Unterlogiker: ‘In logic, there are morals.’ Überlogiker: ‘Hail Gödelian immoralism!’
Arithmetisation of syntax is usually realised through Gödel-encoding with prime factorisation, but it’s more complicated within PA. Try to think of a way to express ‘the exponent of prime p in factorisation of n is k’ as a predicate with three variables (p,n,k) in PA’s language.
Y’all hoes know damn well that spehs like Mozart n’ Scriabin ain’t stand no chance against ts musical masterpiece right here. #WeTheHiHatGODS
Another interpretation of this is that the space Sₖ(Γ₁(N)) has an action of Γ₀(N)/Γ₁(N) ≃ (ℤ/Nℤ)^× on it, which turns it into a ℂ[(ℤ/Nℤ)^×]-module, and Sₖ(N, χ) is just the χ-eigenspace of Sₖ(Γ₁(N)). This same construction frequently appears in Iwasawa theory.
![ModalMetamodel's tweet image. Another interpretation of this is that the space Sₖ(Γ₁(N)) has an action of Γ₀(N)/Γ₁(N) ≃ (ℤ/Nℤ)^× on it, which turns it into a ℂ[(ℤ/Nℤ)^×]-module, and Sₖ(N, χ) is just the χ-eigenspace of Sₖ(Γ₁(N)). This same construction frequently appears in Iwasawa theory.](https://pbs.twimg.com/media/G3A7P16bwAUHnLH.jpg)
Them hoes was tryna figure out if erry compact Hausdorff space wit at least two points and no isolated points got a cardinality of at least 2^ℵ₀. 😸😸😸
X here is externally uninhabited since no ᾶ ∈ ῶ₁ exists such that the truth value of ᾶ ∈ X is all of 2^ℕ, but it’s also uncountable in the sense that the truth value of ‘for any sequence (xᵢ)_{i< ω} of X’s elements, ∃x ∈ X such that ∀i < ω, xᵢ ≠ x’ is all of 2^ℕ.
There’s a subtlety here. Usually, x ∪ {x} is assured to be a different set from x, and iterating this yields a sequence of distinct sets, which isn’t automatically true in ZF⁻, but you can show that the function S(x) = x ∪ {{y ∈ x : y ∉ y}} has this property even in ZF⁻!
Get that musty ass yt people (sped) music outta here and play some All Red instead.
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