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11

"Positive" is what Leibniz and other proponents of the ontological argument called qualities that make something "better" than it is without them (Anselm spoke of "good" as in summum bonum). In particular, Leibniz defined "perfections" ascribed to God as "simple, positive qualities in the highest degree", see What makes Leibniz's definition of perfection ...


8

Gödel’s Incompleteness Theorem is a result about formal systems. Its proof requires certain assumptions about the properties of specific formal system F: basically, about its "expressive capabilities". In a sense that can be specified rigorously, system F must have the capabilities to manufacture the provability predicate for F, i.e. a suitable formula PrF(...


4

While it’s not quite a perfect parallel, a related concern about decision problems had been phrased some years before Gödel’s work: see https://en.m.wikipedia.org/wiki/Entscheidungsproblem . David Hilbert was pretty convinced as to the decidability of arithmetic prior to the Incompleteness/Incomputability proofs, but was aware that the problem remained open.


4

(1) is a special case of the general principle that if you accept a statement, then you accept that the statement is true. If you believe snow is white, then you believe "Snow is white" is true. It remains handwavy until you give a precise explanation of what you mean by "X is true." And that turns out to be a tricky business.


2

SEP itself refers to Platonism and Mathematical Intuition in Kurt Gödel's Thought by Parsons and On the Philosophical Development of Kurt Gödel by van Atten and Kennedy as the sources for this interpretation. Further discussion can be found in van Atten's book Essays on Gödel’s Reception of Leibniz, Husserl, and Brouwer and Conversations with Gödel chapter ...


2

Before Frege, axiomatic systems were not a focus of philosophy, and Goedel is pursuing the immediate upshot of Frege's failure. So, in some sense, no. Nobody cared. Mathematics was grounded in some internal, perfect mental reality and not really based on axioms. Axioms just helped keep things clear. Paradoxes abound throughout the history of philosophy. ...


1

Godel himself said that all he did was formalise the Cretan liar paradox into a formal system. So the idea or notion of undecidable statements was already apparent a long time before Godel but obviously not phrased in such terms. A simple parallel is with arithmetic. Its easy enough to notice that putting things into a group has certain properties which is ...


1

Positivity is a property of unary predicates that Gödel defines implicitly, i.e. he writes axioms for positive properties. These include the one you cite and consequences of positive axioms being positive, so if any positive property were unsatisfiable all properties would be positive, a contradiction. Similarly, Sobel showed Gödel's axioms imply there are ...


1

I recommend the paper: B. Buldt, The Scope of Godel’s First Incompleteness Theorem, Log. Univers. 8 (2014), 499–552 , especially pages 530 - 531.


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