A club set C⊆κ is a Meta-Relative subset of κ(denoted as ℳκ) if it is closed and unbounded in κ and cannot be reached by Meta Relative Extensions by lower sets.This means that:
a.∃N⊆ℳκ,C⊆ℳN b.For any mapping f and α⊆C→f(α)⊆C. c.∃j:M→M,crit(j)<η∧η⊆C⇔j(η)⊆C. d.N⊆C∨N[G]⊆C→N⊆C∧N[G]⊆C. e.If within a formula φ there exists α and β>α,β does not satisfy φ,then{γ|γ⊆ℳκ}∈C. f.∀M, N is ℳκ,{N⋂M∨N⋃M} is ℳκ.
Furthermore, for any elementary embedding j:V→M with crit(j)<c, j(C)⋂κ=C⇒C reflects at κ.
A recursion ℝ is a Boundless Recursion in λ(denoted as ℝλ) if ℝ is closed to all operations over λ({x, μ | μ⊆ℳx}∈ℝ), and:
a.There is a hierarchy of {ℝα}α∈Ord | ∃ℳα, ℝα∉ℳλ.
b.∀f:λ→λ, {γ | γ>λ & f[γ]⊆γ}∈ℝ.
For a Meta-Relative Subset S⊆ζ, S is a Transcendental subset of ζ(denoted as 𝒯ζ) if it is closed to any Boundless Recursions, and:
S is a club set and belongs to a club filter of ζ, and ∃𝒯ζ, ∀𝒯ζ⊆𝒯ζ*, where 𝒯ζ* is also a Transcendental subset.
For every 𝒯ζ, ∃ς∈S, 𝒯ς⊆S∩ς.
∀𝒰 | 𝒰 is an ζ-complete ultrafilter on ζ⇒S∈𝒰.
A Reinhardt cardinal δ is Limitless(δ=𝔈0) if:
For jα:Mα→Mα(In which jα(crit(jα))⫋jα+1 j0:M0→M0 is j:M→M, and jα(Mα)⫋Mα+1), crit(jδ)⫋𝒯δ. Especially, such an embedding jα:Mα→Mα is called an iterated embedding.
∀κ<δ, κ⫋𝒯δ.
∀χ>δ, 𝒯χ∩δ=𝒯δ.
A Limitless Cardinal is Super-Limitless(θ=𝔈∞) if:
There exists a S⊆𝒯θ, S⊆{𝒰|𝒰 is a club filter on θ} in every θ-complete ultrafilter on θ, and ∀ς∈S, 𝒯θ⊆S⋂ς.
For all iterated embeddings jα:Mα→Mα such that ∃H⫋𝒯θ, j(H)∩θ=H, for all θ⊆Mβ. And crit(jα)⫋𝒯θ⫋U(an θ-complete ultrafilter on θ).
ℝ=⋃θ≥αℝα, such that: if μ∈ℳλ, μ⫋𝒯ℝλ. And ∀κ<δ and κ⊆𝒯ς, ς∈ℝx. In addition,∀k:θ→θ and ζ<θ, ∃{γ>ζ|k(γ)⊆γ}∈ℝn.