12 ideas
| 17884 | Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner] |
| Full Idea: There are many coherent stopping points in the hierarchy of increasingly strong mathematical systems, starting with strict finitism, and moving up through predicativism to the higher reaches of set theory. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], Intro) |
| 17893 | 'Reflection principles' say the whole truth about sets can't be captured [Koellner] |
| Full Idea: Roughly speaking, 'reflection principles' assert that anything true in V [the set hierarchy] falls short of characterising V in that it is true within some earlier level. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 2.1) |
| 17894 | We have no argument to show a statement is absolutely undecidable [Koellner] |
| Full Idea: There is at present no solid argument to the effect that a given statement is absolutely undecidable. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 5.3) |
| 17890 | There are at least eleven types of large cardinal, of increasing logical strength [Koellner] |
| Full Idea: Some of the standard large cardinals (in order of increasing (logical) strength) are: inaccessible, Mahlo, weakly compact, indescribable, Erdös, measurable, strong, Wodin, supercompact, huge etc. (...and ineffable). | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) | |
| A reaction: [I don't understand how cardinals can have 'logical strength', but I pass it on anyway] |
| 17887 | PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner] |
| Full Idea: To the extent that we are justified in accepting Peano Arithmetic we are justified in accepting its consistency, and so we know how to expand the axiom system so as to overcome the limitation [of Gödel's Second Theorem]. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.1) | |
| A reaction: Each expansion brings a limitation, but then you can expand again. |
| 17891 | Arithmetical undecidability is always settled at the next stage up [Koellner] |
| Full Idea: The arithmetical instances of undecidability that arise at one stage of the hierarchy are settled at the next. | |
| From: Peter Koellner (On the Question of Absolute Undecidability [2006], 1.4) |
| 11993 | Jones may cease to exist without some simple property, but that doesn't make it essential [Kung] |
| Full Idea: If Jones ceases to be a father, or ceases to be over eight years old, he will cease to exist, yet these properties surely do not belong essentially to him. | |
| From: Joan Kung (Aristotle on Essence and Explanation [1977], II) | |
| A reaction: This seems to correct, though I would doubt whether either of these count as true properties, in the causal sense I prefer. If being 'over 8' is a property, how many 'over n' or 'under m' properties does he have? One for each quantum moment? |
| 11997 | A property may belong essentially to one thing and contingently to another [Kung] |
| Full Idea: It is possible that a property may belong essentially to one thing and contingently to another. | |
| From: Joan Kung (Aristotle on Essence and Explanation [1977], III) | |
| A reaction: Thus a love of blues music may be part of your essence, but only a minor part of me. Sounds right. Spin or charge are part of the essence of an electron, but only contingently part of a child's top. |
| 11992 | Aristotelian essences underlie a thing's existence, explain it, and must belong to it [Kung] |
| Full Idea: Three essentialist claims are labelled 'Aristotelian': the thing would cease to exist without the property; an essential property is explanatory; and it is such that it must belong to everything to which it belongs. | |
| From: Joan Kung (Aristotle on Essence and Explanation [1977], Intro) | |
| A reaction: She says the second one is indispensable, and that it rules out the third one. My working assumption, like hers, is that the second one is the key part of the game, because Aristotle wanted to explain things. |
| 11995 | Some peripheral properties are explained by essential ones, but don't themselves explain properties [Kung] |
| Full Idea: There will be demonstrated properties at the edge of the system, so to speak. They will be explained in terms of the essential properties of the basic entities and principles of the science, but will themselves not be explanatory of further properties. | |
| From: Joan Kung (Aristotle on Essence and Explanation [1977], II) | |
| A reaction: This is an important line of thought which needs clarification. We can't glibly say that essences are what explain the other properties. Some properties do more than others to explain subsequent dependent properties. |
| 11996 | Some non-essential properties may explain more than essential-but-peripheral ones do [Kung] |
| Full Idea: It seems highly likely that some non-essential properties may explain more about the individual or about things of his kind than the peripheral properties. | |
| From: Joan Kung (Aristotle on Essence and Explanation [1977], II) | |
| A reaction: Another important issue, if one is defending the explanatory role of essences. It is not only essences which explain. A key question is whether we endorse individual essences as well as generic ones. I think we should. They explain the details. |
| 16002 | The self is a combination of pairs of attributes: freedom/necessity, infinite/finite, temporal/eternal [Kierkegaard] |
| Full Idea: A human being is essentially spirit, but what is spirit? Spirit is to be a self. But what is the Self? In short, it is a synthesis of the infinite and the finite, of the temporal and the eternal, of freedom and necessity. | |
| From: Sřren Kierkegaard (Sickness unto Death [1849], p.59) | |
| A reaction: The dense language of his first paragraph was to poke fun at fashionable Hegelian writing. The book gets very lucid afterwards! [SY] |