23 ideas
| 21752 | Prior to Gödel we thought truth in mathematics consisted in provability [Gödel, by Quine] |
| Full Idea: Gödel's proof wrought an abrupt turn in the philosophy of mathematics. We had supposed that truth, in mathematics, consisted in provability. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Willard Quine - Forward to Gödel's Unpublished | |
| A reaction: This explains the crisis in the early 1930s, which Tarski's theory appeared to solve. |
| 17835 | Gödel show that the incompleteness of set theory was a necessity [Gödel, by Hallett,M] |
| Full Idea: Gödel's incompleteness results of 1931 show that all axiom systems precise enough to satisfy Hilbert's conception are necessarily incomplete. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Michael Hallett - Introduction to Zermelo's 1930 paper p.1215 | |
| A reaction: [Hallett italicises 'necessarily'] Hilbert axioms have to be recursive - that is, everything in the system must track back to them. |
| 17886 | The limitations of axiomatisation were revealed by the incompleteness theorems [Gödel, by Koellner] |
| Full Idea: The inherent limitations of the axiomatic method were first brought to light by the incompleteness theorems. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Koellner - On the Question of Absolute Undecidability 1.1 |
| 10071 | Second Incompleteness: nice theories can't prove their own consistency [Gödel, by Smith,P] |
| Full Idea: Second Incompleteness Theorem: roughly, nice theories that include enough basic arithmetic can't prove their own consistency. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Smith - Intro to Gödel's Theorems 1.5 | |
| A reaction: On the face of it, this sounds less surprising than the First Theorem. Philosophers have often noticed that it seems unlikely that you could use reason to prove reason, as when Descartes just relies on 'clear and distinct ideas'. |
| 19123 | If soundness can't be proved internally, 'reflection principles' can be added to assert soundness [Gödel, by Halbach/Leigh] |
| Full Idea: Gödel showed PA cannot be proved consistent from with PA. But 'reflection principles' can be added, which are axioms partially expressing the soundness of PA, by asserting what is provable. A Global Reflection Principle asserts full soundness. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Halbach,V/Leigh,G.E. - Axiomatic Theories of Truth (2013 ver) 1.2 | |
| A reaction: The authors point out that this needs a truth predicate within the language, so disquotational truth won't do, and there is a motivation for an axiomatic theory of truth. |
| 17888 | The undecidable sentence can be decided at a 'higher' level in the system [Gödel] |
| Full Idea: My undecidable arithmetical sentence ...is not at all absolutely undecidable; rather, one can always pass to 'higher' systems in which the sentence in question is decidable. | |
| From: Kurt Gödel (On Formally Undecidable Propositions [1931]), quoted by Peter Koellner - On the Question of Absolute Undecidability 1.1 | |
| A reaction: [a 1931 MS] He says the reals are 'higher' than the naturals, and the axioms of set theory are higher still. The addition of a truth predicate is part of what makes the sentence become decidable. |
| 10621 | Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P on Gödel] |
| Full Idea: Where Gödel's First Theorem sabotages logicist ambitions, the Second Theorem sabotages Hilbert's Programme. | |
| From: comment on Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Smith - Intro to Gödel's Theorems 36 | |
| A reaction: Neo-logicism (Crispin Wright etc.) has a strategy for evading the First Theorem. |
| 10132 | There can be no single consistent theory from which all mathematical truths can be derived [Gödel, by George/Velleman] |
| Full Idea: Gödel's far-reaching work on the nature of logic and formal systems reveals that there can be no single consistent theory from which all mathematical truths can be derived. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.8 |
| 10072 | First Incompleteness: arithmetic must always be incomplete [Gödel, by Smith,P] |
| Full Idea: First Incompleteness Theorem: any properly axiomatised and consistent theory of basic arithmetic must remain incomplete, whatever our efforts to complete it by throwing further axioms into the mix. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Smith - Intro to Gödel's Theorems 1.2 | |
| A reaction: This is because it is always possible to formulate a well-formed sentence which is not provable within the theory. |
| 3198 | Gödel showed that arithmetic is either incomplete or inconsistent [Gödel, by Rey] |
| Full Idea: Gödel's theorem states that either arithmetic is incomplete, or it is inconsistent. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Georges Rey - Contemporary Philosophy of Mind 8.7 |
| 9590 | Arithmetical truth cannot be fully and formally derived from axioms and inference rules [Gödel, by Nagel/Newman] |
| Full Idea: The vast continent of arithmetical truth cannot be brought into systematic order by laying down a fixed set of axioms and rules of inference from which every true mathematical statement can be formally derived. For some this was a shocking revelation. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by E Nagel / JR Newman - Gödel's Proof VII.C | |
| A reaction: Good news for philosophy, I'd say. The truth cannot be worked out by mechanical procedures, so it needs the subtle and intuitive intelligence of your proper philosopher (Parmenides is the role model) to actually understand reality. |
| 11069 | Gödel's Second says that semantic consequence outruns provability [Gödel, by Hanna] |
| Full Idea: Gödel's Second Incompleteness Theorem says that true unprovable sentences are clearly semantic consequences of the axioms in the sense that they are necessarily true if the axioms are true. So semantic consequence outruns provability. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Robert Hanna - Rationality and Logic 5.3 |
| 10118 | First Incompleteness: a decent consistent system is syntactically incomplete [Gödel, by George/Velleman] |
| Full Idea: First Incompleteness Theorem: If S is a sufficiently powerful formal system, then if S is consistent then S is syntactically incomplete. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6 | |
