Table of Contents

1. Chapter XXXII. The Correlation of Series

   1. § 249. The infinitesimal and space are no longer required in a statement of principles
   2. § 250. The supposed contradictions of infinity have been resolved
   3. § 251. Correlation of series
   4. § 252. Independent series and series by correlation
   5. § 253. Likeness of relations
   6. § 254. Functions
   7. § 255. Functions of a variable whose values form a series
   8. § 256. Functions which are defined by formulae
   9. § 257. Complete series

2. Chapter XXXIII. Real Numbers

   1. § 258. Real numbers are not limits of series of rationals
   2. § 259. Segments of rationals
   3. § 260. Properties of segments
   4. § 261. Coherent classes in a series

3. Chapter XXXIV. Limits and Irrational Numbers

   1. § 262. Definition of a limit
   2. § 263. Elementary properties of limits
   3. § 264. An arithmetical theory of irrationals is indispensable
   4. § 265. Dedekind's theory of irrationals
   5. § 266. Defects in Dedekind's axiom of continuity
   6. § 267. Objections to his theory of irrationals
   7. § 268. Weierstrass's theory
   8. § 269. Cantor's theory
   9. § 270. Real numbers are segments of rationals

4. Chapter XXXV. Cantor's First Definition of Continuity

   1. § 271. The arithmetical theory of continuity is due to Cantor
   2. § 272. Cohesion
   3. § 273. Perfection
   4. § 274. Defect in Cantor's definition of perfection
   5. § 275. The existence of limits must not be assumed without special grounds

5. Chapter XXXVI. Ordinal Continuity

   1. § 276. Continuity is a purely ordinal notion
   2. § 277. Cantor's ordinal definition of continuity
   3. § 278. Only ordinal notions occur in this definition
   4. § 279. Infinite classes of integers can be arranged in a continuous series
   5. § 280. Segments of general compact series
   6. § 281. Segments defined by fundamental series
   7. § 282. Two compact series may be combined to form a series which is not compact

6. Chapter XXXVII. Transfinite Cardinals

   1. § 283. Transfinite cardinals differ widely from transfinite ordinals
   2. § 284. Definition of cardinals
   3. § 285. Properties of cardinals
   4. § 286. Addition, multiplication, and exponentiation
   5. § 287. The smallest transfinite cardinal a₀
   6. § 288. Other transfinite cardinals
   7. § 289. Finite and transfinite cardinals form a single series by relation to greater and less

7. Chapter XXXVIII. Transfinite Ordinals

   1. § 290. Ordinals are classes of serial relations
   2. § 291. Cantor's definition of the second class of ordinals
   3. § 292. Definition of ω
   4. § 293. An infinite class can be arranged in many types of series
   5. § 294. Addition and subtraction of ordinals
   6. § 295. Multiplication and division
   7. § 296. Well-ordered series
   8. § 297. Series which are not well-ordered
   9. § 298. Ordinal numbers are types of well-ordered series
   10. § 299. Relation-arithmetic
   11. § 300. Proofs of existence-theorems
   12. § 301. There is no maximum ordinal number
   13. § 302. Successive derivatives of a series

8. Chapter XXXIX. The Infinitesimal Calculus

   1. § 303. The infinitesimal has been usually supposed essential to the calculus
   2. § 304. Definition of a continuous function
   3. § 305. Definition of the derivative of a function
   4. § 306. The infinitesimal is not implied in this definition
   5. § 307. Definition of the definite integral
   6. § 308. Neither the infinite nor the infinitesimal is involved in this definition

9. Chapter XL. The Infinitesimal and the Improper Infinite

   1. § 309. A precise definition of the infinitesimal is seldom given
   2. § 310. Definition of the infinitesimal and the improper infinite
   3. § 311. Instances of the infinitesimal
   4. § 312. No infinitesimal segments in compact series
   5. § 313. Orders of infinity and infinitesimality
   6. § 314. Summary

10. Chapter XLI. Philosophical Arguments Concerning the Infinitesimal

    1. § 315. Current philosophical opinions illustrated by Cohen
    2. § 316. Who bases the calculus upon infinitesimals
    3. § 317. Space and motion are here irrelevant
    4. § 318. Cohen regards the doctrine of limits as insufficient for the calculus
    5. § 319. And supposes limits to be essentially quantitative
    6. § 320. To involve infinitesimal differences
    7. § 321. And to introduce a new meaning of equality
    8. § 322. He identifies the inextensive with the intensive
    9. § 323. Consecutive numbers are supposed to be required for continuous change
    10. § 324. Cohen's views are to be rejected

11. Chapter XLII. The Philosophy of the Continuum

    1. § 325. Philosophical sense of continuity not here in question
    2. § 326. The continuum is composed of mutually external units
    3. § 327. Zeno and Weierstrass
    4. § 328. The argument of dichotomy
    5. § 329. The objectionable and the innocent kind of endless regress
    6. § 330. Extensional and intensional definition of a whole
    7. § 331. Achilles and the tortoise
    8. § 332. The arrow
    9. § 333. Change does not involve a state of change
    10. § 334. The argument of the measure
    11. § 335. Summary of Cantor's doctrine of continuity
    12. § 336. The continuum consists of elements

12. Chapter XLIII. The Philosophy of the Infinite

    1. § 337. Historical retrospect
    2. § 338. Positive doctrine of the infinite
    3. § 339. Proof that there are infinite classes
    4. § 340. The paradox of Tristram Shandy
    5. § 341. A whole and a part may be similar
    6. § 342. Whole and part and formal implication
    7. § 343. No immediate predecessor of ω or a₀
    8. § 344. Difficulty as regards the number of all terms, objects, or propositions
    9. § 345. Cantor's first proof that there is no greatest number
    10. § 346. His second proof
    11. § 347. Every class has more sub-classes than terms
    12. § 348. But this is impossible in certain cases
    13. § 349. Resulting contradictions
    14. § 350. Summary of Part V