Reddit Community for fans of the Los Angeles Dodgers.
Welcome to P1ZZ4_M0ZZARELL4, a subreddit made by u/P1ZZ4_M0ZZARELL4. Unfortunately we have closed.
https://www.holdmycosmo.net/ https://linkin.bio/holdmycosmo https://www.mixcloud.com/holdmycosmo/
I have all cards any Tenpai player could ask for EXCEPT: 3rd fenrir, necrovalley, S:P, Black-WINGED Dragon, and some cards for the calamity package
That does mean I have the Kash planet, the set rotation, the tzolkin, the crystal wing, the striker and dharc, the timelord level 10, samurai des, ruddy rose, everything. I’m looking to see what builds I could make. I have an idea for a less conventional build but anytime I deck build for myself I go x-3 at locals, the times I didn’t was either the one time with branded and the one time with a typical Tenpai build (no trident, I just got it yesterday)
The build I was thinking was “if you’re not a starter or an interruption, gtfo except Crossout” which means no prospy, I also opted for a handtrap build. I personally don’t think the whole “oh if u stop opp with 1 handtrap then they have 20” matter because if I also only have the one starter, it goes to grind game to see who can draw a 1 card combo first and I already have a statistical advantage with getting my 6th draw first, and that’s if I don’t see Crossout. Maybe I play called by? Idk pls help. And don’t encourage my idea because “all ideas are good” or “if that’s what you want to go for” cuz I wanna get my invite even if I don’t use it lol
1 Lévy Hierarchy
The Lévy hierarchy is a hierarchy of formulae. For each natural number n, Σ_n, Π_n and Δ_n form classes of formulae. The lowest rank of the Lévy hierarchy is Σ_0 = Π_0 = Δ_0, i.e. recursive. A formula φ is said to be recursive iff φ is equivalent to a formula that can be built from the atomic sentence "x ∈ y" using logical connectives and bounded quantifiers. A quantifier refers to either ∃, the existential quantifier, or ∀, the universal quantifier. A quantifier is "bounded" if its domain is a set: i.e. it's of the form ∃x ∈ y[φ] (≡ ∃x[x ∈ y ∧ φ], "there exists an x in y so that φ is true") or ∀x ∈ y[φ] (≡ ∀x[x ∈ y → y], "for all x in y, φ is true"). An unbounded quantifier quantifies over the whole universe, e.g. ∃x[φ] means "for all x, φ is true". An example of a recursive formula is "a is transitive", which can be written as "∀x ∈ a∀y ∈ x[y ∈ a]". The rest of the ranks of the Lévy hierarchy are defined as follows:
- A formula φ is Σ_n+1 iff φ is equivalent to ∃x₀,...,xₖ[ψ] for some Π_n formula ψ
- A formula φ is Π_n+1 iff φ is equivalent to ∀x₀,...,xₖ[ψ] for some Σ_n formula ψ
- A formula φ is Δ_n iff φ is both Σ_n and Π_n
A Π_n formula is thus a formula with n alternations of quantifiers starting with a universal quantifier, same for Σ_n but starting with an existential quantifier. So an example of a Π_2 formula is: "∀x∃y[x ∈ y]".
Note that all Π_n and Σ_n-formulae φ are both Π_n+1 and Σ_n+1.
Lévy hierarchy - Wikipedia For a transitive set M and a formula φ, M ⊧ φ is true iff φ is true within M: when bounding all unbounded quantifiers in φ to M, the resulting formula is true. For example, ω ⊧ ∀x∃y[x ∈ y] as ∀x ∈ ω∃y ∈ ω[x ∈ y]. I'll refer to these transitive sets as "models". Note that, because all quantifiers in M ⊧ φ are bounded, the formula "M ⊧ φ" is always recursive regardless of the complexity of φ.
Model theory - Wikipedia An important notion in the Lévy hierarchy is absoluteness: a formula φ is said to be absolute iff the truth of φ does not change between models. For models M and N so that the parameters in φ are members of both M and N, M ⊧ φ iff N ⊧ φ. A formula φ is said to be upwards absolute iff the truth of φ "climbs upwards". For models M and N so that M ⊂ N and the parameters in φ are members of both M and N, if M ⊧ φ then N ⊧ φ: if φ is true in a small model M, then it must be true in every larger model N ⊃ M. A formula φ is said to be downwards absolute iff the truth of φ "climbs downwards". For models M and N so that M ⊃ N and the parameters in φ are members of both M and N, if M ⊧ φ then N ⊧ φ: is φ is true in a large model M, then it must be true in every smaller model N ⊂ M (as long as that smaller model still has the parameters of φ).
