Aleph number

In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They are named after the symbol used to denote them, the Hebrew letter aleph ().[1] [2]

Aleph-naught, or aleph-zero, the smallest infinite cardinal number

(Though in older mathematics books, the letter aleph is often printed upside down by accident,[nb 1] partly because a monotype matrix for aleph was mistakenly constructed the wrong way up).[3]

The cardinality of the natural numbers is (read aleph-naught or aleph-zero; the term aleph-null is also sometimes used), the next larger cardinality is aleph-one , then and so on. Continuing in this manner, it is possible to define a cardinal number for every ordinal number , as described below.

The concept and notation are due to Georg Cantor,[4] who defined the notion of cardinality and realized that infinite sets can have different cardinalities.

The aleph numbers differ from the infinity () commonly found in algebra and calculus, in that the alephs measure the sizes of sets, while infinity is commonly defined either as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or as an extreme point of the extended real number line.

Aleph-naught

(aleph-naught, also aleph-zero or aleph-null) is the cardinality of the set of all natural numbers, and is an infinite cardinal. The set of all finite ordinals, called or (where is the lowercase Greek letter omega), has cardinality . A set has cardinality if and only if it is countably infinite, that is, there is a bijection (one-to-one correspondence) between it and the natural numbers. Examples of such sets are

These infinite ordinals: , , , , and are among the countably infinite sets.[5] For example, the sequence (with ordinality ω·2) of all positive odd integers followed by all positive even integers

is an ordering of the set (with cardinality ) of positive integers.

If the axiom of countable choice (a weaker version of the axiom of choice) holds, then is smaller than any other infinite cardinal.

Aleph-one

is the cardinality of the set of all countable ordinal numbers, called or sometimes . This is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore, is distinct from . The definition of implies (in ZF, Zermelo–Fraenkel set theory without the axiom of choice) that no cardinal number is between and . If the axiom of choice is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus is the second-smallest infinite cardinal number. Using the axiom of choice, one can show one of the most useful properties of the set : any countable subset of has an upper bound in (This follows from the fact that the union of a countable number of countable sets is itself countable—one of the most common applications of the axiom of choice.) This fact is analogous to the situation in : every finite set of natural numbers has a maximum which is also a natural number, and finite unions of finite sets are finite.

is actually a useful concept, if somewhat exotic-sounding. An example application is "closing" with respect to countable operations; e.g., trying to explicitly describe the -algebra generated by an arbitrary collection of subsets (see e.g. Borel hierarchy). This is harder than most explicit descriptions of "generation" in algebra (vector spaces, groups, etc.) because in those cases we only have to close with respect to finite operations—sums, products, and the like. The process involves defining, for each countable ordinal, via transfinite induction, a set by "throwing in" all possible countable unions and complements, and taking the union of all that over all of .

Every uncountable coanalytic subset of a Polish space has cardinality or .[6]

Continuum hypothesis

The cardinality of the set of real numbers (cardinality of the continuum) is . It cannot be determined from ZFC (Zermelo–Fraenkel set theory with the axiom of choice) where this number fits exactly in the aleph number hierarchy, but it follows from ZFC that the continuum hypothesis, CH, is equivalent to the identity

[7]

The CH states that there is no set whose cardinality is strictly between that of the integers and the real numbers.[8] CH is independent of ZFC: it can be neither proven nor disproven within the context of that axiom system (provided that ZFC is consistent). That CH is consistent with ZFC was demonstrated by Kurt Gödel in 1940, when he showed that its negation is not a theorem of ZFC. That it is independent of ZFC was demonstrated by Paul Cohen in 1963, when he showed conversely that the CH itself is not a theorem of ZFC—by the (then novel) method of forcing. [7]

Aleph-omega

Aleph-omega is

where the smallest infinite ordinal is denoted ω. That is, the cardinal number is the least upper bound of

.

is the first uncountable cardinal number that can be demonstrated within Zermelo–Fraenkel set theory not to be equal to the cardinality of the set of all real numbers; for any positive integer n we can consistently assume that , and moreover it is possible to assume is as large as we like. We are only forced to avoid setting it to certain special cardinals with cofinality , meaning there is an unbounded function from to it (see Easton's theorem).

Aleph- for general

To define for arbitrary ordinal number , we must define the successor cardinal operation, which assigns to any cardinal number the next larger well-ordered cardinal (if the axiom of choice holds, this is the next larger cardinal).

