maths/pure/ AbstractSetTheory101

Here is a rough overview of the basics of AbstractSetTheory, in particular ZFC and how we construct things like numbers and functions.

What is 'abstract' about Abstract Set Theory?

We never worry about, or ask about, what sets actually are. In the formal language of SetTheory, there is only one non-logical symbol: ∈. Abstract set theory is all about the relation ∈. We begin with a set of a priori assumptions about properties that any sensible notion of ∈ should satisfy. We call these axioms. There are a few different sets of axioms for Set Theory, but the most common is ZFC (Zermelo-Fraenkel with the AxiomOfChoice).

Axioms

  1. Extensionality: two sets are equal if and only if they have the same elements.
  2. Regularity: every non-empty set contains an element e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006c2356069e9d1e79ca924378153cfbbfb4d4416b1f99d41a2940bfdb66c5319db disjoint from e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006b7a56873cd771f2c446d369b649430b65a756ba278ff97ec81bb6f55b2e73569.
  3. Schema of Comprehension: Given a set e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470065f9c4ab08cac7457e9111a30e4664920607ea2c115a1433d7be98e97e64244ca and a formula e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006670671cd97404156226e507973f2ab8330d3022ca96e0c93bdbdb320c41adcaf, the set e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c4700659e19706d51d39f66711c2653cd7eb1291c94d9b55eb14bda74ce4dc636d015a exists.
  4. Pairing: if e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c4700635135aaa6cc23891b40cb3f378c53a17a1127210ce60e125ccf03efcfdaec458 and e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006624b60c58c9d8bfb6ff1886c2fd605d2adeb6ea4da576068201b6c6958ce93f4 exist, then there is a set containing both e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006eb1e33e8a81b697b75855af6bfcdbcbf7cbbde9f94962ceaec1ed8af21f5a50f and e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006e29c9c180c6279b0b02abd6a1801c7c04082cf486ec027aa13515e4f3884bb6b.
  5. Union: the union over the elements of a set exists. That is, if e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006c6f3ac57944a531490cd39902d0f777715fd005efac9a30622d5f5205e7f6894 then the set e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c4700686e50149658661312a9e0b35558d84f6c6d3da797f552a9657fe0558ca40cdef exists.
  6. Replacement: informally if e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470069f14025af0065b30e47e23ebb3b491d39ae8ed17d33739e5ff3827ffb3634953 is a function definable by a formula e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c4700676a50887d8f1c2e9301755428990ad81479ee21c25b43215cf524541e0503269, e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470067a61b53701befdae0eeeffaecc73f14e20b537bb0f8b91ad7c2936dc63562b25 is a set, and e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006aea92132c4cbeb263e6ac2bf6c183b5d81737f179f21efdc5863739672f0f470 exists for all e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470060b918943df0962bc7a1824c0555a389347b4febdc7cf9d1254406d80ce44e3f9, then the set e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006d59eced1ded07f84c145592f65bdf854358e009c5cd705f5215bf18697fed103 exists.
  7. Infinity: there is a set containing e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470063d914f9348c9cc0ff8a79716700b9fcd4d2f3e711608004eb8f138bcba7f14d9 and closed under the successor operation e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c4700673475cb40a568e8da8a045ced110137e159f890ac4da883b6b17dc651b3a8049.
  8. Power set: if e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c4700644cb730c420480a0477b505ae68af508fb90f96cf0ec54c6ad16949dd427f13a is a set, then so is the set of all subsets of e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c4700671ee45a3c0db9a9865f7313dd3372cf60dca6479d46261f3542eb9346e4a04d6, which we call the power set of e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006811786ad1ae74adfdd20dd0372abaaebc6246e343aebd01da0bfc4c02bf0106c and denote it by e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c4700625fc0e7096fc653718202dc30b0c580b8ab87eac11a700cba03a7c021bc35b0c.
  9. Choice (AC): the cartesian product of a collection of nonempty sets is nonempty.
  10. Or equivalent to AC: for every set e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c4700631489056e0916d59fe3add79e63f095af3ffb81604691f21cad442a85c7be617 there is a binary relation on e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c4700698010bd9270f9b100b6214a21754fd33bdc8d41b2bc9f9dd16ff54d3c34ffd71 which makes e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470060e17daca5f3e175f448bacace3bc0da47d0655a74c8dd0dc497a3afbdad95f1f well-ordered.