[Lemma 1.1] (1) All recursive formulae are absolute. (2) All Σ_1 formulae are upwards absolute. (3) All Π_1 formulae are downwards absolute.
A proof of this lemma is left as an exercise for the reader.
Absoluteness (logic) - Wikipedia)
2 Constructible Hierarchy
Gödel's constructible hierarchy is a cumulative hierarchy like the von Neumann hierarchy. The constructible hierarchy is defined as follows:
- L_0 = ø is the empty set
- L_α+1 = {{x ∈ L_α L_α ⊧ φx} φ a formula} is the set of definable subsets of L_α
- L_α = ∪{L_β β < α} for limit α, is the union of all previous ranks
Note that L_α+1 only has all definable subsets of L_α, and not all subsets of L_α. Since finite sets are always definable, the constructible hierarchy corresponds with the von Neumann hierarchy in the interval [0, ω]. From this point, they start to diverge, and L_α is always a subset of V_α. For example, L_ω+1 does not contain a set of Gödel numbers of all and only formulae that are true in L_ω, but V_ω+1 does. To get an idea of how the constructible hierarchy works, you can try to prove the following lemma:
[Lemma 2.1] For infinite α, the cardinality of L_α is equal to the cardinality α (i.e. there is a bijection between L_α and α).
As opposed to the von Neumann hierarchy, which always has the cardinality of V_ω+α equal to ℶ_α.
Constructible universe - Wikipedia 3 Recursive Analogues
[Definition 3.1] C_Ω(a,b) is the closure of b ∪ {0} under addition, x ↦ Ωω^x and ψ with arguments restricted to < a. ψ_Ω(a) is the least b for which C(a,b) ∩ Ω = b.
Usually, Ω here is chosen to be the least uncountable, ω₁. This is because ω₁ has a curtained property: it is regular, i.e. for all f: ω₁ → ω₁, there is some a < ω₁ so that for all x < a, f(x) < a. This means that "for all a, ψ_ω₁(a) is strictly smaller than ω₁" is easy to prove.
However, Ω can also be chosen to be a much smaller ordinal, i.e. the Church Kleene ordinal, CK. This ordinal is recursively regular, i.e. for all
recursive f: L_CK → L_CK, there is some a < CK so that for all x ∈ L_ a, f(x) ∈ L_a. "for all a, ψ_ω₁(a) is strictly smaller than ω₁" is still true, but a bit more difficult to prove.
The Church Kleene ordinal is a
recursive analogue to the least uncountable: they behave roughly the same, but CK is much smaller than ω₁.
Recursive analogues are useful as they can be proven to exist in ZFC, while their non-recursive analogues cannot. For example, a recursively inaccessible (recursively regular limit of recursively regulars) can be proven to exist in ZFC, while an inaccessible cardinal (regular strong limit cardinal) cannot.
In this post, I'll be focusing on recursive analogues.
4 Reflecting Ordinals
[Definition 4.1] Let a be an ordinal, let Φ be a set of formulae and let A be a class of ordinals. a is said to be Φ-reflecting on A iff for all φ ∈ Φ, if L_a ⊧ φ, then there is some b < a so that b ∈ A and L_b ⊧ φ (i.e. φ "reflects" to some member of A) (it should be understood that the parameters of φ must be in L_b). Φ may be omitted if it's the set of all formulae. A may be omitted if it's the class of all ordinals.
For example, for Π_2-reflecting a, if L_a ⊧ ∀x∃y[x ∈ y] then there must be some b < a so that L_b ⊧ ∀x∃y[x ∈ y].
If a is Π_0-reflecting, then a must be a limit ordinal: let x < a be some ordinal. Then, the Π_0-sentence "¬x ∈ x" must reflect to some b < a. So a b for which x < b < a is found, meaning that a is a limit ordinal.
Using downwards absoluteness, the following lemma can be proven:
[Lemma 4.2] For ordinal a and class of ordinals A, all of the following are equivalent:
- a is Π_0-reflecting on A
- a is Π_1-reflecting on A
- a is a limit of A
A proof of this lemma is left as an exercise for the reader.
