementary submodel of $L_\kappa[0^\#]$, code that into a real, note that "There is a real coding a well-founded model of $V=L[0^\#]$" is a $\Sigma^1_2$ statement, remember what Shoen …

One of my favourite applications is proving that $\sqrt2$ is irrational using ultraproducts. This requires knowing a nontrivial fact, that there are infinitely many prime numbers $p$ such that $x^2\no …