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12

Gödel's theism is discussed by Franzen in Gödel’s Theorem: An Incomplete Guideto Its Use and Abuse. He penned a version of the ontological argument, and in 1961 ranked the worldviews “according to the degree and the manner of their affinity to or, respectively, turning away from metaphysics (or religion)... Skepticism, materialism, and positivism stand on ...


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(...


5

The origin is with the so-called Whiteley Sentence. See C.Whiteley, “Minds, Machines and Gödel: A Reply to Mr. Lucas (1962)”, Philosophy 37:61-62 : It is possible to devise a formula which will trap a human mind —say, Mr Lucas's— in the same way that his application of Gödel traps the machine. Take, for instance, the formula 'This formula ...


4

ZFC is susceptible to Gödel's incompleteness as was the Principia Mathematica. See Gödel’s Incompleteness Theorems for an introduction to the theorem : Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor ...


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

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)...


2

Goedel's 1930 completeness theorem showed that first-order predicate calculus is complete in the sense that every valid formula is a theorem. There are many calculi that have as theorems all and only the tautologies, that is, the valid formulas of propositional logic, as pointed out in the answer given previously by Kjos-Hanssen. Furthermore, as stated in ...


2

It is quite common to assume ZFC as the unspoken system in which mathematical claims are to be understood. In that case, it would be unnecessary to specify "well-orderable", because the axiom of choice is (over ZF) equivalent to the statement that every set is well-orderable. The typical proof of the completeness theorem proceeds by constructing a model "by ...


2

Gödel's theorems don't have agency, or cause any effects. There's nothing magic about them. What they are is a clever way of demonstrating that there are pre-existing, intrinsic limits to what can be computed. You might imagine being in a field with a fence. The field is computing, the fence is the boundaries of what can be computed, and all the problems ...


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 think you are misunderstanding Godel's incompleteness theorem: The incompleteness theorem says if you have a complete system (i.e., a complete "axiomatic" system), then there are true statement such that they are unprovable in that system. So no, the true statements are not the consequence of foundational assumptions (axioms). What the theorem actually ...


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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