39 ideas
| 10147 | The Axiom of Choice is consistent with the other axioms of set theory [Feferman/Feferman] |
| Full Idea: In 1938 Gödel proved that the Axiom of Choice is consistent with the other axioms of set theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: Hence people now standardly accept ZFC, rather than just ZF. |
| 10148 | Axiom of Choice: a set exists which chooses just one element each of any set of sets [Feferman/Feferman] |
| Full Idea: Zermelo's Axiom of Choice asserts that for any set of non-empty sets that (pairwise) have no elements in common, then there is a set that 'simultaneously chooses' exactly one element from each set. Note that this is an existential claim. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: The Axiom is now widely accepted, after much debate in the early years. Even critics of the Axiom turn out to be relying on it. |
| 10149 | Platonist will accept the Axiom of Choice, but others want criteria of selection or definition [Feferman/Feferman] |
| Full Idea: The Axiom of Choice seems clearly true from the Platonistic point of view, independently of how sets may be defined, but is rejected by those who think such existential claims must show how to pick out or define the object claimed to exist. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: The typical critics are likely to be intuitionists or formalists, who seek for both rigour and a plausible epistemology in our theory. |
| 10150 | The Trichotomy Principle is equivalent to the Axiom of Choice [Feferman/Feferman] |
| Full Idea: The Trichotomy Principle (any number is less, equal to, or greater than, another number) turned out to be equivalent to the Axiom of Choice. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) | |
| A reaction: [He credits Sierpinski (1918) with this discovery] |
| 10146 | Cantor's theories needed the Axiom of Choice, but it has led to great controversy [Feferman/Feferman] |
| Full Idea: The Axiom of Choice is a pure existence statement, without defining conditions. It was necessary to provide a foundation for Cantor's theory of transfinite cardinals and ordinal numbers, but its nonconstructive character engendered heated controversy. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int I) |
| 10158 | A structure is a 'model' when the axioms are true. So which of the structures are models? [Feferman/Feferman] |
| Full Idea: A structure is said to be a 'model' of an axiom system if each of its axioms is true in the structure (e.g. Euclidean or non-Euclidean geometry). 'Model theory' concerns which structures are models of a given language and axiom system. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: This strikes me as the most interesting aspect of mathematical logic, since it concerns the ways in which syntactic proof-systems actually connect with reality. Tarski is the central theoretician here, and his theory of truth is the key. |
| 10162 | Tarski and Vaught established the equivalence relations between first-order structures [Feferman/Feferman] |
| Full Idea: In the late 1950s Tarski and Vaught defined and established basic properties of the relation of elementary equivalence between two structures, which holds when they make true exactly the same first-order sentences. This is fundamental to model theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: This is isomorphism, which clarifies what a model is by giving identity conditions between two models. Note that it is 'first-order', and presumably founded on classical logic. |
| 10160 | Löwenheim-Skolem says if the sentences are countable, so is the model [Feferman/Feferman] |
| Full Idea: The Löwenheim-Skolem Theorem, the earliest in model theory, states that if a countable set of sentences in a first-order language has a model, then it has a countable model. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) | |
| A reaction: There are 'upward' (sentences-to-model) and 'downward' (model-to-sentences) versions of the theory. |
| 10159 | Löwenheim-Skolem Theorem, and Gödel's completeness of first-order logic, the earliest model theory [Feferman/Feferman] |
| Full Idea: Before Tarski's work in the 1930s, the main results in model theory were the Löwenheim-Skolem Theorem, and Gödel's establishment in 1929 of the completeness of the axioms and rules for the classical first-order predicate (or quantificational) calculus. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) |
| 10161 | If a sentence holds in every model of a theory, then it is logically derivable from the theory [Feferman/Feferman] |
| Full Idea: Completeness is when, if a sentences holds in every model of a theory, then it is logically derivable from that theory. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int V) |
| 10156 | 'Recursion theory' concerns what can be solved by computing machines [Feferman/Feferman] |
| Full Idea: 'Recursion theory' is the subject of what can and cannot be solved by computing machines | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Ch.9) | |
| A reaction: This because 'recursion' will grind out a result step-by-step, as long as the steps will 'halt' eventually. |