Relations and functions

Given sets e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470061a6562590ef19d1045d06c4055742d38288e9e6dcd71ccde5cee80f1d5a774eb and e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006031b4af5197ec30a926f48cf40e11a7dbc470048a21e4003b7a3c07c5dab1baa, the cartesian product of e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c4700641cfc0d1f2d127b04555b7246d84019b4d27710a3f3aff6e7764375b1e06e05d and e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470062858dcd1057d3eae7f7d5f782167e24b61153c01551450a628cee722509f6529, denoted e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470062fca346db656187102ce806ac732e06a62df0dbb2829e511a770556d398e1a6e is the set of all pairs e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c4700602d20bbd7e394ad5999a4cebabac9619732c343a4cac99470c03e23ba2bdc2bc such that e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470067688b6ef52555962d008fff894223582c484517cea7da49ee67800adc7fc8866 and e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006c837649cce43f2729138e72cc315207057ac82599a59be72765a477f22d14a54. That is: e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470066208ef0f7750c111548cf90b6ea1d0d0a66f6bff40dbef07cb45ec436263c7d6. Similarly we define the cartesian product of many sets e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470063e1e967e9b793e908f8eae83c74dba9bcccce6a5535b4b462bd9994537bfe15c. The cartesian product of an infinite collection of sets e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c4700639fa9ec190eee7b6f4dff1100d6343e10918d044c75eac8f9e9a2596173f80c9 where e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006d029fa3a95e174a19934857f535eb9427d967218a36ea014b70ad704bc6c8d1c is an infinite index set, is more fiddly to define (essentially elements of the cartesian product of an infinite collection of sets are functions e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c4700681b8a03f97e8787c53fe1a86bda042b6f0de9b0ec9c09357e107c99ba4d6948a such that for all e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006da4ea2a5506f2693eae190d9360a1f31793c98a1adade51d93533a6f520ace1c we have e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006a68b412c4282555f15546cf6e1fc42893b7e07f271557ceb021821098dd66c1b).

A e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006108c995b953c8a35561103e2014cf828eb654a99e310f87fab94c2f4b7d2a04f-ary relation on a set e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470063ada92f28b4ceda38562ebf047c6ff05400d4c572352a1142eedfef67d21e662 is a subset of e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c4700649d180ecf56132819571bf39d9b7b342522a2ac6d23c1418d3338251bfe469c8 where the cartesian product is of e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006a21855da08cb102d1d217c53dc5824a3a795c1c1a44e971bf01ab9da3a2acbbf copies of e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006c75cb66ae28d8ebc6eded002c28a8ba0d06d3a78c6b5cbf9b2ade051f0775ac4.

A function e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006ff5a1ae012afa5d4c889c50ad427aaf545d31a4fac04ffc1c4d03d403ba4250a is a relation e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470067f2253d7e228b22a08bda1f09c516f6fead81df6536eb02fa991a34bb38d9be8 on e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470068722616204217eddb39e7df969e0698aed8e599ba62ed2de1ce49b03ade0fede, such that e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c4700696061e92f58e4bdcdee73df36183fe3ac64747c81c26f6c83aada8d2aabb1864 associates an element of e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006eb624dbe56eb6620ae62080c10a273cab73ae8eca98ab17b731446a31c79393a to at most one element of e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006f369cb89fc627e668987007d121ed1eacdc01db9e28f8bb26f358b7d8c4f08ac. That is, if e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006f74efabef12ea619e30b79bddef89cffa9dda494761681ca862cff2871a85980 and e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006a88a7902cb4ef697ba0b6759c50e8c10297ff58f942243de19b984841bfe1f73 then necessarily e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006349c41201b62db851192665c504b350ff98c6b45fb62a8a2161f78b6534d8de9. When e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c4700698a3ab7c340e8a033e7b37b6ef9428751581760af67bbab2b9e05d4964a8874a defines a function, we use the notation e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c4700648449a14a4ff7d79bb7a1b6f3d488eba397c36ef25634c111b49baf362511afc.

Properties of functions

A function is injective if different inputs give different outputs. That is, if e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470065316ca1c5ddca8e6ceccfce58f3b8540e540ee22f6180fb89492904051b3d531 and e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006a46e37632fa6ca51a13fe39a567b3c23b28c2f47d8af6be9bd63e030e214ba38 then e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006bbb965ab0c80d6538cf2184babad2a564a010376712012bd07b0af92dcd3097d.

A function is surjective if its codomain is equal to its range. That is, if e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c4700644c8031cb036a7350d8b9b8603af662a4b9cdbd2f96e8d5de5af435c9c35da69 then for all e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006b4944c6ff08dc6f43da2e9c824669b7d927dd1fa976fadc7b456881f51bf5ccc there is e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006434c9b5ae514646bbd91b50032ca579efec8f22bf0b4aac12e65997c418e0dd6 with e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006bdd2d3af3a5a1213497d4f1f7bfcda898274fe9cb5401bbc0190885664708fc2.

A function is bijective if it is both injective and surjective.

