## Global Reflection Principles (2012)

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@MISC{Welch12globalreflection,

author = {P. D. Welch},

title = {Global Reflection Principles},

year = {2012}

}

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### Abstract

Reflection Principles are commonly thought to produce only strong axioms of infinity consistent with V = L. It would be desirable to have some notion of strong reflection to remedy this, and we have proposed Global Reflection Principles based on a somewhat Cantorian view of the universe. Such principles justify the kind of cardinals needed for, inter alia, Woodin’s Ω-Logic. 1 To say that the universe of all sets is an unfinished totality does not mean objective undeterminateness, but merely a subjective inability to finish it. Gödel, in Wang, [17] 1 Reflection Principles in Set Theory Historically reflection principles are associated with attempts to say that no one notion, idea, or statement can capture our whole view of the universe of sets V = ⋃ α∈On Vα where On is the class of all ordinals. That no one idea can pin down the universe of all sets has firm historical roots (see the quotation from Cantor later or the following): The Universe of sets cannot be uniquely characterized (i.e. distinguished from all its initial segments) by any internal structural property of the membership relation in it, which is expressible in any logic of finite or transfinite type, including infinitary logics of any cardinal number. Gödel: Wang- ibid. Indeed once set theory was formalized by the (first order version of) the axioms and schemata of Zermelo with the additions of Skolem and Fraenkel, it was seen that reflection of first order formulae ϕ(v0, , vn) in the language of set theory L∈ ˙ could actually be proven:

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Citation Context ...be falsified in set generic forcing models if there are a proper class of such cardinals. Such absoluteness or fixity results are symptomatic of very general facts due again to Woodin: 1) (Woodin, cf =-=[18]-=-) Assume there is a proper class of Woodin cardinals. Then Th(L(R)) is fixed: no set forcing notion can change Th(L(R)), and in particular the truth value of any sentence about reals in the language o... |

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Citation Context ...ion Principle Das Absolute kann nur anerkannt, aber nie erkannt, auch nicht annähernd erkannt werden. (Cantor: Über unendliche, lineare Punktmannigfaltigkeiten. Mathematische Annalen 1883, Anmerk. 2, =-=[3]-=- p.587.) Instead of ‘formally projecting V ’ à la Reinhardt, let us turn the whole argument all around and generalise to obtain a stronger reflection principle. Let us take at first a somewhat naïve C... |

31 |
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Citation Context ... fact a threshold: general reflection principles were either weaker than the existence of κ(ω) - the first such Erdos cardinal or else inconsistent. 2. It is interesting to see where this ends up. In =-=[15]-=- (Intro.): Extendability is briefly motivated by asserting that there is ‘resemblance’ between different ranks Vα, Vβ within the V hierarchy. In Sect 5, extendible cardinals are motivated by reflectin... |

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Citation Context ...ter alia, Woodin’s Ω-Logic. 1 To say that the universe of all sets is an unfinished totality does not mean objective undeterminateness, but merely a subjective inability to finish it. Gödel, in Wang, =-=[17]-=- 1 Reflection Principles in Set Theory Historically reflection principles are associated with attempts to say that no one notion, idea, or statement can capture our whole view of the universe of sets ... |

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Citation Context ...further can be proven in ZFC alone. However PU becomes a theorem if we add to ZFC the assumpion of the existence of infinitely many Woodin cardinals. Why be concerned about PD? Because (Woodin again, =-=[19]-=-) PD is something of a ‘complete theory’ of countable sets, much as PA is something of a ‘complete theory’ of the finite natural numbers, in the sense that we have no examples of sentences σ about HC ... |

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Citation Context ...discussions of Ackermann’s set theory, which Levy, Vaught and Reinhardt investigated. Reinhardt’s mathematical work on this theory appeared in his thesis which was for the most part then published as =-=[12]-=-. Briefly: Ackermann’s set theory A provided for a universe with extensionally determined entities (classes) and a predicate V˙ for set-hood: “x ∈ V ”. Besides axioms for extensionality, a class const... |

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Citation Context ...nditions for a collection only turn on that it must be sufficiently sharply delimited what belongs to a collection and what does not belong to it. But now the concept of set is wide open.” (Ackermann =-=[1]-=- p.337) 3 , Reinhardt, whilst working from the premise that Ackermann considered the concept of set itself as not sharply delimited ([12] p190), surmises that the intuition behind Ackermann’s principl... |

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Citation Context ...of NBG. Hence if j preserves these Π 2 1 assertions then (V , ∈ , C) will be an NBG model. That done, we may then define Sat 1 for the first order (i.e. ZF) part of the language within NBG (see e.g., =-=[10]-=-). Now, if we wished to formalise the assertion of j (or J) as a fully Σ ω 1 -elementary preserving embedding, we may wish to add to the language a ternary primitive relation symbol Sat 2 with the req... |

