17 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) |
| 14590 | If we accept scattered objects such as archipelagos, why not think of cars that way? [Hawthorne] |
| Full Idea: In being willing to countenance archipelagos, one embraces scattered objects. Why not then embrace the 'archipelago' of my car and the Eiffel Tower? | |
| From: John Hawthorne (Three-Dimensionalism v Four-Dimensionalism [2008], 2.1) | |
| A reaction: This is a beautifully simple and striking point. Language is full of embracing terms like 'the furniture', but that doesn't mean we assume the furniture is unified. The archipelago is less of an 'object' if you live on one of the islands. |
| 14591 | Four-dimensionalists say instantaneous objects are more fundamental than long-lived ones [Hawthorne] |
| Full Idea: Self-proclaimed four-dimensionalists typically adopt a picture that reckons instantaneous objects (and facts about them) to be more fundamental than long-lived ones. | |
| From: John Hawthorne (Three-Dimensionalism v Four-Dimensionalism [2008], 2.2) | |
| A reaction: A nice elucidation. As in Idea 14588, this seems motivated by a desire for some sort of foundationalism or atomism. Why shouldn't a metaphysic treat the middle-sized or temporally extended as foundational, and derive the rest that way? |
| 14589 | A modal can reverse meaning if the context is seen differently, so maybe context is all? [Hawthorne] |
| Full Idea: One person says 'He can't dig a hole; he hasn't got a spade', and another says 'He can dig a hole; just give him a spade', and both uses of the modal 'can' will be true. So some philosophers say that all modal predications are thus context-dependent. | |
| From: John Hawthorne (Three-Dimensionalism v Four-Dimensionalism [2008], 1.2) | |
| A reaction: Quine is the guru for this view of modality. Hawthorne's example seems to me to rely too much on the linguistic feature of contrasting 'can' and 'can't'. The underlying assertion in the propositions says something real about the possibilities. |
| 8130 | Qualities of experience are just representational aspects of experience ('Representationalism') [Harman, by Burge] |
| Full Idea: Harman defended what came to be known as 'representationalism' - the view that qualitative aspects of experience are nothing other than representational aspects. | |
| From: report of Gilbert Harman (The Intrinsic Quality of Experience [1990]) by Tyler Burge - Philosophy of Mind: 1950-2000 p.459 | |
| A reaction: Functionalists like Harman have a fairly intractable problem with the qualities of experience, and this may be clutching at straws. What does 'represent' mean? How is the representation achieved? Why that particular quale? |
| 14588 | Modern metaphysicians tend to think space-time points are more fundamental than space-time regions [Hawthorne] |
| Full Idea: Nowadays it is common for metaphysicians to hold both that space-time regions are less fundamental than the space-time points that compose them, and that facts about the regions are less fundamental than facts about the points and their arrangements. | |
| From: John Hawthorne (Three-Dimensionalism v Four-Dimensionalism [2008], 1) | |
| A reaction: I'm not quite sure what a physicist would make of this. It seems to be motivated by some a priori preference for atomism, and for system-building from minimal foundations. |