We can then define the aleph numbers as follows:

and for λ, an infinite limit ordinal,

The α-th infinite initial ordinal is written . Its cardinality is written . In ZFC, the aleph function is a bijection from the ordinals to the infinite cardinals.[9]

Fixed points of omega

For any ordinal α we have

In many cases is strictly greater than α. For example, for any successor ordinal α this holds. There are, however, some limit ordinals which are fixed points of the omega function, because of the fixed-point lemma for normal functions. The first such is the limit of the sequence

Any weakly inaccessible cardinal is also a fixed point of the aleph function.[10] This can be shown in ZFC as follows. Suppose is a weakly inaccessible cardinal. If were a successor ordinal, then would be a successor cardinal and hence not weakly inaccessible. If were a limit ordinal less than , then its cofinality (and thus the cofinality of ) would be less than and so would not be regular and thus not weakly inaccessible. Thus and consequently which makes it a fixed point.

Role of axiom of choice

The cardinality of any infinite ordinal number is an aleph number. Every aleph is the cardinality of some ordinal. The least of these is its initial ordinal. Any set whose cardinality is an aleph is equinumerous with an ordinal and is thus well-orderable.

Each finite set is well-orderable, but does not have an aleph as its cardinality.

The assumption that the cardinality of each infinite set is an aleph number is equivalent over ZF to the existence of a well-ordering of every set, which in turn is equivalent to the axiom of choice. ZFC set theory, which includes the axiom of choice, implies that every infinite set has an aleph number as its cardinality (i.e. is equinumerous with its initial ordinal), and thus the initial ordinals of the aleph numbers serve as a class of representatives for all possible infinite cardinal numbers.

When cardinality is studied in ZF without the axiom of choice, it is no longer possible to prove that each infinite set has some aleph number as its cardinality; the sets whose cardinality is an aleph number are exactly the infinite sets that can be well-ordered. The method of Scott's trick is sometimes used as an alternative way to construct representatives for cardinal numbers in the setting of ZF. For example, one can define card(S) to be the set of sets with the same cardinality as S of minimum possible rank. This has the property that card(S) = card(T) if and only if S and T have the same cardinality. (The set card(S) does not have the same cardinality of S in general, but all its elements do.)

gollark: What?
gollark: To prevent this, we recommend doing `++choose one two three four [...]`.
gollark: ++choose 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524
gollark: The first number is the number of times to choose.
gollark: <@402456897812168705> Working as intended.

See also

Notes

  1. For example, in (Sierpiński 1958, p.402) the letter aleph appears both the right way up and upside down

Citations

  1. "Comprehensive List of Set Theory Symbols". Math Vault. 2020-04-11. Retrieved 2020-08-12.
  2. Weisstein, Eric W. "Aleph". mathworld.wolfram.com. Retrieved 2020-08-12.
  3. Swanson, Ellen; O'Sean, Arlene Ann; Schleyer, Antoinette Tingley (1999) [1979], Mathematics into type: Copy editing and proofreading of mathematics for editorial assistants and authors (updated ed.), Providence, R.I.: American Mathematical Society, p. 16, ISBN 0-8218-0053-1, MR 0553111
  4. Jeff Miller. "Earliest Uses of Symbols of Set Theory and Logic". jeff560.tripod.com. Retrieved 2016-05-05. Miller quotes Joseph Warren Dauben (1990). Georg Cantor:His Mathematics and Philosophy of the Infinite. ISBN 9780691024479. : "His new numbers deserved something unique. ... Not wishing to invent a new symbol himself, he chose the aleph, the first letter of the Hebrew alphabet...the aleph could be taken to represent new beginnings..."
  5. Jech, Thomas (2003), Set Theory, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag
  6. Dales H.G., Dashiell F.K., Lau A.TM., Strauss D. (2016) Introduction. In: Banach Spaces of Continuous Functions as Dual Spaces. CMS Books in Mathematics (Ouvrages de mathématiques de la SMC). Springer, Cham
  7. Szudzik, Mattew (31 July 2018). "Continuum Hypothesis". Wolfram Mathworld. Wolfram Web Resources. Retrieved 15 August 2018.
  8. Weisstein, Eric W. "Continuum Hypothesis". mathworld.wolfram.com. Retrieved 2020-08-12.
  9. aleph numbers at PlanetMath.org.
  10. Harris, Kenneth (April 6, 2009). "Math 582 Intro to Set Theory, Lecture 31" (PDF). Department of Mathematics, University of Michigan. Archived from the original (PDF) on March 4, 2016. Retrieved September 1, 2012.

References

  • Sierpiński, Wacław (1958), Cardinal and ordinal numbers., Polska Akademia Nauk Monografie Matematyczne, 34, Warsaw: Państwowe Wydawnictwo Naukowe, MR 0095787


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