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Citation Context ...r fixed points, (in the work of Levy and Bernays) and thence to indescribability properties expressed as sentences in higher order languages is well known and we do not re-tell that here (see Bernays =-=[2]-=-.) However the Global Reflection Property (GRP) we had proposed (whilst we had thought originally of something coming from weakening the third order notion of sub-compact cardinal) was actually much c... |

6 |
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Citation Context ...are absolutely no new mathematical or set-theoretical results here: the derivation of the proper class of measurable Woodin cardinals from the principle is an exercise that can be done by a reader of =-=[6]-=-. The point is that we wish to obtain a reasonable, and proper class, principle in the form of a reflection.)4 Section 2 2 Reinhardt’s Aims We first set the scene with the briefest of discussions of ... |

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Citation Context ...y leaning towards that of Cantor. We seek to distance ourselves thus from the Reinhardtian viewpoint(s). 2 We were spurred to think about these questions by reading an early draft of Peter Koellner’s =-=[7]-=-, where he considered the general problem of reflection, and the particular suggestions of Tait [16] for strengthened higher order languages which admitted limited forms of higher order parameters. Th... |

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Citation Context ... by reflecting upon a transfinite theory of types over and above the universe V . “With the natural reflection down into the world of sets we have the concept of an extendible cardinal. (As Reinhardt =-=[13]-=- points out, however this sort of internalisation within V rather begs the question if we want to discuss fundamental issues about the nature of V and Ω.)”Reflection Principles in Set Theory 3 Howeve... |

4 |
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(Show Context)
Citation Context ...t(s). 2 We were spurred to think about these questions by reading an early draft of Peter Koellner’s [7], where he considered the general problem of reflection, and the particular suggestions of Tait =-=[16]-=- for strengthened higher order languages which admitted limited forms of higher order parameters. These limitations were known since Reinhardt’s investigations, as the latter provided a simple counter... |

3 |
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(Show Context)
Citation Context ...of a suitable pairing function to render the list X1, ,Xm as 〈X1, ,Xm〉.) Given Global Choice we may even define Skolem functions and obtain the above for fully second order Σ n 1 formulae (see, e.g., =-=[4]-=-). We may thus for any n ∈ ω define a Σn 1 formula Sat n 2 (v0, v1, , vk,Y1, , Ym) so that, provably in NBG + Global Choice ∀h ∈ ω 2 V ∀X1, ∀Xm[Satn(�ϕ�, k,m, 〈h0, , hk−1〉, 〈X1, ,Xm〉) ↔ ϕ(h, KX1, , Xm... |

2 |
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Citation Context ...e class Card of cardinals, as they are not sets, so they are not mathematical objects: they are the absolute infinities or inconsistent multiplicities of Cantor (depending on when he was writing (see =-=[5]-=-). We swallow the Cantorian pill that there are two types of objects: the mathematical-discourse or set objects, and the absolute infinities. Let us imagine that C is the collection of all such absolu... |

2 |
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Citation Context ...to other imaginary realms. (In later terms it is On that is the ‘extendable cardinal’ not On ′ ; the emphasis is on the extendability of the domain.) The difference between this treatment and that of =-=[11]-=- is that the latter took the alternatives to the actual universe of sets to be in terms of representations of V in which was itself a set In [14] he discussed a number of then current set theories due... |

2 |
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Citation Context ...ree with Koellner here, as elsewhere in his analysis of Reinhardt, and rather than reiterating his points refer the reader to the final part of his [7]. We turn to the later appearing papers [13] and =-=[14]-=-. In [13] Reinhardt again imagines the possibility of “getting outside” of Cantor’s V = VOn and so thinks of “ordinals” such as “On + 1” and further “sets” such as VOn+On etc. He thinks of this as aki... |

1 |
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(Show Context)
Citation Context ...(V , ∈ , C). Initially at least we presuppose very little about C and its members. We may, if we wish, tell a mereological story about the whole mathematical universe (which is told in more detail in =-=[4]-=-). As above we think of sets as the sole mathematical objects. However sets together with the absolute infinities are parts of the whole realm of mathematical discourse V (V itself is, again, not a ma... |

1 |
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- 1961
(Show Context)
Citation Context ...e sets”), it contained the following crucial principle: • (Ackermann’s Main Principle) If X ⊆ V is definable using only set parameters, and not using the predicate V ˙ , then X ∈ V . • (Levy, Vaught) =-=[9]-=- Let A ∗ be A with the addition of Foundation. Then A ∗ is consistent relative to A, and proves the existence of the classes: {V }, P(V ), PP(V ) • Levy ([8]): A∗ is L∈˙ - conservative over ZF: A ⊢σV ... |