A "witness" of a formula ∃x[φx] is an x for which φx is true. Using the notion of a witness, the following lemma can be proven:
[Lemma 4.3] For ordinal a and class of ordinals A, a is Σ_n+1-reflecting on A iff a is Π_n-reflecting on A
One might see that a formula ∀x∃y[φ] is true if there is a function f mapping x to y so that φ always holds for these x and y. Using this, the following can be proven:
[Lemma 4.4] An ordinal a is Π_2 reflecting iff a is recursively regular, i.e. for all recursive f: L_a → L_a, there is some b < a so that for all x ∈ L_b, f(x) ∈ L_b.
A Π_n-class is a class of ordinals A for which there exists a Π_n-formula φ so that φ defines A: for all a, a ∈ A iff L_a ⊧ φ. A Π^1_0-class is a class of ordinals A that is Π_n for some n.
For positive natural n, and Π^1_0-class A, the sentence "the universe is Π_n-reflecting on A" is Π_n+1. The class of ordinals that are Π_n-reflecting on A is thus a Π_n+1-class. It's left as an exercise to the reader to figure out why it is Π_n+1, and why this wouldn't always work when n = 0 or when A isn't a Π^1_0-class.
To give an example, the class of limits of recursively regulars (Π_1 on Π_2, see lemma 4.2 and 4.4) is a Π_2-class: the class of Π_2-reflecting ordinals is Π_3, thus Π^1_0, thus the class of Π_1-reflecting ordinals on Π_2-reflecting ordinals is Π_2.
Using this (the general case, not the example above), the following lemma can be proven:
[Lemma 4.5] Let a be an ordinal, let n be a positive natural, let A be a Π^1_0-class and let B be a Π_n+1-class. Suppose a is Π_n+1-reflecting on A and a is a member of B. Let C be the class of ordinals that are Π_n-reflecting on the intersection of A and B. a ∈ C and C is a Π_n+1-class.
This lemma is a bit more difficult to prove than the previous ones, but should still be doable.
For example, when a is Π_2-reflecting and Π_1-reflecting on the class of Π_2-reflecting ordinals (Π_2 and Π_1 on Π_2), it can be concluded that a is Π_1-reflecting on the class of ordinals that are Π_1-reflecting on the class of Π_2-reflecting ordinals (Π_1 on Π_1 on Π_2). So a is Π_2 and Π_1 on Π_1 on Π_2, so it is Π_1 on Π_1 on Π_1 on Π_2, etc. This is an analogue to how inaccessible cardinals are limits of regular cardinals, limits of limits of regular cardinals, limits of limits of limits of regular cardinals, etc.
Let n be a positive natural and let f: Ord → P(Ord) be a recursive function mapping ordinals to Π_n-classes. To be precise, f is characterized by a Π_n formula φ so that b ∈ f(a) iff L_b contains all parameters in φ, a < b and L_b ⊧ φ(a). The class of ordinals {b ∀a < b[b ∈ f(a)]}, which can be viewed as a "diagonal intersection" of f, is Π_n as well: it is characterized by the Π_n formula ∀a[φ(a)]. If the range of f is restricted to some ordinal x, then we can simply take the intersection ∪{f(a) a < x} (restricted to ordinals b > x) which'd be a Π_n-class.
For example, f(a) for a < ω can be defined as the class of ordinals that are a-ply Π_1-reflecting (a-ply means that reflection is iterated a times), each of these classes are Π_2, so the resulting intersection (ω-ply Π_1-reflecting ordinals, i.e. ordinals of the form ω^ω) is a Π_2-class as well.
To give another example, f(a) can be defined as the class of ordinals that are Π_1-ref on a-ply (Π_2-ref and Π_1-ref on ...), i.e. limits of recursively a-inaccessibles. Each of these classes are Π_2, so the diagonal intersection {b ∀a < b[b ∈ f(a)]}, which is the class of recursively pseudo hyper-inaccessibles, is Π_2 as well.
To get an idea of the size of Π_2-reflecting ordinals, one can use the idea of diagonal intersections to prove the following:
[Lemma 4.6] If a is Π_2-reflecting then a is an epsilon number (i.e. a-ply Π_1-reflecting).
In fact, a then is a ζ-number, strongly critical (a Γ-number), and much much more.
