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 c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178bc2356069e9d1e79ca924378153cfbbfb4d4416b1f99d41a2940bfdb66c5319db disjoint from c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178bb7a56873cd771f2c446d369b649430b65a756ba278ff97ec81bb6f55b2e73569.
  3. Schema of Comprehension: Given a set c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b5f9c4ab08cac7457e9111a30e4664920607ea2c115a1433d7be98e97e64244ca and a formula c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b670671cd97404156226e507973f2ab8330d3022ca96e0c93bdbdb320c41adcaf, the set c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b59e19706d51d39f66711c2653cd7eb1291c94d9b55eb14bda74ce4dc636d015a exists.
  4. Pairing: if c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b35135aaa6cc23891b40cb3f378c53a17a1127210ce60e125ccf03efcfdaec458 and c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b624b60c58c9d8bfb6ff1886c2fd605d2adeb6ea4da576068201b6c6958ce93f4 exist, then there is a set containing both c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178beb1e33e8a81b697b75855af6bfcdbcbf7cbbde9f94962ceaec1ed8af21f5a50f and c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178be29c9c180c6279b0b02abd6a1801c7c04082cf486ec027aa13515e4f3884bb6b.
  5. Union: the union over the elements of a set exists. That is, if c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178bc6f3ac57944a531490cd39902d0f777715fd005efac9a30622d5f5205e7f6894 then the set c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b86e50149658661312a9e0b35558d84f6c6d3da797f552a9657fe0558ca40cdef exists.
  6. Replacement: informally if c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b9f14025af0065b30e47e23ebb3b491d39ae8ed17d33739e5ff3827ffb3634953 is a function definable by a formula c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b76a50887d8f1c2e9301755428990ad81479ee21c25b43215cf524541e0503269, c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b7a61b53701befdae0eeeffaecc73f14e20b537bb0f8b91ad7c2936dc63562b25 is a set, and c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178baea92132c4cbeb263e6ac2bf6c183b5d81737f179f21efdc5863739672f0f470 exists for all c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b0b918943df0962bc7a1824c0555a389347b4febdc7cf9d1254406d80ce44e3f9, then the set c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178bd59eced1ded07f84c145592f65bdf854358e009c5cd705f5215bf18697fed103 exists.
  7. Infinity: there is a set containing c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b3d914f9348c9cc0ff8a79716700b9fcd4d2f3e711608004eb8f138bcba7f14d9 and closed under the successor operation c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b73475cb40a568e8da8a045ced110137e159f890ac4da883b6b17dc651b3a8049.
  8. Power set: if c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b44cb730c420480a0477b505ae68af508fb90f96cf0ec54c6ad16949dd427f13a is a set, then so is the set of all subsets of c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b71ee45a3c0db9a9865f7313dd3372cf60dca6479d46261f3542eb9346e4a04d6, which we call the power set of c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b811786ad1ae74adfdd20dd0372abaaebc6246e343aebd01da0bfc4c02bf0106c and denote it by c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b25fc0e7096fc653718202dc30b0c580b8ab87eac11a700cba03a7c021bc35b0c.
  9. Choice (AC): the cartesian product of a collection of nonempty sets is nonempty.
  10. Or equivalent to AC: for every set c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b31489056e0916d59fe3add79e63f095af3ffb81604691f21cad442a85c7be617 there is a binary relation on c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b98010bd9270f9b100b6214a21754fd33bdc8d41b2bc9f9dd16ff54d3c34ffd71 which makes c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b0e17daca5f3e175f448bacace3bc0da47d0655a74c8dd0dc497a3afbdad95f1f well-ordered.