Ordinals

We can construct the natural numbers by identifying e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470068b940be7fb78aaa6b6567dd7a3987996947460df1c668e698eb92ca77e425349 with the empty set e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006cd70bea023f752a0564abb6ed08d42c1440f2e33e29914e55e0be1595e24f45a, and the proceeding to define the successor e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c4700669f59c273b6e669ac32a6dd5e1b2cb63333d8b004f9696447aee2d422ce63763. This way we get an ordinal for each natural number e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470061da51b8d8ff98f6a48f80ae79fe3ca6c26e1abb7b7d125259255d6d2b875ea08. These are the finite ordinals. We can take the union of all finite ordinals to get the first infinite ordinal, which we denote e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470068241649609f88ccd2a0a5b233a07a538ec313ff6adf695aa44a969dbca39f67d. We call these limit ordinals. Once we have e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470066e4001871c0cf27c7634ef1dc478408f642410fd3a444e2a88e301f5c4a35a4d, we can construct e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006e3d6c4d4599e00882384ca981ee287ed961fa5f3828e2adb5e9ea890ab0d0525), which we denote e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006ad48ff99415b2f007dc35b7eb553fd1eb35ebfa2f2f308acd9488eeb86f71fa8, and then e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470067b1a278f5abe8e9da907fc9c29dfd432d60dc76e17b0fabab659d2a508bc65c4 and so on. Then we can take the union of all the e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006d6d824abba4afde81129c71dea75b8100e96338da5f416d2f69088f1960cb091 to get e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c4700629db0c6782dbd5000559ef4d9e953e300e2b479eed26d887ef3f92b921c06a67, which we denote e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470068c1f1046219ddd216a023f792356ddf127fce372a72ec9b4cdac989ee5b0b455. In this way, we can create bigger and bigger ordinals. Either using the successor operation or taking unions to produce limit ordinals. Ordinals are well-ordered. That is, a set of ordinals does not contain an infinite descending sequence.

Cardinals

The cardinality of a set is the 'number' of elements in it. It can be infinite. Two sets have the same cardinality, that is, they are equinumerous if, and only if, there exists a bijective function between them.

The Schröder-Bernstein Theorem says that if there exists an injection e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006ad57366865126e55649ecb23ae1d48887544976efea46a48eb5d85a6eeb4d306 and an injection e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c4700616dc368a89b428b2485484313ba67a3912ca03f2b2b42429174a4f8b3dc84e44 then there exists a bijection e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c4700637834f2f25762f23e1f74a531cbe445db73d6765ebe60878a7dfbecd7d4af6e1.

We denote the cardinality of a set e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006454f63ac30c8322997ef025edff6abd23e0dbe7b8a3d5126a894e4a168c1b59b by e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470065ef6fdf32513aa7cd11f72beccf132b9224d33f271471fff402742887a171edf, and we identify e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470061253e9373e781b7500266caa55150e08e210bc8cd8cc70d89985e3600155e860 with the smallest ordinal e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006482d9673cfee5de391f97fde4d1c84f9f8d6f2cf0784fcffb958b4032de7236c such that e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470063346f2bbf6c34bd2dbe28bd1bb657d0e9c37392a1d5ec9929e6a5df4763ddc2d is equinumerous with e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470069537f32ec7599e1ae953af6c9f929fe747ff9dadf79a9beff1f304c550173011. The cardinal of the set of all finite ordinals is denoted e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470060fd42b3f73c448b34940b339f87d07adf116b05c0227aad72e8f0ee90533e699, (aleph null), and (assuming AC) is the smallest infinite ordinal.

If we assume the Axiom of Choice, or equivalently, the well-ordering principle, then every set is equinumerous with some ordinal, and so the cardinals are linearly ordered.

Countability

A set is countable if either it is finite or else its cardinality is e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470069bdb2af6799204a299c603994b8e400e4b1fd625efdb74066cc869fee42c9df3. Equivalently, a set X is countable if it is the image of some function from e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006f6e0a1e2ac41945a9aa7ff8a8aaa0cebc12a3bcc981a929ad5cf810a090e11ae. That is, if e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006b1556dea32e9d0cdbfed038fd7787275775ea40939c146a64e205bcb349ad02f.

The power set e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470066c658ee83fb7e812482494f3e416a876f63f418a0b8a1f5e76d47ee4177035cb of a set e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470069f1f9dce319c4700ef28ec8c53bd3cc8e6abe64c68385479ab89215806a5bdd6 is always of a larger cardinality than e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c4700628dae7c8bde2f3ca608f86d0e16a214dee74c74bee011cdfdd46bc04b655bc14, as can be proved by Cantor's Diagonal Argument. The real numbers e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c47006e5b861a6d8a966dfca7e7341cd3eb6be9901688d547a72ebed0b1f5e14f3d08d are at least as large as the power set e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c470062ac878b0e2180616993b4b6aa71e61166fdc86c28d47e359d0ee537eb11d46d3 of the natural numbers, so are strictly larger (in cardinality terms), than e34d955f086593171531592e6a78030da78dd67e6326646142781e1062c4700685daaf6f7055cd5736287faed9603d712920092c4f8fd0097ec3b650bf27530e.