Combining Texts

All the ideas for 'On the Question of Absolute Undecidability', 'Instrospection' and 'fragments/reports'

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8 ideas

4. Formal Logic / F. Set Theory ST / 1. Set Theory

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)

5. Theory of Logic / K. Features of Logics / 5. Incompleteness

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)

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity

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]

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic

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.

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic

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)

16. Persons / C. Self-Awareness / 1. Introspection

5692	Introspection is not perception, because there are no extra qualities apart from the mental events themselves [Rosenthal]
	Full Idea: Introspection cannot be a form of perceiving, since that invariably involves sensory qualities, and no qualities occur in introspection other than those of the sensations and perceptions we introspect; there are no additional qualities.
	From: David M. Rosenthal (Instrospection [1998])
	A reaction: This sounds pretty conclusive. Presumably introspection is best described as meta-thought rather than perception, which means that it involves beliefs and judgements, rather than new perceptual qualities. It has to be conceptual, and probably linguistic.

23. Ethics / C. Virtue Theory / 3. Virtues / a. Virtues

6004	The cardinal virtues are theoretical (based on knowledge), and others are 'non-theoretical' [Hecato, by Dorandi]
	Full Idea: Hecato defined the cardinal virtues as 'theoretical', that is, based on knowledge, and to these he opposed those that are 'non-theoretical', for example, health, beauty, strength of spirit, and courage.
	From: report of Hecato (fragments/reports [c.70 BCE]) by Tiziano Dorandi - Hecato of Rhodes
	A reaction: Mostly these are Aristotle's external and non-external virtues, except that courage is here included among the former, implying, presumably, that it is more of a natural gift than an intellectual achievement.