Relations and functions

Given sets c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b1a6562590ef19d1045d06c4055742d38288e9e6dcd71ccde5cee80f1d5a774eb and c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b031b4af5197ec30a926f48cf40e11a7dbc470048a21e4003b7a3c07c5dab1baa, the cartesian product of c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b41cfc0d1f2d127b04555b7246d84019b4d27710a3f3aff6e7764375b1e06e05d and c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b2858dcd1057d3eae7f7d5f782167e24b61153c01551450a628cee722509f6529, denoted c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b2fca346db656187102ce806ac732e06a62df0dbb2829e511a770556d398e1a6e is the set of all pairs c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b02d20bbd7e394ad5999a4cebabac9619732c343a4cac99470c03e23ba2bdc2bc such that c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b7688b6ef52555962d008fff894223582c484517cea7da49ee67800adc7fc8866 and c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178bc837649cce43f2729138e72cc315207057ac82599a59be72765a477f22d14a54. That is: c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b6208ef0f7750c111548cf90b6ea1d0d0a66f6bff40dbef07cb45ec436263c7d6. Similarly we define the cartesian product of many sets c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b3e1e967e9b793e908f8eae83c74dba9bcccce6a5535b4b462bd9994537bfe15c. The cartesian product of an infinite collection of sets c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b39fa9ec190eee7b6f4dff1100d6343e10918d044c75eac8f9e9a2596173f80c9 where c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178bd029fa3a95e174a19934857f535eb9427d967218a36ea014b70ad704bc6c8d1c is an infinite index set, is more fiddly to define (essentially elements of the cartesian product of an infinite collection of sets are functions c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b81b8a03f97e8787c53fe1a86bda042b6f0de9b0ec9c09357e107c99ba4d6948a such that for all c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178bda4ea2a5506f2693eae190d9360a1f31793c98a1adade51d93533a6f520ace1c we have c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178ba68b412c4282555f15546cf6e1fc42893b7e07f271557ceb021821098dd66c1b).

A c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b108c995b953c8a35561103e2014cf828eb654a99e310f87fab94c2f4b7d2a04f-ary relation on a set c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b3ada92f28b4ceda38562ebf047c6ff05400d4c572352a1142eedfef67d21e662 is a subset of c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b49d180ecf56132819571bf39d9b7b342522a2ac6d23c1418d3338251bfe469c8 where the cartesian product is of c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178ba21855da08cb102d1d217c53dc5824a3a795c1c1a44e971bf01ab9da3a2acbbf copies of c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178bc75cb66ae28d8ebc6eded002c28a8ba0d06d3a78c6b5cbf9b2ade051f0775ac4.

A function c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178bff5a1ae012afa5d4c889c50ad427aaf545d31a4fac04ffc1c4d03d403ba4250a is a relation c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b7f2253d7e228b22a08bda1f09c516f6fead81df6536eb02fa991a34bb38d9be8 on c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b8722616204217eddb39e7df969e0698aed8e599ba62ed2de1ce49b03ade0fede, such that c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b96061e92f58e4bdcdee73df36183fe3ac64747c81c26f6c83aada8d2aabb1864 associates an element of c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178beb624dbe56eb6620ae62080c10a273cab73ae8eca98ab17b731446a31c79393a to at most one element of c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178bf369cb89fc627e668987007d121ed1eacdc01db9e28f8bb26f358b7d8c4f08ac. That is, if c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178bf74efabef12ea619e30b79bddef89cffa9dda494761681ca862cff2871a85980 and c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178ba88a7902cb4ef697ba0b6759c50e8c10297ff58f942243de19b984841bfe1f73 then necessarily c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b349c41201b62db851192665c504b350ff98c6b45fb62a8a2161f78b6534d8de9. When c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b98a3ab7c340e8a033e7b37b6ef9428751581760af67bbab2b9e05d4964a8874a defines a function, we use the notation c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b48449a14a4ff7d79bb7a1b6f3d488eba397c36ef25634c111b49baf362511afc.

Properties of functions

A function is injective if different inputs give different outputs. That is, if c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b5316ca1c5ddca8e6ceccfce58f3b8540e540ee22f6180fb89492904051b3d531 and c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178ba46e37632fa6ca51a13fe39a567b3c23b28c2f47d8af6be9bd63e030e214ba38 then c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178bbbb965ab0c80d6538cf2184babad2a564a010376712012bd07b0af92dcd3097d.

A function is surjective if its codomain is equal to its range. That is, if c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b44c8031cb036a7350d8b9b8603af662a4b9cdbd2f96e8d5de5af435c9c35da69 then for all c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178bb4944c6ff08dc6f43da2e9c824669b7d927dd1fa976fadc7b456881f51bf5ccc there is c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b434c9b5ae514646bbd91b50032ca579efec8f22bf0b4aac12e65997c418e0dd6 with c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178bbdd2d3af3a5a1213497d4f1f7bfcda898274fe9cb5401bbc0190885664708fc2.

