#mathematical logic

A condition p ∈ P iff p ∈ L(S) ∩ M and p is a closed subset of S.

The forcing poset (Skibidi) forces constructibility; take p < q iff p \subseteq q. (Both are sequences of elements, which is very important in ZF) Let p ∈ P and (it) assume[s] p ⊨ ṡ to be constructible, with ṡ being a sequence or embedding.

S in condition 1 is allowed to be non-ω_1.

Theorem 1. If f is a continuous real-valued function on X, then f^{-1}[0] is a G_δ. The set {0} is in the space of all reals.

My first proof of this theorem was of the following: The real numbers ℝ have a countable base, and from this it needs to be shown that the image of f^{-1}: ℝ -> {0} is G_δ. If f^{-1}{0} were somehow not G_δ, then there will be only be a finite intersection of open sets U_α in R covering f^{-1}[0] in the form f^{-1}[0] ⊂ {∩ U_α}. However, one can easily select other open sets (let's label it U_β for now) that will cover f^{-1}[0] ⊂ {∩ U_α} ⊂ {∩ U_β}; this can go on ad nauseum to infinity, as R has a countable base.

I also came up with another proof. For some B_1 and B_2 ∈ ℝ and every point x ∈ B_1 ∩ B_2, there exists a B_3 ∈ B such that x ∈ B_3 ⊂ B_1 ∩ B_2. For f^{-1}[0] in particular, and for f{B_1], f[B_2] ∈ B and every point f^{-1}[0] ∈ f[B_1] ∩ f[B_2], there is a f[B_3] ∈ B that is a subset of f[B_1] ∩ f[B_2] which contains f^{-1}[0] and is smaller than the original B_1's and B_2's. From this, it may be deduced that there is a {∩ B_α} of open sets covering f^{-1}[0], in which α is countable.

how can i find out more about it / can you do a big post or set of posts introducing it and building it up from foundations?

I would recommend the following books:

Set Theory by Kunen (the newer version that just says, “Set Theory”)

The Higher Infinite by Akihiro Kanamori

Classical Descriptive Set Theory by Kechris

Set Theory by Thomas Jech (although this is more of a reference work than anything)

The core model is a special type of model invented by Jensen in order to preserve the covering lemma if there is no inner model that has a measurable cardinal. This seems oddly specific. Wonder what the motivation for that could be.

Anyway, the model in particular is called the "Dodd-Jensen core model". It is also not absolute, especially with regards to different kinds of sharps. It is also the basis of "mice" named that way because the core model is often thought of as the union of mice, with mice essentially being "nice" models of $L^U$. More specifically, as $L_U$ is an inner model with $\kappa$ being a measurable, and $P(\kappa) \cap K$, in iterating $L^U$ to models $L^{U_i}$ with $U_i$ normal on $\kappa_i$;

\begin{equation}

K = \bigcap_{i < \aleph_0} L_{U_i}.

\end{equation}

You might wonder if we live in a simulation, and mathematics helps explore this idea seriously. Mathematical logic frames what reality means, but computational limits challenge any fully detailed simulation of the universe. Quantum mechanics introduces paradoxes complicating simple models, while algorithmic information theory measures the complexity behind universe encoding. Probability theory…

Today I Learned about an E-M (Ehrenfeucht-Mostowski Set) formulation of Zero Sharp. Basically, a set Σ of formulas of B is an E-M set if and only if there is a model A = ⟨A,E⟩ of BSL + V = L and an infinite set H ⊆ On^A which is A-indiscernible, such that Σ is the set of all ℒ-formulae which are valid in A on increasing sequences of indiscernibles from H. BSL is essentially an inner model of KP consisting of finitely many instances of the Σ_0-collection schema of KP such that L_λ satisfies BSL for any limit ordinal λ > ω, and that if M is a transitive model for BSL, then (L_α)^M = L_α for any alpha ∈ M.

The unique, well-founded, remarkable, and cofinal E-M set is called Zero Sharp or just 0^#.

Tbh I think this is very interesting, considering that Zero Sharp is typically given in terms of indiscernables anyway and EM blueprints (at least in Kanamori), whereas in this formulation we use E-M sets in a way such that we have more regard for, say, inner models of set theory?