In the following chapters, we'll build stronger and stronger OCFs, eventually reaching the PTO of KP + Π^1_0-reflection.
5 Recursively Inaccessible OCF
Let I denote the least recursively inaccessible (Π_2-ref limit of Π_2-refs). Let ε(I) denote the least epsilon after I. Let R denote the set of recursively regulars at most I. a and b always denote ordinals < ε(I), κ always denotes a member of R.
[Definition 5.1] H maps pairs of ordinals < ε(I) to sets of ordinals < ε(I). ψ maps pairs of ordinals ∈ R × ε(I) to ordinals < I. H_a(b) and ψ_κ(a) are defined by induction on a. H_a(b) is the closure of b ∪ {0} under addition, x ↦ x⁺ (where x⁺ is the least recursively regular above x), x ↦ Iω^x and ψ with the second argument restricted to < a. ψ_κ(a) is defined iff κ ∈ H_κ(a). ψ_κ(a) is the least b for which κ ∈ H_b(a) and H_b(a) ∪ κ = b.
For example, ψ_I(0) = ω_ω^CK is the least limit of admissible ordinals (admissible = recursively regular or ω).
[Lemma 5.2] For ordinals κ, λ ∈ R and a, b < ε(I) so that κ, a ∈ H_a(κ) and λ, b ∈ H_b(λ), ψ_κ(a) < ψ_λ(b) iff (1) for κ ≠ I and λ ≠ I: κ < λ or [κ = λ and a < b], (2) for κ ≠ I and λ = I: κ ≤ ψ_λ(b), (3) for κ = I and λ ≠ I: ψ_κ(a) < λ, (4) for κ = I and λ = I: a < b.
[Lemma 5.3] For ordinals x ∈ R ∪ {0} and a < ε(I) for which x⁺ ∈ H_a(x⁺), x < ψ_x(a) < x⁺. For a < ε(I), ψ_I(a) < I.
These lemmas can be proven using the lemmas in chapter 4.
6 Recursively Mahlo OCF
Let M denote the least recursively Mahlo (Π_2-ref on Π_2-ref). Let ε(M) denote the least epsilon after M. Let R denote the set of recursively regulars below. a and b always denote ordinals < ε(M), κ always denotes a member of R.
We now have two "degrees": Π_2-reflection, and Π_2-reflection on Π_2-reflection. Each of these degrees requires its own set of OCFs. For the simple Π_2-reflecting ordinals, I'll define a collapsing function ψ_κ similar to the one in chapter 5. For Π_2-ref on Π_2-ref, I'll define a collapsing function χ_M collapsing ordinals > M to Π_2-reflecting ordinals < M. Generally, if κ is Π_2-reflecting on a Π^1_0 class X, then a collapsing function ψ^X_κ can be defined collapsing ordinals > κ to ordinals ∈ X ∩ κ (in X and below κ).
[Definition 6.1] H maps pairs of ordinals < ε(M) to sets of ordinals < ε(M). ψ maps pairs of ordinals ∈ R × ε(M) to ordinals < M. χ_M maps ordinals < ε(M) to ordinals ∈ R. H_a(b), ψ_κ(a) and χ_M(a) are defined by induction on a. H_a(b) is the closure of b ∪ {0} under addition, x ↦ Mω^x, ψ with the second argument restricted to < a and χ with arguments restricted to < a. ψ_κ(a) is defined iff κ ∈ H_a(κ). ψ_κ(a) is the least b for which κ ∈ H_a(b) and H_a(b) ∩ κ = b. χ_M(a) is the least κ for which H_a(κ) ∩ M = κ.
For example, χ_M(M^2) is the least recursively 2-inaccessible: Π_2-ref and limits of rec inacs.
[Lemma 6.2] For ordinals a, κ so that κ ∈ H_a(κ), ψ_κ(a) < κ. χ_M(a) < M.
7 Recursively Weakly Compact OCF
Let K be the least Π_3 reflecting ordinal (recursively weakly compact). Let ε(K) denote the least epsilon after K. For Π^1_0-class X, let Π_2[X] denote the set of ordinals that are Π_2-reflecting on X. a, b, d, κ always denote ordinals < ε(K).
