### Chapter XXXII. The Correlation of Series

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

### Chapter XXXIII. Real Numbers

### Chapter XXXIV. Limits and Irrational Numbers

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

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

### Chapter XXXVI. Ordinal Continuity

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

### Chapter XXXVII. Transfinite Cardinals

- § 283. Transfinite cardinals differ widely from transfinite ordinals
- § 284. Definition of cardinals
- § 285. Properties of cardinals
- § 286. Addition, multiplication, and exponentiation
- § 287. The smallest transfinite cardinal
`a`_{0} - § 288. Other transfinite cardinals
- § 289. Finite and transfinite cardinals form a single series by relation to greater and less

### Chapter XXXVIII. Transfinite Ordinals

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

### Chapter XXXIX. The Infinitesimal Calculus

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

### Chapter XL. The Infinitesimal and the Improper Infinite

### Chapter XLI. Philosophical Arguments Concerning the Infinitesimal

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

### Chapter XLII. The Philosophy of the Continuum

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

### Chapter XLIII. The Philosophy of the Infinite

- § 337. Historical retrospect
- § 338. Positive doctrine of the infinite
- § 339. Proof that there are infinite classes
- § 340. The paradox of Tristram Shandy
- § 341. A whole and a part may be similar
- § 342. Whole and part and formal implication
- § 343. No immediate predecessor of
`ω`or`a`_{0} - § 344. Difficulty as regards the number of all terms, objects, or propositions
- § 345. Cantor's first proof that there is no greatest number
- § 346. His second proof
- § 347. Every class has more sub-classes than terms
- § 348. But this is impossible in certain cases
- § 349. Resulting contradictions
- § 350. Summary of Part V

The Principles of Mathematics was written by Bertrand Russell, and published in in 1903. It is now available in the Public Domain.