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**1 - 5**of**5**### Global Reflection Principles

, 2012

"... 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 princi ..."

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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:

### The Formalization of Syntax-Based Mathematical Algorithms Using Quotation and Evaluation ⋆

, 2013

"... Abstract. Algorithms like those for differentiating functional expressions manipulate the syntactic structure of mathematical expressions in a mathematically meaningful way. A formalization of such an algorithm should include a specification of its computational behavior, a specification of its math ..."

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Abstract. Algorithms like those for differentiating functional expressions manipulate the syntactic structure of mathematical expressions in a mathematically meaningful way. A formalization of such an algorithm should include a specification of its computational behavior, a specification of its mathematical meaning, and a mechanism for applying the algorithm to actual expressions. Achieving these goals requires the ability to integrate reasoning about the syntax of the expressions with reasoning about what the expressions mean. A syntax framework is a mathematical structure that is an abstract model for a syntax reasoning system. It contains a mapping of expressions to syntactic values that represent the syntactic structures of the expressions; a language for reasoning about syntactic values; a quotation mechanism to refer to the syntactic value of an expression; and an evaluation mechanism to refer to the value of the expression represented by a syntactic value. We present and compare two approaches, based on instances of a syntax framework, to formalize a syntax-based mathematical algorithm in a formal theory T. In the first approach the syntactic values for the expressions manipulated by the algorithm are members of an inductive type in T, but quotation and evaluation are functions defined in the metatheory of T. In the second approach every expression in T is represented by a syntactic value, and quotation and evaluation are operators in T itself. 1

### Absolute Infinity ∗

, 2012

"... This article is concerned with reflection principles in the context of Cantor’s conception of the set theoretic universe. We argue that within a Cantorian conception of the set theoretic universe reflection principles can be formulated that confer intrinsic plausibility to strong axioms of infinity. ..."

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This article is concerned with reflection principles in the context of Cantor’s conception of the set theoretic universe. We argue that within a Cantorian conception of the set theoretic universe reflection principles can be formulated that confer intrinsic plausibility to strong axioms of infinity. How can I talk to you, I have no words... Virgin Prunes, I am God 1

### Conceptualism: sets and classes

, 2012

"... We outline an extension of Martin’s view of a conceptual realism, to a Cantorian realm of absolute infinities. We then formulate a strong reflection principle within this framework to obtain extra-constructible large cardinals. 1 ..."

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We outline an extension of Martin’s view of a conceptual realism, to a Cantorian realm of absolute infinities. We then formulate a strong reflection principle within this framework to obtain extra-constructible large cardinals. 1