| 10155 | Both Principia Mathematica and Peano Arithmetic are undecidable [Feferman/Feferman] |
| Full Idea: In 1936 Church showed that Principia Mathematica is undecidable if it is ω-consistent, and a year later Rosser showed that Peano Arithmetic is undecidable, and any consistent extension of it. | |
| From: Feferman / Feferman (Alfred Tarski: life and logic [2004], Int IV) |
| 19400 | Possibles demand existence, so as many of them as possible must actually exist [Leibniz] |
| Full Idea: From the conflict of all the possibles demanding existence, this at once follows, that there exists that series of things by which as many of them as possible exist. | |
| From: Gottfried Leibniz (Exigency to Exist in Essences [1690], p.91) | |
| A reaction: I'm in tune with a lot of Leibniz, but my head swims with this one. He seems to be a Lewisian about possible worlds - that they are concrete existing entities (with appetites!). Could Lewis include Leibniz's idea in his system? |
| 19401 | God's sufficient reason for choosing reality is in the fitness or perfection of possibilities [Leibniz] |
| Full Idea: The sufficient reason for God's choice can be found only in the fitness (convenance) or in the degree of perfection that the several worlds possess. | |
| From: Gottfried Leibniz (Exigency to Exist in Essences [1690], p.92) | |
| A reaction: The 'fitness' of a world and its 'perfection' seem very different things. A piece of a jigsaw can have wonderful fitness, without perfection. Occasionally you get that sinking feeling with metaphysicians that they just make it up. |
| 19402 | The actual universe is the richest composite of what is possible [Leibniz] |
| Full Idea: The actual universe is the collection of the possibles which forms the richest composite. | |
| From: Gottfried Leibniz (Exigency to Exist in Essences [1690], p.92) | |
| A reaction: 'Richest' for Leibniz means a maximum combination of existence, order and variety. It's rather like picking the best starting team from a squad of footballers. |
| 20547 | We should respect the right of people to live in their own way, even if it is irrational [Swift] |
| Full Idea: Forcing people to do what is rational involves a lack of respect, a failure to respect the value of her living her life in her own (irrational) way. | |
| From: Adam Swift (Political Philosophy (3rd ed) [2014], 2 'Resisting' 6) | |
| A reaction: Up to a point. Irrationally eccentric is one thing, and irrationally self-destructive is another. You can sit back and watch your children embrace a life less happy than the one you wanted for them - but not a life of utter misery. |
| 20564 | Anti-colonial movements usually invoke the right of their 'people' to self-determination [Swift] |
| Full Idea: Nationalist movements seeking to throw off the yoke of colonial rule are often motivated by a sense that their 'people' have the right to self-determination. | |
| From: Adam Swift (Political Philosophy (3rd ed) [2014], 5 'Intrinsic 1') | |
| A reaction: In 2017, Basques, Catalans and Kurds come to mind. The whole of Africa was an example of this c.1950-80, but there was uncertainty about states, tribes and language groups. |
| 20535 | Isn't it more rational to maximise the average position, but with a safety net? [Swift] |
| Full Idea: Wouldn't it be more rational to choose principles that would maximize the average position, perhaps subject to some 'floor' level beneath which they would not want to take the risk of sinking? | |
| From: Adam Swift (Political Philosophy (3rd ed) [2014], 1 'Rawls') | |
| A reaction: The criticism is that Rawls's prediction is over-cautious, and that people will take mild risks in what they choose, as long as there is no danger of disaster. (Just as you should allow small children to risk injury, but not death). |
| 20537 | Hypothetical contracts have no binding force [Swift] |
| Full Idea: A common objection to Rawls is that hypothetical contracts, unlike real ones, have no binding force. | |
| From: Adam Swift (Political Philosophy (3rd ed) [2014], 1 'Rawls') | |
| A reaction: [I think Dworkin made this point] 'Contract' may be metaphorical. Perhaps it is just an 'initial agreement' or a 'working arrangement', |
| 20542 | Cosmopolitans reject the right of different states to distribute resources in different ways [Swift] |
| Full Idea: Cosmopolitans who claim that the same distributive principles should apply to all human beings seem to be denying that different states may make different judgements about how they want to allocate resources among their members. | |
| From: Adam Swift (Political Philosophy (3rd ed) [2014], 1 'Social') | |
| A reaction: If you want to be a citizen of the world, you have to face up to the pluralistic character of cultures. Do you thereby want to be a citizen of both California and Saudi Arabia? Or are you actually just becoming a citizen of nowhere? |
| 20559 | Democracy is bad, but the other systems are worse [Swift] |
| Full Idea: During WW2 Winston Churchill famously said that democracy is the worst form of government, except for all the others that have been tried. | |
| From: Adam Swift (Political Philosophy (3rd ed) [2014], 5 'Procedures') | |
