80 ideas
21752 | Prior to Gödel we thought truth in mathematics consisted in provability [Gödel, by Quine] |
17835 | Gödel show that the incompleteness of set theory was a necessity [Gödel, by Hallett,M] |
9406 | A class is natural when everybody can spot further members of it [Quinton] |
17886 | The limitations of axiomatisation were revealed by the incompleteness theorems [Gödel, by Koellner] |
10071 | Second Incompleteness: nice theories can't prove their own consistency [Gödel, by Smith,P] |
19123 | If soundness can't be proved internally, 'reflection principles' can be added to assert soundness [Gödel, by Halbach/Leigh] |
10621 | Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme [Smith,P on Gödel] |
17888 | The undecidable sentence can be decided at a 'higher' level in the system [Gödel] |
10132 | There can be no single consistent theory from which all mathematical truths can be derived [Gödel, by George/Velleman] |
3198 | Gödel showed that arithmetic is either incomplete or inconsistent [Gödel, by Rey] |
10072 | First Incompleteness: arithmetic must always be incomplete [Gödel, by Smith,P] |
9590 | Arithmetical truth cannot be fully and formally derived from axioms and inference rules [Gödel, by Nagel/Newman] |
11069 | Gödel's Second says that semantic consequence outruns provability [Gödel, by Hanna] |
10118 | First Incompleteness: a decent consistent system is syntactically incomplete [Gödel, by George/Velleman] |
10122 | Second Incompleteness: a decent consistent system can't prove its own consistency [Gödel, by George/Velleman] |
10611 | There is a sentence which a theory can show is true iff it is unprovable [Gödel, by Smith,P] |
10867 | 'This system can't prove this statement' makes it unprovable either way [Gödel, by Clegg] |
8747 | Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Gödel, by Shapiro] |
15730 | Extreme nominalists say all classification is arbitrary convention [Quinton] |
15728 | The naturalness of a class depends as much on the observers as on the objects [Quinton] |
9407 | Properties imply natural classes which can be picked out by everybody [Quinton] |
15729 | Uninstantiated properties must be defined using the instantiated ones [Quinton] |
8520 | An individual is a union of a group of qualities and a position [Quinton, by Campbell,K] |
3192 | Basic logic can be done by syntax, with no semantics [Gödel, by Rey] |
19956 | True goodness is political, and consists of love of and submission to the laws [Montesquieu] |
19961 | Primitive people would be too vulnerable and timid to attack anyone, so peace would reign [Montesquieu] |
19962 | Men do not desire to subjugate one another; domination is a complex and advanced idea [Montesquieu] |
19963 | People are drawn into society by needs, shared fears, pleasure, and knowledge [Montesquieu] |
20008 | People are guided by a multitude of influences, from which the spirit of a nation emerges [Montesquieu] |
19993 | In small republics citizens identify with the public good, and abuses are fewer [Montesquieu] |
19992 | In a large republic there is too much wealth for individuals to manage it [Montesquieu] |
20005 | The rich would never submit to a lottery deciding which part of their society should be slaves [Montesquieu] |
19995 | All states aim at preservation, and then have distinctive individual purposes [Montesquieu] |
19964 | The natural power of a father suggests rule by one person, but that authority can be spread [Montesquieu] |
19986 | Monarchies can act more quickly, because one person is in charge [Montesquieu] |
19972 | The nobility are an indispensable part of a monarchy [Montesquieu] |
19974 | Monarchs must not just have links to the people; they need a body which maintains the laws [Montesquieu] |
19976 | Ambition is good in a monarchy, because the monarch can always restrain it [Montesquieu] |
19978 | In monarchies, men's actions are judged by their grand appearance, not their virtues [Montesquieu] |
19985 | In a monarchy, the nobility must be hereditary, to bind them together [Montesquieu] |
19988 | A despot's agents must be given power, so they inevitably become corrupt [Montesquieu] |
19977 | Despotism and honour are incompatible, because honour scorns his power, and lives by rules [Montesquieu] |
20007 | Tyranny is either real violence, or the imposition of unpopular legislation [Montesquieu] |
19975 | Despots are always lazy and ignorant, so they always delegate their power to a vizier [Montesquieu] |
19989 | The will of a despot is an enigma, so magistrates can only follow their own will [Montesquieu] |
19970 | If the nobility is numerous, the senate is the artistocracy, and the nobles are a democracy [Montesquieu] |
19971 | Aristocracy is democratic if they resemble the people, but not if they resemble the monarch [Montesquieu] |
19984 | Great inequality between aristocrats and the rest is bad - and also among aristocrats themselves [Montesquieu] |
19980 | If a government is to be preserved, it must first be loved [Montesquieu] |
19996 | A government has a legislature, an international executive, and a domestic executive [Montesquieu] |
19997 | The judiciary must be separate from the legislature, to avoid arbitrary power [Montesquieu] |
19965 | The fundamental laws of a democracy decide who can vote [Montesquieu] |
19968 | It is basic to a democracy that the people themselves must name their ministers [Montesquieu] |
19969 | Voting should be public, so the lower classes can be influenced by the example of notable people [Montesquieu] |
19999 | All citizens (apart from the very humble poor) should choose their representatives [Montesquieu] |
19967 | In a democracy the people should manage themselves, and only delegate what they can't do [Montesquieu] |
19966 | A democratic assembly must have a fixed number, to see whether everyone has spoken [Montesquieu] |
19998 | If deputies represent people, they are accountable, but less so if they represent places [Montesquieu] |
20003 | Slaves are not members of the society, so no law can forbid them to run away [Montesquieu] |
20000 | Slavery is entirely bad; the master abandons the virtues, and they are pointless in the slave [Montesquieu] |
20006 | The demand for slavery is just the masters' demand for luxury [Montesquieu] |
20009 | Freedom of speech and writing, within the law, is essential to preserve liberty [Montesquieu] |
19994 | Freedom in society is ability to do what is right, and not having to do what is wrong [Montesquieu] |
19981 | No one even thinks of equality in monarchies and despotism; they all want superiority [Montesquieu] |
19991 | Equality is not command by everyone or no one, but command and obedience among equals [Montesquieu] |
19990 | Democracy is corrupted by lack of equality, or by extreme equality (between rulers and ruled) [Montesquieu] |
19983 | Democracies may sometimes need to restrict equality [Montesquieu] |
19982 | Some equality can be achieved by social categories, combined with taxes and poor relief [Montesquieu] |
19959 | Prior to positive laws there is natural equity, of obedience, gratitude, dependence and merit [Montesquieu] |
19960 | Sensation gives animals natural laws, but knowledge can make them break them [Montesquieu] |
20002 | The death penalty is permissible, because its victims enjoyed the protection of that law [Montesquieu] |
20010 | If religion teaches determinism, penalties must be severe; if free will, then that is different [Montesquieu] |
20001 | The only right victors have over captives is the protection of the former [Montesquieu] |
19973 | The clergy are essential to a monarchy, but dangerous in a republic [Montesquieu] |
19987 | Religion has the most influence in despotic states, and reinforces veneration for the ruler [Montesquieu] |
20011 | Religion can support the state when the law fails to do so [Montesquieu] |
20004 | French slavery was accepted because it was the best method of religious conversion [Montesquieu] |
19979 | In monarchies education ennobles people, and in despotisms it debases them [Montesquieu] |
19957 | Teaching is the best practice of the general virtue that leads us to love everyone [Montesquieu] |
19958 | Laws are the necessary relations that derive from the nature of things [Montesquieu] |