Titlebook: Diagonalization in Formal Mathematics; Paulo Guilherme Santos Book 2020 The Editor(s) (if applicable) and The Author(s), under exclusive l

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Modern Approaches in Solid Earth Sciencesen a relation .(.), it is used to refer to the possibility of .(.) being the case for a given element .. It is now immediate how fixed point theorems are a case of diagonalization: given a function . , to say that . has a fixed point is equivalent to say that the relation .(.) defined by . (.) = . is diagonalisable (has a fixed point).

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OVERT

Service Industry and Public Utilities,In this chapter, we will use the Diagonalization Lemma for two purposes: to present natural properties related to self-reference that are not decidable, and to argue that one cannot prove the Strong Diagonalization Lemma using the Diagonalization Lemma, i.e., that diagonalization of term is substantially different from diagonalization of formulas.

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Transportation and Communications,In this chapter, we are going to present a result by Smullyan that is behind several important diagonalization phenomena, and we are going to present a general approach to several paradoxes. The main ideas of this chapter are based on a paper by the author: [SK17].

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Suzanne M. Spencer-Wood,Stacey Lynn CampIn this chapter we will study several examples of diagonalization from Mathematics and we will show that all of them are a particular case of the reasoning of the GDT.

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Two uses of the Diagonalization Lemma,In this chapter, we will use the Diagonalization Lemma for two purposes: to present natural properties related to self-reference that are not decidable, and to argue that one cannot prove the Strong Diagonalization Lemma using the Diagonalization Lemma, i.e., that diagonalization of term is substantially different from diagonalization of formulas.

Between

,Yablo’s Paradox and Self-Reference,Yablo in [Yab93] presented a paradox that, according to him, should not depend on self-reference.

daredevil

,Smullyan’s Theorem, Löb’s Theorem, and a General Approach to Paradoxes,In this chapter, we are going to present a result by Smullyan that is behind several important diagonalization phenomena, and we are going to present a general approach to several paradoxes. The main ideas of this chapter are based on a paper by the author: [SK17].