Ideas from 'Knowledge and the Philosophy of Number' by Keith Hossack [2020], by Theme Structure

[found in 'Knowledge and the Philosophy of Number' by Hossack, Keith [Routledge 2021,978-1-350-27796-0]].

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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Naïve logical sets

23623

Predicativism says only predicated sets exist

4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets

23624

The iterative conception has to appropriate Replacement, to justify the ordinals

4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size

23625

Limitation of Size justifies Replacement, but then has to appropriate Power Set

5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / d. and

23628

The connective 'and' can have an order-sensitive meaning, as 'and then'

5. Theory of Logic / E. Structures of Logic / 6. Relations in Logic

23627

'Before' and 'after' are not two relations, but one relation with two orders

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity

23626

Transfinite ordinals are needed in proof theory, and for recursive functions and computability

6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory

23621

Numbers are properties, not sets (because numbers are magnitudes)

6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism

23622

We can only mentally construct potential infinities, but maths needs actual infinities