25 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) |
| 17486 | Supervenience is simply modally robust property co-variance [Hendry] |
| Full Idea: Supervenience is not an ontological relationship, being just modally robust property co-variance. | |
| From: Robin F. Hendry (Chemistry [2008], 'Ontol') | |
| A reaction: I take supervenience to be nothing more than an interesting phenomenon that requires explanation. I suppose Humean Supervenience is a priori metaphysics, since you could hardly explain it. |
| 16643 | Accidents always remain suited to a subject [Bonaventura] |
| Full Idea: An accident's aptitudinal relationship to a subject is essential, and this is never taken away from accidents….for it is true to say that they are suited to a subject. | |
| From: Bonaventura (Commentary on Sentences [1252], IV.12.1.1.1c) | |
| A reaction: This is the compromise view that allows accidents to be separated, for Transubstantiation, while acknowledging that we identify them with their subjects. |
| 16696 | Successive things reduce to permanent things [Bonaventura] |
| Full Idea: Everything successive reduces to something permanent. | |
| From: Bonaventura (Commentary on Sentences [1252], II.2.1.1.3 ad 5), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 18.2 | |
| A reaction: Avicenna first took successive entities seriously, but Bonaventure and Aquinas seem to have rejected them, or given reductive accounts of them. It resembles modern actualists versus modal realists. |
| 17481 | Nuclear charge (plus laws) explains electron structure and spectrum, but not vice versa [Hendry] |
| Full Idea: Given relevant laws of nature (quantum mechanics, the exclusion principle) nuclear charge determines and explains electronic structure and spectroscopic behaviour, but not vice versa. | |
| From: Robin F. Hendry (Chemistry [2008], 'Micro') | |
| A reaction: I argue that the first necessary condition for essentialism is a direction of explanation, and here we seem to have one. |
| 17478 | Maybe two kinds are the same if there is no change of entropy on isothermal mixing [Hendry] |
| Full Idea: One suggestion is that any two different substance, however alike, exhibit a positive entropy change on mixing. So absence of entropy change on isothermal mixing provides a criterion of sameness of kind. | |
| From: Robin F. Hendry (Chemistry [2008], 'Micro') | |
| A reaction: [He cites Paul Needham 2000] This sounds nice, because at a more amateur level we can say that stuff is the same if mixing two samples of it produces no difference. I call it the Upanishads Test. |
| 17484 | Maybe the nature of water is macroscopic, and not in the microstructure [Hendry] |
| Full Idea: Some deny that that microstructure is what makes it water; substance identity and difference should be determined instead by macroscopic similarities and differences. | |
| From: Robin F. Hendry (Chemistry [2008], 'Micro') | |
| A reaction: Very plausible. Is the essential nature of human beings to be found in the structure of our cells? |
| 17479 | The nature of an element must survive chemical change, so it is the nucleus, not the electrons [Hendry] |
| Full Idea: Whatever earns something membership of the extension of 'krypton' must be a property that can survive chemical change and, therefore, the gain and loss of electrons. Hence what makes it krypton must be a nuclear property. | |
| From: Robin F. Hendry (Chemistry [2008], 'Micro') | |
| A reaction: A very nice illuminating example of essentialism in chemistry. The 'nature' is what survives through change, just like what Aristotle said, innit? |
| 17485 | Maybe water is the smallest part of it that still counts as water (which is H2O molecules) [Hendry] |
| Full Idea: If they do count as water, individual H2O molecules are the smallest items that can qualify as water on their own account. Hydroxyl ions and protons, in contrast, qualify as water only as part of a larger body. | |
| From: Robin F. Hendry (Chemistry [2008], 'Micro') | |
| A reaction: As Aristotle might say, this is the homoeomerous aspect of water. This is Hendry's own proposal, and seems rather good. |
| 17482 | Compounds can differ with the same collection of atoms, so structure matters too [Hendry] |
| Full Idea: The distinctness of the isomers ethanol (CH3CH2OH, boiling at 78.4°) and dimethyl ether (CH3OCH3, boiling at -24.9°) must lie in their different molecular structures. ...But structure has continuously varying quantities, like bond length and angle. | |
| From: Robin F. Hendry (Chemistry [2008], 'Micro') | |
| A reaction: [compressed] This seems to imply that what matters is an idealised abstraction of the structure (i.e. its topology), which is a reason for denying that chemistry is reducible to mere physics. |
| 17483 | Water continuously changes, with new groupings of molecules [Hendry] |
| Full Idea: Macroscopic bodies of water are complex and dynamic congeries of different molecular species, in which there is a constant dissociation of individual molecules, re-association of ions, and formation, growth and disassociation of oligomers. | |
| From: Robin F. Hendry (Chemistry [2008], 'Micro') | |
| A reaction: The point is that these activities are needed to explain the behaviour of water (such as its conductivity). |
| 17476 | Elements survive chemical change, and are tracked to explain direction and properties [Hendry] |
| Full Idea: Elements survive chemical change, and chemical explanations track them from one composite substance to another, thereby explaining both the direction of the chemical change, and the properties of the substances they compose. | |
| From: Robin F. Hendry (Chemistry [2008], Intro) | |
| A reaction: [The 16,000th idea of this database, entered on Guy Fawkes' Day 2013] |
| 17477 | Defining elements by atomic number allowed atoms of an element to have different masses [Hendry] |
| Full Idea: In 1923 elements were defined as populations of atoms with the same nuclear charge (i.e. atomic number), allowing that atoms of the same element may have different masses. | |
| From: Robin F. Hendry (Chemistry [2008], 'Chem') | |
| A reaction: The point is that it allowed isotopes of the same element to come under one heading. This is fine for the heavier elements, but a bit dubious for the very light ones (where an isotope makes a bigger difference). |
| 17480 | Generally it is nuclear charge (not nuclear mass) which determines behaviour [Hendry] |
| Full Idea: In general, nuclear charge is the overwhelming determinant of an element's chemical behaviour, while nuclear mass is a negligible factor. | |
| From: Robin F. Hendry (Chemistry [2008], 'Micro') | |
| A reaction: The exception is the isotopes of very light elements light hydrogen. |