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10

"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 ...


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

Short version: No, this doesn't work. First, note that you haven't stated Godel's theorem correctly - rather, it is: Every consistent computably axiomatizable theory "containing enough arithmetic" is incomplete. "Computably axiomatizable" here basically means that the theory isn't so complicated as to be impossible to describe; for ...


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.


3

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

I think the title and body questions are subtly different. Here I'm going to address the title question, which I'll paraphrase for clarity as: What sort of "mathematical truth" can a non-Platonist make sense of? I think this is less strange than it may first appear, since there is an existing parallel: "sharp" vs. "fuzzy" referents in natural language. ...


2

See e.g. Gödel’s Incompleteness Theorems : Diagonalization: it is a general result of FOL that : Let A(x) be an arbitrary formula of the language of F with only one free variable. Then a sentence D can be mechanically constructed such that F ⊢ D ≡ A(⌈D⌉). In Heck's paper, page 2, the author applies this general result to formula (9) above (of system PA)...


1

Probably, what he's trying to get at is that there is a way to encode in arithmetic the notion of 𐌵 being a provable sentence in K so that, if arithmetic was complete, then that sentence could be proved or disproved in arithmetic. You are absolutely correct. Godel showed (via his β-lemma) that one can encode finite sequences of natural numbers as natural ...


1

This is certainly an issue: whereas the consistency of weak axiom systems (like no axioms at all, just the inference rules of first-order logic) can be demonstrated in a very strong sense (basically, if we can exhibit a model and verify that those axioms are satisfied in it in an "appropriately constructive" way), this isn't satisfactory at all since per ...


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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