| A reaction: [Actually a speech in 1947, which began 'it has been said that....'] Aristotle thought an intelligent and benevolent dictatorship was the best form, but held little hope of achieving it. Getting rid of bad rulers is the big virtue. |
| 20561 | Since all opinions are treated as equal in democracy, it implies there are no right answers [Swift] |
| Full Idea: If there were moral knowledge about political matters, democracy would be a very strange way of reaching it. Democratic law-making means treating each person's view as equally good, which only makes sense if there is nothing to be right or wrong about. | |
| From: Adam Swift (Political Philosophy (3rd ed) [2014], 5 'Subjectivism') | |
| A reaction: Ah, I suddenly grasp that the modern fad for a rather gormless blanket relativism is rooted in the modern desire to take democracy really seriously. Important to remember Condorcet's point here. |
| 20562 | Design your democracy to treat citizens equally, or to produce better citizens? [Swift] |
| Full Idea: If your main reason for being a democrat is that democratic procedures respect citizens equally, then you may want a different kind of democracy from those who favour it because they think it tends to produce better citizens? | |
| From: Adam Swift (Political Philosophy (3rd ed) [2014], 5 'Values') | |
| A reaction: [Combine this with Idea 20563] |
| 20563 | Design your democracy to yield political stability, or good decisions? [Swift] |
| Full Idea: If you value democracy because it yields political stability, then you will probably worry about different aspects of the procedure from those who care about its producing good decisions. | |
| From: Adam Swift (Political Philosophy (3rd ed) [2014], 5 'Values') | |
| A reaction: [Combine this with Idea 20562] Surely the primary aim must be good decisions? The other three options are the result of pessimism about any method achieving that. Instability, inequality and dud citizens are bars to good decisions. |
| 20560 | Teledemocracy omits debate and deliberation, which are important parts of good decisions [Swift] |
| Full Idea: We are averse to teledemocracy because it misses out some important parts of a good decision-making procedure, such as debate and deliberation. | |
| From: Adam Swift (Political Philosophy (3rd ed) [2014], 5 'Procedures') | |
| A reaction: Perhaps you should be sent a short info pack, and only allowed to vote when you have passed a factual multiple choice test about the topic. Or one pack from each political party. Maybe compulsory online discussion as well. |
| 20554 | Multiculturalism is a barrier to the whole state being a community [Swift] |
| Full Idea: For those wanting to regard the state itself as a community, multiculturalism can be a problem. | |
| From: Adam Swift (Political Philosophy (3rd ed) [2014], 4 'Liberalism') | |
| A reaction: A very important idea. A certain type of aggressive patriot passionately wants the whole country to be a close-bound community, and becomes deeply frustrated by the impossibility of this in a complex and fluid modern world. |
| 20553 | Liberals mistakenly think individuals choose their values, without reference to the community [Swift] |
| Full Idea: The two core liberal mistakes (according to communitarians) are that people choose their values, and that they do so in some way detached from their communities. | |
| From: Adam Swift (Political Philosophy (3rd ed) [2014], 4 'Correcting') | |
| A reaction: I think I might be a communitarian liberal, meaning that extreme individualism is both incorrect and pernicious, but that communities should only exist to promote the varied lives of individuals within them. |
| 20556 | The best way to build a cohesive community is to be involved in a war [Swift] |
| Full Idea: There is nothing like a war to build a sense of common purpose, of being in the same boat, and to generate the kind of interaction between people that breaks down divisive social boundaries. | |
| From: Adam Swift (Political Philosophy (3rd ed) [2014]) | |
| A reaction: A nice warning to those who over-do or simplify communitarianism. Alternatively, the greatest sign of health in a community is that citizens have almost no interest in one another? |
| 20557 | Membership and inclusion in a community implies non-membership and exclusion [Swift] |
| Full Idea: Community is about membership and inclusion. But that means it is also about non-membership and exclusion. | |
| From: Adam Swift (Political Philosophy (3rd ed) [2014], 4 'Conc') | |
| A reaction: I'm a fan of communitarianism (focused on Aristotle's life of individual virtue for each citizen), but I'm beginning to see that it has a poisonous cousin travelling under the same name. The cousin's rallying cries focus on aliens and enemies. |
| 20555 | Liberals are concerned to protect individuals from too much community [Swift] |
| Full Idea: Liberals are concerned to protect individuals from too much community - from practices that stifle the individual's freedom to choose for herself how she lives her life. | |
| From: Adam Swift (Political Philosophy (3rd ed) [2014], 4 'Liberalism') | |
