23 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) |
| 23029 | Knowledge is secured by the relations between its parts, through differences and identities [Green,TH, by Muirhead] |
| Full Idea: What gives reality and stability to our knowledge is the reality and stability of the relations established between its parts..…by the differences and identities with other things which have similarly achieved comparative fixity and substantiality. | |
| From: report of T.H. Green (Lectures on the Principles of Political Obligation [1882]) by John H. Muirhead - The Service of the State I | |
| A reaction: Although I don't sympathise with Green's idealist metaphysics, and nevertheless think that this internalist account of knowledge is correct. |
| 7861 | Libet says the processes initiated in the cortex can still be consciously changed [Libet, by Papineau] |
| Full Idea: Libet himself points out that the conscious decisions still have the power to 'endorse' or 'cancel', so to speak, the processes initiated by the earlier cortical activity: no action will result if the action's execution is consciously countermanded. | |
| From: report of Benjamin Libet (Unconscious Cerebral Initiative [1985]) by David Papineau - Thinking about Consciousness 1.4 | |
| A reaction: This is why Libet's findings do not imply 'epiphenomenalism'. It seems that part of a decisive action is non-conscious, undermining the all-or-nothing view of consciousness. Searle tries to smuggle in free will at this point (Idea 3817). |
| 6660 | Libet found conscious choice 0.2 secs before movement, well after unconscious 'readiness potential' [Libet, by Lowe] |
| Full Idea: Libet found that a subject's conscious choice to move was about a fifth of a second before movement, and thus later than the onset of the brain's so-called 'readiness potential', which seems to imply that unconscious processes initiates action. | |
| From: report of Benjamin Libet (Unconscious Cerebral Initiative [1985]) by E.J. Lowe - Introduction to the Philosophy of Mind Ch.9 | |
| A reaction: Of great interest to philosophers! It seems to make conscious choices epiphenomenal. The key move, I think, is to give up the idea of consciousness as being all-or-nothing. My actions are still initiated by 'me', but 'me' shades off into unconsciousness. |
| 23046 | States only have full authority if they heed the claims of human fellowship [Green,TH] |
| Full Idea: The claim of the state is only absolutely paramount on the supposition that in its commands and prohibitions it takes account of all the claims that arise out of human fellowship. | |
| From: T.H. Green (Lectures on the Principles of Political Obligation [1882], §146), quoted by John H. Muirhead - The Service of the State III | |
| A reaction: He rejects the idea of the general will in ordinary political activity, so it is not clear how this condition could ever be met in practice. Hideous governments just pay lip service to 'human fellowship'. How could you tell whether they believe it? |
| 23051 | Equality also implies liberty, because equality must be of opportunity as well as possessions [Green,TH] |
| Full Idea: Liberty was essential, not only as a means to equality, but as part of it. …because the opportunity which was to be equalised was not merely to have and to be happy, but to do and to realise. It was 'the right of man to make the best of himself'. | |
| From: T.H. Green (Lectures on the Principles of Political Obligation [1882]), quoted by John H. Muirhead - The Service of the State IV | |
| A reaction: This nicely identifies the core idea of civilised liberalism (as opposed to the crazy self-seeking kind). I think 'give people the right to make the best of themselves' makes a good slogan, because it implies ensuring that they have the means. |
| 23028 | The highest political efforts express our deeper social spirit [Green,TH, by Muirhead] |
| Full Idea: Political effort in all its highest forms is the expression of a belief in the reality of the social spirit as the deeper element in the individual. | |
| From: report of T.H. Green (Lectures on the Principles of Political Obligation [1882]) by John H. Muirhead - The Service of the State I | |
| A reaction: Although Green is rather literally spiritual, if we express it as a central aspect of human nature, this idea strikes me as correct. Writing in 2021, I am totally bewildered by the entire absence of any 'higher' forms of political expression. |
| 23054 | Communism is wrong because it restricts the freedom of individuals to contribute to the community [Green,TH, by Muirhead] |
| Full Idea: Green condemned pure communism, not in the name of any abstract rights of the individual, but of the right of the community itself to the best that individuals can contribute through the free and spontaneous exercise of their powers of self-expression. | |
| From: report of T.H. Green (Lectures on the Principles of Political Obligation [1882]) by John H. Muirhead - The Service of the State IV | |
| A reaction: Interesting. In a very authoritarian communist state it does seem that citizens are less able to contribute to the general good. But extreme liberty seems also to undermine the general good. Hm. |
| 23047 | Original common ownership is securing private property, not denying it [Green,TH, by Muirhead] |
| Full Idea: Common ownership in early societies is not the denial of a man's private property in the products of his own labour, but the only way under the circumstances of securing it. | |
| From: report of T.H. Green (Lectures on the Principles of Political Obligation [1882], §218) by John H. Muirhead - The Service of the State III | |
| A reaction: This is announced with some confidence, but it is very speculative. I think there is some truth in Locke's thought that putting work into a creation creates natural ownership. But who owns the raw materials? Why is work valued highly? |
| 23042 | National spirit only exists in the individuals who embody it [Green,TH, by Muirhead] |
| Full Idea: A national spirit cannot exist apart from the individuals who embody it. | |
| From: report of T.H. Green (Lectures on the Principles of Political Obligation [1882]) by John H. Muirhead - The Service of the State II | |
| A reaction: We see this in football supporters. They are thrilled by the glory of a great victory, but the reality is just the thrill of the players, and the exuberance in each supporter's mind. There is no further entity called the 'glory'. Green was a liberal. |
| 23048 | The ground of property ownership is not force but the power to use it for social ends [Green,TH, by Muirhead] |
| Full Idea: It is not the power of forcible tenure but the power of utilisation for social ends that is the ground of the permanent recognition that constitutes a right to property. | |
| From: report of T.H. Green (Lectures on the Principles of Political Obligation [1882]) by John H. Muirhead - The Service of the State III | |
| A reaction: Tell that to the aristocratic owners of British grouse moors! This just seems to be wishful thinking. Does that mean that I have no right to property if my ends are not 'social'? |
| 23049 | Property is needed by all citizens, to empower them to achieve social goods [Green,TH] |
| Full Idea: The rationale of property is that every one should be secured by society in the power of getting and keeping the means of realising a will which in possibility is a will directed to social good. | |
| From: T.H. Green (Lectures on the Principles of Political Obligation [1882], §220), quoted by John H. Muirhead - The Service of the State III | |
| A reaction: An interesting argument. If you want free citizens in a liberal society to be capable of achieving social good, you must allow them the right to acquire the means of doing so. |