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

Ordinals

We can construct the natural numbers by identifying c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b8b940be7fb78aaa6b6567dd7a3987996947460df1c668e698eb92ca77e425349 with the empty set c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178bcd70bea023f752a0564abb6ed08d42c1440f2e33e29914e55e0be1595e24f45a, and the proceeding to define the successor c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b69f59c273b6e669ac32a6dd5e1b2cb63333d8b004f9696447aee2d422ce63763. This way we get an ordinal for each natural number c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b1da51b8d8ff98f6a48f80ae79fe3ca6c26e1abb7b7d125259255d6d2b875ea08. These are the finite ordinals. We can take the union of all finite ordinals to get the first infinite ordinal, which we denote c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b8241649609f88ccd2a0a5b233a07a538ec313ff6adf695aa44a969dbca39f67d. We call these limit ordinals. Once we have c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b6e4001871c0cf27c7634ef1dc478408f642410fd3a444e2a88e301f5c4a35a4d, we can construct c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178be3d6c4d4599e00882384ca981ee287ed961fa5f3828e2adb5e9ea890ab0d0525), which we denote c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178bad48ff99415b2f007dc35b7eb553fd1eb35ebfa2f2f308acd9488eeb86f71fa8, and then c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b7b1a278f5abe8e9da907fc9c29dfd432d60dc76e17b0fabab659d2a508bc65c4 and so on. Then we can take the union of all the c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178bd6d824abba4afde81129c71dea75b8100e96338da5f416d2f69088f1960cb091 to get c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b29db0c6782dbd5000559ef4d9e953e300e2b479eed26d887ef3f92b921c06a67, which we denote c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b8c1f1046219ddd216a023f792356ddf127fce372a72ec9b4cdac989ee5b0b455. 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 c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178bad57366865126e55649ecb23ae1d48887544976efea46a48eb5d85a6eeb4d306 and an injection c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b16dc368a89b428b2485484313ba67a3912ca03f2b2b42429174a4f8b3dc84e44 then there exists a bijection c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b37834f2f25762f23e1f74a531cbe445db73d6765ebe60878a7dfbecd7d4af6e1.

We denote the cardinality of a set c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b454f63ac30c8322997ef025edff6abd23e0dbe7b8a3d5126a894e4a168c1b59b by c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b5ef6fdf32513aa7cd11f72beccf132b9224d33f271471fff402742887a171edf, and we identify c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b1253e9373e781b7500266caa55150e08e210bc8cd8cc70d89985e3600155e860 with the smallest ordinal c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b482d9673cfee5de391f97fde4d1c84f9f8d6f2cf0784fcffb958b4032de7236c such that c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b3346f2bbf6c34bd2dbe28bd1bb657d0e9c37392a1d5ec9929e6a5df4763ddc2d is equinumerous with c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b9537f32ec7599e1ae953af6c9f929fe747ff9dadf79a9beff1f304c550173011. The cardinal of the set of all finite ordinals is denoted c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b0fd42b3f73c448b34940b339f87d07adf116b05c0227aad72e8f0ee90533e699, (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 c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b9bdb2af6799204a299c603994b8e400e4b1fd625efdb74066cc869fee42c9df3. Equivalently, a set X is countable if it is the image of some function from c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178bf6e0a1e2ac41945a9aa7ff8a8aaa0cebc12a3bcc981a929ad5cf810a090e11ae. That is, if c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178bb1556dea32e9d0cdbfed038fd7787275775ea40939c146a64e205bcb349ad02f.

The power set c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b6c658ee83fb7e812482494f3e416a876f63f418a0b8a1f5e76d47ee4177035cb of a set c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b9f1f9dce319c4700ef28ec8c53bd3cc8e6abe64c68385479ab89215806a5bdd6 is always of a larger cardinality than c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b28dae7c8bde2f3ca608f86d0e16a214dee74c74bee011cdfdd46bc04b655bc14, as can be proved by Cantor's Diagonal Argument. The real numbers c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178be5b861a6d8a966dfca7e7341cd3eb6be9901688d547a72ebed0b1f5e14f3d08d are at least as large as the power set c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b2ac878b0e2180616993b4b6aa71e61166fdc86c28d47e359d0ee537eb11d46d3 of the natural numbers, so are strictly larger (in cardinality terms), than c5f8daf3e51dcfd9f15f57a872d5a1926a990b6fd3e6233e7ffba913f96d178b85daaf6f7055cd5736287faed9603d712920092c4f8fd0097ec3b650bf27530e.