Ideas from 'works' by Kurt Gödel [1930], by Theme Structure

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2. Reason / A. Nature of Reason / 1. On Reason

17892

For clear questions posed by reason, reason can also find clear answers

			Full Idea: I uphold the belief that for clear questions posed by reason, reason can also find clear answers.
			From: Kurt Gödel (works [1930]), quoted by Peter Koellner - On the Question of Absolute Undecidability 1.5
			A reaction: [written in 1961] This contradicts the implication normally taken from his much earlier Incompleteness Theorems.

5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic

9188	Gödel proved that first-order logic is complete, and second-order logic incomplete

			Full Idea: Gödel proved the completeness of standard formalizations of first-order logic, including Frege's original one. However, an implication of his famous theorem on the incompleteness of arithmetic is that second-order logic is incomplete.
			From: report of Kurt Gödel (works [1930]) by Michael Dummett - The Philosophy of Mathematics 3.1
			A reaction: This must mean that it is impossible to characterise arithmetic fully in terms of first-order logic. In which case we can only characterize the features of abstract reality in general if we employ an incomplete system. We're doomed.

5. Theory of Logic / I. Semantics of Logic / 2. Formal Truth

10620

Originally truth was viewed with total suspicion, and only demonstrability was accepted

			Full Idea: At that time (c.1930) a concept of objective mathematical truth as opposed to demonstrability was viewed with greatest suspicion and widely rejected as meaningless.
			From: Kurt Gödel (works [1930]), quoted by Peter Smith - Intro to Gödel's Theorems 28.2
			A reaction: [quoted from a letter] This is the time of Ramsey's redundancy account, and before Tarski's famous paper of 1933. It is also the high point of Formalism, associated with Hilbert.

5. Theory of Logic / K. Features of Logics / 5. Incompleteness

17883

Gödel's Theorems did not refute the claim that all good mathematical questions have answers

			Full Idea: Gödel was quick to point out that his original incompleteness theorems did not produce instances of absolute undecidability and hence did not undermine Hilbert's conviction that for every precise mathematical question there is a discoverable answer.
			From: report of Kurt Gödel (works [1930]) by Peter Koellner - On the Question of Absolute Undecidability Intro
			A reaction: The normal simplistic view among philosophes is that Gödel did indeed decisively refute the optimistic claims of Hilbert. Roughly, whether Hilbert is right depends on which axioms of set theory you adopt.

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic

17885

Gödel eventually hoped for a generalised completeness theorem leaving nothing undecidable

			Full Idea: Eventually Gödel ...expressed the hope that there might be a generalised completeness theorem according to which there are no absolutely undecidable sentences.
			From: report of Kurt Gödel (works [1930]) by Peter Koellner - On the Question of Absolute Undecidability Intro
			A reaction: This comes as a bit of a shock to those who associate him with the inherent undecidability of reality.

10614

The real reason for Incompleteness in arithmetic is inability to define truth in a language

			Full Idea: The concept of truth of sentences in a language cannot be defined in the language. This is the true reason for the existence of undecidable propositions in the formal systems containing arithmetic.
			From: Kurt Gödel (works [1930]), quoted by Peter Smith - Intro to Gödel's Theorems 21.6
			A reaction: [from a letter by Gödel] So they key to Incompleteness is Tarski's observations about truth. Highly significant, as I take it.