--- anointed. (adjective) 1. smeared or rubbed with oil, typically as part of a religious ceremony. --- ## Std Foundations ### Or: The Fishy People ### Or: The Annointed Ones ### Or: The Oily Ones ### Or: Fat y ácid ### Or: Fish Oil Descendents of Gödel, the ωs are the keepers of the "Standard Foundations" of mathematics: - Set theory - Proof theory[^1] - Model theory[^2] - Recursion theory[^3] ## The People ### Thomas Jech Not really a Λω or even really a full ω, he's an ω in the sense of set theory with a measurable touch of Ω. ### Kanamori and Kunen Kanamori and Kunen are pure Ω, though Kunen's _Foundations of Mathematics_ has a sizable amount of ω too while presenting a view of "foundations" with almost no Λ. ### Peter Hinman One of the best examples of pure Λω culture: the "standard foundations of mathematics." Hinman's massive Mathematical Logic book was written over a span of 20 years, and it's clear that its goal is to cover the standard foundations in all their forms except proof theory, which is fine because proof theory really is the red-headed stepchild of the four subfields of ω, and even most ωs don't know much proof theory anyway. But Hinman succeeds in his goal: Cover as much as possible of 20th century logic as it appears from the viewpoint of the early 21st. He covers about as much of the standard foundations as one could hope for, with a notation designed to painstakingly distinguish the metalanguage from the formal language. And as jarring as that notation might feel at first glance (so many dots), it's essential to have a book like this in the field because nobody else in the field does that as consistently as Hinman. Hinman's book is thus a clarifying example of a fundamentally important distinction that most other authors gloss over with a brief wave of the hand. [^1]: Basically Λ minus logic, with the occasional joyride up to ε₀ or higher. [^2]: Sort of `αγ | sed s/polynomials/wffs/g`. Doesn't feel particularly Λ, and is more or less the practice of using "vibes math" to find "mathematical structures" (where did they come from?) that satisfy formal sentences. In other words, vibes math just exists, and produces all these structures which are just laying around with the same ontological status as chairs, and those structures are then picked up and used to analyze these odd formal languages and formal theories that the Λs came up with for some reason. Basically the opposite of foundational, imo. [^3]: This is just Π, but the ωs claim it's theirs. Sort of a "Taiwan vs Mainland" situation, ISWIM.