Now, we have a myriad of degrees: Π_2-reflecting, Π_2-ref on Π_2-refs, hyper-Π_2-refs (using diagonal intersections), (1 @ ω)-Π_2-ref (I'm not going to explain what that means). We thus need more OCFs that can handle these degrees. I thus introduce a new argument, d, called the degree: ψ^d_κ(a), where κ is Π_2-reflecting on ordinals with degree d, collapses to ordinals with degree d. Here, degree d roughly refers to d-ply Π_2-reflecting ordinals (for small d, for large d, it also refers to hyper-Π_2-ref, etc, through the use of collapsing). To know what degree what ordinal is, we'll define a
thinning hierarchy M^d. A thinning hierarchy is a hierarchy of sets of ordinals that, well, thins: for a > b, M^a ⊆ M^b. M^0 would be the set of all ordinals < ε(K), M^1 is the set of recursively regulars < ε(K), etc.
[Definition 7.1] H maps pairs of ordinals < ε(K) to sets of ordinals < ε(K). M maps ordinals < ε(K) to sets of ordinals < ε(K). ψ maps triplets of ordinals < ε(K) to ordinals < ε(K). H_a(b), M^a and ψ^d_κ(a) are defined by induction on a. H_a(b) is the closure of b ∪ {0}, x ↦ Kω^x and ψ with the third argument restricted to < a. M^d is the set of b < ε(M) so that, for all d₀ ∈ H_d(b) ∩ d, b ∈ Π_2[M^d₀]. ψ^d_κ(a) is defined iff d ≤ a, κ ∈ H_a(κ) and κ ∈ Π_2[M^d]. ψ^d_κ(a) is the least b so that κ ∈ H_a(b), b ∈ M^d and H_a(b) ∩ κ = b.
For example, ψ^{K^ω}_K(K^ω) is the least (1 @ ω)-Π_2-reflecting ordinal.
[Lemma 7.2] For ordinals d, a, κ so that d ≤ a and κ ∈ H_a(κ), ψ^d_κ(a) < κ.
8 Reflecting OCF
To extend the OCF in chapter 7 to full reflection, more complicated degrees are needed. Traditionally, these degrees were complicated structures involving tuples of ordinals and other degrees. However, one can see that an exponential structure arises from these degrees, and just use base Ξ Cantor normal form as degrees.
Let Ξ denote the least reflecting ordinal. For natural n ≥ 2 and Π^1_0-class X, let Π_n[X] denote the class of ordinals that are Π_n-reflecting on X. Let ε(Ξ) denote the least epsilon number after Ξ. a, b, κ always denote ordinals ≤ Ξ, d always denotes an ordinal < ε(Ξ), n always denotes a natural number ≥ 2. I'll write d =nf d₀ + Ξ^d₁ × d₂ if d = d₀ + Ξ^d₁ × d₂, d₀ is a multiple of Ξ^{d₁+1} (d₀ = 0 is allowed) and d₂ < Ξ.
[Definition 8.1] H maps pairs of ordinals < Ξ to sets of ordinals < ε(Ξ). M maps pairs consisting of an ordinal < ε(Ξ) and a natural ≥ 2 to sets of ordinals < Ξ. ψ maps triplets of ordinals d,a,κ to ordinals < Ξ. H_a(b), M^d_n and ψ^d_κ(a) are defined by induction on a (in the case of M^d_n, a is the largest base Ξ CNF component of d). H_a(b) is the closure of b ∪ {0} under addition, x ↦ Ξω^x and ψ with the third argument restricted to < a. For d = 0, M^d_n is the set of all ordinals < Ξ, otherwise for d =nf d₀ + Ξ^d₁ × d₂, M^d_n is the set of ordinals b < ε(Ξ) for which b ∈ M^d_{n+1}(a) and for all d₂' ∈ H_d₂(b) ∩ d₂, b ∈ Π_n[M^{d₀ + Ξ^d₁ × d₂'}_n]. ψ^d_κ(a) is defined iff the largest base Ξ CNF component of d is at most a and d,κ ∈ H_a(κ) and κ ∈ Π_2[M^d_2]. ψ^d_κ(a) is the least b for which d,κ ∈ H_a(b), b ∈ M^d_2 and H_a(b) ∩ κ = b.
For example, ψ^{Ξ2}_Ξ(0) is the least Π_3-reflecting ordinal that is Π_2-reflecting on the set of Π_3-reflecting ordinals.
I'm tired. Bye!
.(˙˘˙)‘/’