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| 13011 | New axioms are being sought, to determine the size of the continuum |
| Full Idea: In current set theory, the search is on for new axioms to determine the size of the continuum. | |||
| From: Penelope Maddy (Believing the Axioms I [1988], §0) | |||
| A reaction: This sounds the wrong way round. Presumably we seek axioms that fix everything else about set theory, and then check to see what continuum results. Otherwise we could just pick our continuum, by picking our axioms. |
| 13014 | Extensional sets are clearer, simpler, unique and expressive |
| Full Idea: The extensional view of sets is preferable because it is simpler, clearer, and more convenient, because it individuates uniquely, and because it can simulate intensional notions when the need arises. | |||
| From: Penelope Maddy (Believing the Axioms I [1988], §1.1) | |||
| A reaction: [She cites Fraenkel, Bar-Hillet and Levy for this] The difficulty seems to be whether the extensional notion captures our ordinary intuitive notion of what constitutes a group of things, since that needs flexible size and some sort of unity. |
| 13013 | The Axiom of Extensionality seems to be analytic |
| Full Idea: Most writers agree that if any sense can be made of the distinction between analytic and synthetic, then the Axiom of Extensionality should be counted as analytic. | |||
| From: Penelope Maddy (Believing the Axioms I [1988], §1.1) | |||
| A reaction: [Boolos is the source of the idea] In other words Extensionality is not worth discussing, because it simply tells you what the world 'set' means, and there is no room for discussion about that. The set/class called 'humans' varies in size. |
| 13021 | The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics |
| Full Idea: The Axiom of Infinity is a simple statement of Cantor's great breakthrough. His bold hypothesis that a collection of elements that had lurked in the background of mathematics could be infinite launched modern mathematics. | |||
| From: Penelope Maddy (Believing the Axioms I [1988], §1.5) | |||
| A reaction: It also embodies one of those many points where mathematics seems to depart from common sense - but then most subjects depart from common sense when they get more sophisticated. Look what happened to art. |
| 13022 | Infinite sets are essential for giving an account of the real numbers |
| Full Idea: If one is interested in analysis then infinite sets are indispensable since even the notion of a real number cannot be developed by means of finite sets alone. | |||
| From: Penelope Maddy (Believing the Axioms I [1988], §1.5) | |||
| A reaction: [Maddy is citing Fraenkel, Bar-Hillel and Levy] So Cantor's great breakthrough (Idea 13021) actually follows from the earlier acceptance of the real numbers, so that's where the departure from common sense started. |
| 13023 | The Power Set Axiom is needed for, and supported by, accounts of the continuum |
| Full Idea: The Power Set Axiom is indispensable for a set-theoretic account of the continuum, ...and in so far as those attempts are successful, then the power-set principle gains some confirmatory support. | |||
| From: Penelope Maddy (Believing the Axioms I [1988], §1.6) | |||
| A reaction: The continuum is, of course, notoriously problematic. Have we created an extra problem in our attempts at solving the first one? |
| 13024 | Efforts to prove the Axiom of Choice have failed |
| Full Idea: Jordain made consistent and ill-starred efforts to prove the Axiom of Choice. | |||
| From: Penelope Maddy (Believing the Axioms I [1988], §1.7) | |||
| A reaction: This would appear to be the fate of most axioms. You would presumably have to use a different system from the one you are engaged with to achieve your proof. |
| 13025 | Modern views say the Choice set exists, even if it can't be constructed |
| Full Idea: Resistance to the Axiom of Choice centred on opposition between existence and construction. Modern set theory thrives on a realistic approach which says the choice set exists, regardless of whether it can be defined, constructed, or given by a rule. | |||
| From: Penelope Maddy (Believing the Axioms I [1988], §1.7) | |||
| A reaction: This seems to be a key case for the ontology that lies at the heart of theory. Choice seems to be an invaluable tool for proofs, so it won't go away, so admit it to the ontology. Hm. So the tools of thought have existence? |
| 13026 | A large array of theorems depend on the Axiom of Choice |
| Full Idea: Many theorems depend on the Axiom of Choice, including that a countable union of sets is countable, and results in analysis, topology, abstract algebra and mathematical logic. | |||
| From: Penelope Maddy (Believing the Axioms I [1988], §1.7) | |||
| A reaction: The modern attitude seems to be to admit anything if it leads to interesting results. It makes you wonder about the modern approach of using mathematics and logic as the cutting edges of ontological thinking. |
| 13019 | The Iterative Conception says everything appears at a stage, derived from the preceding appearances |
| Full Idea: The Iterative Conception (Zermelo 1930) says everything appears at some stage. Given two objects a and b, let A and B be the stages at which they first appear. Suppose B is after A. Then the pair set of a and b appears at the immediate stage after B. | |||
| From: Penelope Maddy (Believing the Axioms I [1988], §1.3) | |||
| A reaction: Presumably this all happens in 'logical time' (a nice phrase I have just invented!). I suppose we might say that the existence of the paired set is 'forced' by the preceding sets. No transcendental inferences in this story? |
| 13018 | Limitation of Size is a vague intuition that over-large sets may generate paradoxes |
| Full Idea: The 'limitation of size' is a vague intuition, based on the idea that being too large may generate the paradoxes. | |||
| From: Penelope Maddy (Believing the Axioms I [1988], §1.3) | |||
| A reaction: This is an intriguing idea to be found right at the centre of what is supposed to be an incredibly rigorous system. |