12 ideas
17832 | Zermelo showed that the ZF axioms in 1930 were non-categorical [Zermelo, by Hallett,M] |
Full Idea: Zermelo's paper sets out to show that the standard set-theoretic axioms (what he calls the 'constitutive axioms', thus the ZF axioms minus the axiom of infinity) have an unending sequence of different models, thus that they are non-categorical. | |
From: report of Ernst Zermelo (On boundary numbers and domains of sets [1930]) by Michael Hallett - Introduction to Zermelo's 1930 paper p.1209 | |
A reaction: Hallett says later that Zermelo is working with second-order set theory. The addition of an Axiom of Infinity seems to have aimed at addressing the problem, and the complexities of that were pursued by Gödel. |
13028 | Replacement was added when some advanced theorems seemed to need it [Zermelo, by Maddy] |
Full Idea: Zermelo included Replacement in 1930, after it was noticed that the sequence of power sets was needed, and Replacement gave the ordinal form of the well-ordering theorem, and justification for transfinite recursion. | |
From: report of Ernst Zermelo (On boundary numbers and domains of sets [1930]) by Penelope Maddy - Believing the Axioms I §1.8 | |
A reaction: Maddy says that this axiom suits the 'limitation of size' theorists very well, but is not so good for the 'iterative conception'. |
18812 | Split out the logical vocabulary, make an assignment to the rest. It's logical if premises and conclusion match [Tarski, by Rumfitt] |
Full Idea: Tarski made a division of logical and non-logical vocabulary. He then defined a model as a non-logical assignment satisfying the corresponding sentential function. Then a conclusion follows logically if every model of the premises models the conclusion. | |
From: report of Alfred Tarski (The Concept of Logical Consequence [1936]) by Ian Rumfitt - The Boundary Stones of Thought 3.2 | |
A reaction: [compressed] This is Tarski's account of logical consequence, which follows on from his account of truth. 'Logical validity' is then 'true in every model'. Rumfitt doubts whether Tarski has given the meaning of 'logical consequence'. |
13344 | X follows from sentences K iff every model of K also models X [Tarski] |
Full Idea: The sentence X follows logically from the sentences of the class K if and only if every model of the class K is also a model of the sentence X. | |
From: Alfred Tarski (The Concept of Logical Consequence [1936], p.417) | |
A reaction: [see Idea 13343 for his account of a 'model'] He is offering to define logical consequence in general, but this definition fits what we now call 'semantic consequence', written |=. This it is standard practice to read |= as 'models'. |
13343 | A 'model' is a sequence of objects which satisfies a complete set of sentential functions [Tarski] |
Full Idea: An arbitrary sequence of objects which satisfies every sentential function of the sentences L' will be called a 'model' or realization of the class L of sentences. There can also be a model of a single sentence is this way. | |
From: Alfred Tarski (The Concept of Logical Consequence [1936], p.417) | |
A reaction: [L' is L with the constants replaced by variables] Tarski is the originator of model theory, which is central to modern logic. The word 'realization' is a helpful indicator of what he has in mind. A model begins to look like a possible world. |
17626 | The antinomy of endless advance and of completion is resolved in well-ordered transfinite numbers [Zermelo] |
Full Idea: Two opposite tendencies of thought, the idea of creative advance and of collection and completion (underlying the Kantian 'antinomies') find their symbolic representation and their symbolic reconciliation in the transfinite numbers based on well-ordering. | |
From: Ernst Zermelo (On boundary numbers and domains of sets [1930], §5) | |
A reaction: [a bit compressed] It is this sort of idea, from one of the greatest set-theorists, that leads philosophers to think that the philosophy of mathematics may offer solutions to metaphysical problems. As an outsider, I am sceptical. |
13345 | Sentences are 'analytical' if every sequence of objects models them [Tarski] |
Full Idea: A class of sentences can be called 'analytical' if every sequence of objects is a model of it. | |
From: Alfred Tarski (The Concept of Logical Consequence [1936], p.418) | |
A reaction: See Idea 13344 and Idea 13343 for the context of this assertion. |
6613 | The natural kinds are objects, processes and properties/relations [Ellis] |
Full Idea: There are three hierarchies of natural kinds: objects or substances (substantive universals), events or processes (dynamic universals), and properties or relations (tropic universals). | |
From: Brian Ellis (Katzav on limitations of dispositions [2005], 91) | |
A reaction: Most interesting here is the identifying of natural kinds with universals, making universals into the families of nature. Universals are high-level sets of natural kinds. To grasp universals you must see patterns, and infer the underlying order. |
6616 | Least action is not a causal law, but a 'global law', describing a global essence [Ellis] |
Full Idea: The principle of least action is not a causal law, but is what I call a 'global law', which describes the essence of the global kind, which every object in the universe necessarily instantiates. | |
From: Brian Ellis (Katzav on limitations of dispositions [2005]) | |
A reaction: As a fan of essentialism I find this persuasive. If I inherit part of my essence from being a mammal, I inherit other parts of my essence from being an object, and all objects would share that essence, so it would look like a 'law' for all objects. |
6615 | A species requires a genus, and its essence includes the essence of the genus [Ellis] |
Full Idea: A specific universal can exist only if the generic universal of which it is a species exists, but generic universals don't depend on species; …the essence of any genus is included in its species, but not conversely. | |
From: Brian Ellis (Katzav on limitations of dispositions [2005], 91) | |
A reaction: Thus the species 'electron' would be part of the genus 'lepton', or 'human' part of 'mammal'. The point of all this is to show how individual items connect up with the rest of the universe, giving rise to universal laws, such as Least Action. |
6614 | A hierarchy of natural kinds is elaborate ontology, but needed to explain natural laws [Ellis] |
Full Idea: The hierarchy of natural kinds proposed by essentialism may be more elaborate than is strictly required for purposes of ontology, but it is necessary to explain the necessity of the laws of nature, and the universal applicability of global principles. | |
From: Brian Ellis (Katzav on limitations of dispositions [2005], 91) | |
A reaction: I am all in favour of elaborating ontology in the name of best explanation. There seem, though, to be some remaining ontological questions at the point where the explanations of essentialism run out. |
6612 | Without general principles, we couldn't predict the behaviour of dispositional properties [Ellis] |
Full Idea: It is objected to dispositionalism that without the principle of least action, or some general principle of equal power, the specific dispositional properties of things could tell us very little about how these things would be disposed to behave. | |
From: Brian Ellis (Katzav on limitations of dispositions [2005], 90) | |
A reaction: Ellis attempts to meet this criticism, by placing dispositional properties within a hierarchy of broader properties. There remains a nagging doubt about how essentialism can account for space, time, order, and the existence of essences. |