| A reaction: Gödel found a single sentence, effectively saying 'I am unprovable in S', which is neither provable nor refutable in S. |
| 10122 | Second Incompleteness: a decent consistent system can't prove its own consistency [Gödel, by George/Velleman] |
| Full Idea: Second Incompleteness Theorem: If S is a sufficiently powerful formal system, then if S is consistent then S cannot prove its own consistency | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6 | |
| A reaction: This seems much less surprising than the First Theorem (though it derives from it). It was always kind of obvious that you couldn't use reason to prove that reason works (see, for example, the Cartesian Circle). |
| 10611 | There is a sentence which a theory can show is true iff it is unprovable [Gödel, by Smith,P] |
| Full Idea: The original Gödel construction gives us a sentence that a theory shows is true if and only if it satisfies the condition of being unprovable-in-that-theory. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Peter Smith - Intro to Gödel's Theorems 20.5 |
| 10867 | 'This system can't prove this statement' makes it unprovable either way [Gödel, by Clegg] |
| Full Idea: An approximation of Gödel's Theorem imagines a statement 'This system of mathematics can't prove this statement true'. If the system proves the statement, then it can't prove it. If the statement can't prove the statement, clearly it still can't prove it. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.15 | |
| A reaction: Gödel's contribution to this simple idea seems to be a demonstration that formal arithmetic is capable of expressing such a statement. |
| 8747 | Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Gödel, by Shapiro] |
| Full Idea: Gödel defended impredicative definitions on grounds of ontological realism. From that perspective, an impredicative definition is a description of an existing entity with reference to other existing entities. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Stewart Shapiro - Thinking About Mathematics 5.3 | |
| A reaction: This is why constructivists must be absolutely precise about definition, where realists only have to do their best. Compare building a car with painting a landscape. |
| 16062 | A necessary relation between fact-levels seems to be a further irreducible fact [Lynch/Glasgow] |
| Full Idea: It seems unavoidable that the facts about logically necessary relations between levels of facts are themselves logically distinct further facts, irreducible to the microphysical facts. | |
| From: Lynch,MP/Glasgow,JM (The Impossibility of Superdupervenience [2003], C) | |
| A reaction: I'm beginning to think that rejecting every theory of reality that is proposed by carefully exposing some infinite regress hidden in it is a rather lazy way to do philosophy. Almost as bad as rejecting anything if it can't be defined. |
| 16061 | If some facts 'logically supervene' on some others, they just redescribe them, adding nothing [Lynch/Glasgow] |
| Full Idea: Logical supervenience, restricted to individuals, seems to imply strong reduction. It is said that where the B-facts logically supervene on the A-facts, the B-facts simply re-describe what the A-facts describe, and the B-facts come along 'for free'. | |
| From: Lynch,MP/Glasgow,JM (The Impossibility of Superdupervenience [2003], C) | |
| A reaction: This seems to be taking 'logically' to mean 'analytically'. Presumably an entailment is logically supervenient on its premisses, and may therefore be very revealing, even if some people think such things are analytic. |
| 16060 | Nonreductive materialism says upper 'levels' depend on lower, but don't 'reduce' [Lynch/Glasgow] |
| Full Idea: The root intuition behind nonreductive materialism is that reality is composed of ontologically distinct layers or levels. …The upper levels depend on the physical without reducing to it. | |
| From: Lynch,MP/Glasgow,JM (The Impossibility of Superdupervenience [2003], B) | |
| A reaction: A nice clear statement of a view which I take to be false. This relationship is the sort of thing that drives people fishing for an account of it to use the word 'supervenience', which just says two things seem to hang out together. Fluffy materialism. |
| 16064 | The hallmark of physicalism is that each causal power has a base causal power under it [Lynch/Glasgow] |
| Full Idea: Jessica Wilson (1999) says what makes physicalist accounts different from emergentism etc. is that each individual causal power associated with a supervenient property is numerically identical with a causal power associated with its base property. | |
| From: Lynch,MP/Glasgow,JM (The Impossibility of Superdupervenience [2003], n 11) | |
| A reaction: Hence the key thought in so-called (serious, rather than self-evident) 'emergentism' is so-called 'downward causation', which I take to be an idle daydream. |
| 3102 | Why don't we experience or remember going to sleep at night? [Magee] |
| Full Idea: As a child it was incomprehensible to me that I did not experience going to sleep, and never remembered it. When my sister said 'Nobody remembers that', I just thought 'How does she know?' | |
| From: Bryan Magee (Confessions of a Philosopher [1997], Ch.I) | |
| A reaction: This is actually evidence for something - that we do not have some sort of personal identity which is separate from consciousness, so that "I am conscious" would literally mean that an item has a property, which it can lose. |
| 3192 | Basic logic can be done by syntax, with no semantics [Gödel, by Rey] |
| Full Idea: Gödel in his completeness theorem for first-order logic showed that a certain set of syntactically specifiable rules was adequate to capture all first-order valid arguments. No semantics (e.g. reference, truth, validity) was necessary. | |
| From: report of Kurt Gödel (On Formally Undecidable Propositions [1931]) by Georges Rey - Contemporary Philosophy of Mind 8.2 | |
| A reaction: This implies that a logic machine is possible, but we shouldn't raise our hopes for proper rationality. Validity can be shown for purely algebraic arguments, but rationality requires truth as well as validity, and that needs propositions and semantics. |