## Principia Mathematica, Volume 1 |

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### Contents

1AGE | 1 |

SERIES continued 250 Elementary properties of wellordered series | 4 |

251 Ordinal numbers | 18 |

252 Segments of wellordered aeries | 27 |

253 Sectional relations of wellordered series | 32 |

254 Qreater and less among wellordered series | 44 |

255 Greater and less among ordinal numbers | 58 |

256 The series of ordinals | 73 |

306 Addition of simple ratios | 289 |

307 Generalized ratios | 296 |

308 Addition of generalized ratios | 299 |

309 Multiplication of generalized ratios | 309 |

310 The series of real numbers 316 | 316 |

311 Addition of concordant real numbers | 320 |

312 Algebraic addition of real number | 327 |

313 Multiplication of real numbers | 333 |

257 The transfinite ancestnvl relation | 81 |

258 Zermelos theorem | 96 |

259 Inductively defined correlations | 102 |

SECTION E FINITE AND INFINITE SERIES AND ORDINALS | 108 |

260 On finite intervals in a series | 109 |

261 Finite and infinite series | 118 |

262 Finite ordinals | 131 |

263 Progressions | 143 |

264 Derivatives of wellordered series | 156 |

265 The series of alephs | 169 |

SECTION F COMPACT SERIES RATIONAL SERIES AND CONTINUOUS SERIES | 179 |

270 Compact series | 180 |

271 Median classes in series _ | 186 |

272 Similarity of position | 191 |

273 Rational series | 199 |

274 On series of finite subclasses of a series | 207 |

275 Continuous series | 218 |

276 On series of infinite subulasses of a series | 221 |

QUANTITY | 231 |

Summary of Part VI | 233 |

SECTION A GENERALIZATION OF NUMBER | 234 |

300 Positive and negative integers and numerical relations 235 | 235 |

301 Numerically defined powers of relations | 244 |

302 On relative primes | 251 |

303 Ratios | 260 |

304 The series of ratios | 278 |

305 Multiplication of simple ratios | 283 |

314 Real numbers as relations | 336 |

SECTION B VECTORFAMILIES | 339 |

330 Elementary properties of vectorfamilies | 350 |

331 Connected families | 360 |

332 On the representative of relation in a family | 367 |

333 Open families | 376 |

334 Serial families | 383 |

335 Initial families | 390 |

336 The seriea of vectors | 393 |

337 Multiples and aubroultiplca of vectors | 403 |

MEASUREMENT | 407 |

350 Ratios of members of a family | 412 |

351 Submultipliable families | 419 |

352 Rational multiples of a given vector | 423 |

353 Rational families | 431 |

354 Rational nets | 436 |

356 Measurement by real numbers | 443 |

359 Existencetheorems for vectorfamilies | 452 |

CYCLIC FAMILIES | 457 |

370 Elementary properties of cyclic families | 462 |

371 The series of vectors | 466 |

372 Integral sections of the series of vectors | 470 |

373 Submultiplns of identity | 475 |

374 Principal submultiples 435 | 485 |

375 Principal ration | 487 |

### Common terms and phrases

a e NC axiom of infinity Bord Cantor Cl induct'C'P Cls induct comp connected family correlator Dedekindian defined definition DK Prop e Cls induct existence-theorems existent sub-class field finite ordinals FM ap conx FM conx FM cycl following propositions ft infin Hence hypothesis inductive cardinals Infin ax infinite KfFM last term less Q mathematical induction median class multiplicative axiom NC induct Nr'ft Nr'P Nr'Q ordinal number ordinally similar present number Prog proof proper fractions proper section properties prove Q less rational series ratios real numbers Rel num id relation Rl'P Rl'Q sect'P serial Similarly h smof smor Q submultiple Transp typically indefinite v e NC ind v e NC induct vector vector-family well-ordered series whence Ye Rat