Combining Texts

All the ideas for 'Philosophical Fragments', 'Logicism, Some Considerations (PhD)' and 'On boundary numbers and domains of sets'

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9 ideas

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets

17832

Zermelo showed that the ZF axioms in 1930 were non-categorical [Zermelo, by Hallett,M]

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII

13028

Replacement was added when some advanced theorems seemed to need it [Zermelo, by Maddy]

5. Theory of Logic / L. Paradox / 3. Antinomies

17626

The antinomy of endless advance and of completion is resolved in well-ordered transfinite numbers [Zermelo]

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers

13412

Obtaining numbers by abstraction is impossible - there are too many; only a rule could give them, in order [Benacerraf]

13413

We must explain how we know so many numbers, and recognise ones we haven't met before [Benacerraf]

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers

13411

If numbers are basically the cardinals (Frege-Russell view) you could know some numbers in isolation [Benacerraf]

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism

13415

An adequate account of a number must relate it to its series [Benacerraf]

7. Existence / A. Nature of Existence / 5. Reason for Existence

16007

I assume existence, rather than reasoning towards it [Kierkegaard]

10. Modality / A. Necessity / 2. Nature of Necessity

16013

Nothing necessary can come into existence, since it already 'is' [Kierkegaard]