| A reaction: The phrase 'too much community' is an excellent warning to communitarians. I'm happy to be enmeshed in a community, as long as it is composed of highly liberal and easy-going individuals. Avoid too much bad community. |
| 20540 | Redistributing wealth treats some people as means, rather than as ends [Swift] |
| Full Idea: Treating people as means seems like a fairly accurate description of what is involved when the state coercively redistributes resources from some to others. | |
| From: Adam Swift (Political Philosophy (3rd ed) [2014], 1 'Nozick') | |
| A reaction: The objection comes from Nozick, and alludes to Kant's desire to treat everyone as an end in themselves. Personally I don't mind at all being treated as a means, when my wife asks me to make her a cup of tea. Or paying my taxes to help the community. |
| 20551 | Men have had the power to structure all of our social institutions [Swift] |
| Full Idea: The problem for feminists is that men have had the power to structure all our social institutions - family, economy, polity - in ways that suit them. | |
| From: Adam Swift (Political Philosophy (3rd ed) [2014], 3 'Gender') | |
| A reaction: An interesting question is whether masculine domination runs even deeper than that, into our value system, our metaphysics, our science, our epistemology, our language. How do you tell? If women take over half the masculine roles, does that solve it? |
| 20545 | Maybe a freedom is from a restraint, and also in order to do something [Swift] |
| Full Idea: Maybe freedom is a triadic relation, involving an agent, freedom from a contraint, and in order to act towards some goal. | |
| From: Adam Swift (Political Philosophy (3rd ed) [2014], 2 'Two') | |
| A reaction: [He cites Gerald MacCallum for this thought] The point is that this makes freedom both negative and positive, contrary to Isaiah Berlin's claim. But on the first day of the school holidays you are 'free', with nothing in particular in mind. |
| 20550 | Opportunity should ignore extraneous factors, or foster competence, or ignore all disadvantages [Swift] |
| Full Idea: The minimal conception of equality of opportunity is that race or gender or religion should not affect chances of a good job or education. The conventional conception needs equality in acquiring competences. Radical views ignore inborn disadvantages. | |
| From: Adam Swift (Political Philosophy (3rd ed) [2014], 3 'Equality') | |
| A reaction: [my summary of Swift] The strong version only says the less talented should have access to large rewards. The whole idea has strong capitalist assumptions. |
| 20536 | Inequalities are needed, as incentives to do the most important jobs [Swift] |
| Full Idea: Without inequalities, people will have no incentive to do one job rather than another - to do the kind of work which is most useful. | |
| From: Adam Swift (Political Philosophy (3rd ed) [2014], 1 'Rawls') | |
| A reaction: The reality is that the lowest pay goes to the jobs that no one wants to do, and all the really nice jobs are usually well paid. Which is a conspiracy, because all the salaries are set by the people with the nice jobs. |
| 20548 | A person can desire redistibution of wealth, without it being for reasons of equality [Swift] |
| Full Idea: Someone who rejects equality can care passionately that resources should be transferred from the rich to the poor. They are just rejecting a particular reason that might be offered to justify the redistribution. | |
| From: Adam Swift (Political Philosophy (3rd ed) [2014], 3 'Intro') | |
| A reaction: For example, it might be for utilitarian reasons, which usually only seek maximised happiness, not equal happiness. And one may love many forms of equality, without economic equality being one of them. |
| 20541 | You can't necessarily sell your legitimate right to something, even if you produced it [Swift] |
| Full Idea: Ownership is a complicated idea. I have a right to the office photocopier, but I can's sell the right to others. If people have absolute rights over what they produce, why can't parents sell their children into slavery? | |
| From: Adam Swift (Political Philosophy (3rd ed) [2014], 1 'Nozick') | |
| A reaction: If I make a car from stolen parts, does constructing it make it mine? Etc. Do birds own their nests? Swift goes on to ask if we 'own' our bodies. |
| 20546 | Libertarians about property ignore the fact that private property is a denial of freedoms [Swift] |
| Full Idea: Libertarians say that they care about freedom, and argue for private property rights on freedom grounds. But they don't sem to care about, or even notice, the unfreedom implied by the existence of private property rights. | |
| From: Adam Swift (Political Philosophy (3rd ed) [2014], 2 'Freedom') | |
| A reaction: When I pass some vast country estate totally surrounded by a high wall, I certainly don't think how wonderful it is that someone has the right to own this property as private land. On the contrary.... |
| 20533 | Justice can be seen as fairness or entitlement or desert [Swift] |
| Full Idea: The three influential conceptions of justice are as fairness (Rawls), as entitlement (Nozick), and as desert. | |
| From: Adam Swift (Political Philosophy (3rd ed) [2014], 1 'Concept') |