Combining Texts

All the ideas for 'On the Question of Absolute Undecidability', 'What are Sets and What are they For?' and 'Properties and Predicates'

expand these ideas     |    start again     |     specify just one area for these texts


21 ideas

4. Formal Logic / F. Set Theory ST / 1. Set Theory
Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner]
'Reflection principles' say the whole truth about sets can't be captured [Koellner]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
The empty set is usually derived from Separation, but it also seems to need Infinity [Oliver/Smiley]
The empty set is something, not nothing! [Oliver/Smiley]
We don't need the empty set to express non-existence, as there are other ways to do that [Oliver/Smiley]
Maybe we can treat the empty set symbol as just meaning an empty term [Oliver/Smiley]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / c. Unit (Singleton) Sets
The unit set may be needed to express intersections that leave a single member [Oliver/Smiley]
5. Theory of Logic / G. Quantification / 6. Plural Quantification
If you only refer to objects one at a time, you need sets in order to refer to a plurality [Oliver/Smiley]
We can use plural language to refer to the set theory domain, to avoid calling it a 'set' [Oliver/Smiley]
5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
Logical truths are true no matter what exists - but predicate calculus insists that something exists [Oliver/Smiley]
5. Theory of Logic / K. Features of Logics / 5. Incompleteness
We have no argument to show a statement is absolutely undecidable [Koellner]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
If mathematics purely concerned mathematical objects, there would be no applied mathematics [Oliver/Smiley]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
There are at least eleven types of large cardinal, of increasing logical strength [Koellner]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
Arithmetical undecidability is always settled at the next stage up [Koellner]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Sets might either represent the numbers, or be the numbers, or replace the numbers [Oliver/Smiley]
8. Modes of Existence / B. Properties / 2. Need for Properties
A property is merely a constituent of laws of nature; temperature is just part of thermodynamics [Mellor]
8. Modes of Existence / B. Properties / 10. Properties as Predicates
There is obviously a possible predicate for every property [Mellor]
8. Modes of Existence / D. Universals / 2. Need for Universals
We need universals for causation and laws of nature; the latter give them their identity [Mellor]
8. Modes of Existence / E. Nominalism / 3. Predicate Nominalism
If properties were just the meanings of predicates, they couldn't give predicates their meaning [Mellor]
26. Natural Theory / C. Causation / 8. Particular Causation / e. Probabilistic causation
Singular causation requires causes to raise the physical probability of their effects [Mellor]