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 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01e629fa6598d732768f7c726b4b621285f9c3b85303900aa912017db7617d8bdb disjoint from 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01b17ef6d19c7a5b1ee83b907c595526dcb1eb06db8227d650d5dda0a9f4ce8cd9.
  3. Schema of Comprehension: Given a set 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c014523540f1504cd17100c4835e85b7eefd49911580f8efff0599a8f283be6b9e3 and a formula 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c014ec9599fc203d176a301536c2e091a19bc852759b255bd6818810a42c5fed14a, the set 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c019400f1b21cb527d7fa3d3eabba93557a18ebe7a2ca4e471cfe5e4c5b4ca7f767 exists.
  4. Pairing: if 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01f5ca38f748a1d6eaf726b8a42fb575c3c71f1864a8143301782de13da2d9202b and 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c016f4b6612125fb3a0daecd2799dfd6c9c299424fd920f9b308110a2c1fbd8f443 exist, then there is a set containing both 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01785f3ec7eb32f30b90cd0fcf3657d388b5ff4297f2f9716ff66e9b69c05ddd09 and 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01535fa30d7e25dd8a49f1536779734ec8286108d115da5045d77f3b4185d8f790.
  5. Union: the union over the elements of a set exists. That is, if 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01c2356069e9d1e79ca924378153cfbbfb4d4416b1f99d41a2940bfdb66c5319db then the set 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01b7a56873cd771f2c446d369b649430b65a756ba278ff97ec81bb6f55b2e73569 exists.
  6. Replacement: informally if 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c015f9c4ab08cac7457e9111a30e4664920607ea2c115a1433d7be98e97e64244ca is a function definable by a formula 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01670671cd97404156226e507973f2ab8330d3022ca96e0c93bdbdb320c41adcaf, 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c0159e19706d51d39f66711c2653cd7eb1291c94d9b55eb14bda74ce4dc636d015a is a set, and 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c0135135aaa6cc23891b40cb3f378c53a17a1127210ce60e125ccf03efcfdaec458 exists for all 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01624b60c58c9d8bfb6ff1886c2fd605d2adeb6ea4da576068201b6c6958ce93f4, then the set 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01eb1e33e8a81b697b75855af6bfcdbcbf7cbbde9f94962ceaec1ed8af21f5a50f exists.
  7. Infinity: there is a set containing 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01e29c9c180c6279b0b02abd6a1801c7c04082cf486ec027aa13515e4f3884bb6b and closed under the successor operation 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01c6f3ac57944a531490cd39902d0f777715fd005efac9a30622d5f5205e7f6894.
  8. Power set: if 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c0186e50149658661312a9e0b35558d84f6c6d3da797f552a9657fe0558ca40cdef is a set, then so is the set of all subsets of 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c019f14025af0065b30e47e23ebb3b491d39ae8ed17d33739e5ff3827ffb3634953, which we call the power set of 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c0176a50887d8f1c2e9301755428990ad81479ee21c25b43215cf524541e0503269 and denote it by 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c017a61b53701befdae0eeeffaecc73f14e20b537bb0f8b91ad7c2936dc63562b25.
  9. Choice (AC): the cartesian product of a collection of nonempty sets is nonempty.
  10. Or equivalent to AC: for every set 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01aea92132c4cbeb263e6ac2bf6c183b5d81737f179f21efdc5863739672f0f470 there is a binary relation on 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c010b918943df0962bc7a1824c0555a389347b4febdc7cf9d1254406d80ce44e3f9 which makes 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01d59eced1ded07f84c145592f65bdf854358e009c5cd705f5215bf18697fed103 well-ordered.

Relations and functions

Given sets 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c013d914f9348c9cc0ff8a79716700b9fcd4d2f3e711608004eb8f138bcba7f14d9 and 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c0173475cb40a568e8da8a045ced110137e159f890ac4da883b6b17dc651b3a8049, the cartesian product of 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c0144cb730c420480a0477b505ae68af508fb90f96cf0ec54c6ad16949dd427f13a and 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c0171ee45a3c0db9a9865f7313dd3372cf60dca6479d46261f3542eb9346e4a04d6, denoted 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01811786ad1ae74adfdd20dd0372abaaebc6246e343aebd01da0bfc4c02bf0106c is the set of all pairs 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c0125fc0e7096fc653718202dc30b0c580b8ab87eac11a700cba03a7c021bc35b0c such that 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c0131489056e0916d59fe3add79e63f095af3ffb81604691f21cad442a85c7be617 and 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c0198010bd9270f9b100b6214a21754fd33bdc8d41b2bc9f9dd16ff54d3c34ffd71. That is: 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c010e17daca5f3e175f448bacace3bc0da47d0655a74c8dd0dc497a3afbdad95f1f. Similarly we define the cartesian product of many sets 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c011a6562590ef19d1045d06c4055742d38288e9e6dcd71ccde5cee80f1d5a774eb. The cartesian product of an infinite collection of sets 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01031b4af5197ec30a926f48cf40e11a7dbc470048a21e4003b7a3c07c5dab1baa where 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c0141cfc0d1f2d127b04555b7246d84019b4d27710a3f3aff6e7764375b1e06e05d is an infinite index set, is more fiddly to define (essentially elements of the cartesian product of an infinite collection of sets are functions 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c012858dcd1057d3eae7f7d5f782167e24b61153c01551450a628cee722509f6529 such that for all 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c012fca346db656187102ce806ac732e06a62df0dbb2829e511a770556d398e1a6e we have 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c0102d20bbd7e394ad5999a4cebabac9619732c343a4cac99470c03e23ba2bdc2bc).

A 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c017688b6ef52555962d008fff894223582c484517cea7da49ee67800adc7fc8866-ary relation on a set 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01c837649cce43f2729138e72cc315207057ac82599a59be72765a477f22d14a54 is a subset of 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c016208ef0f7750c111548cf90b6ea1d0d0a66f6bff40dbef07cb45ec436263c7d6 where the cartesian product is of 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c013e1e967e9b793e908f8eae83c74dba9bcccce6a5535b4b462bd9994537bfe15c copies of 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c0139fa9ec190eee7b6f4dff1100d6343e10918d044c75eac8f9e9a2596173f80c9.

A function 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01d029fa3a95e174a19934857f535eb9427d967218a36ea014b70ad704bc6c8d1c is a relation 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c0181b8a03f97e8787c53fe1a86bda042b6f0de9b0ec9c09357e107c99ba4d6948a on 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01da4ea2a5506f2693eae190d9360a1f31793c98a1adade51d93533a6f520ace1c, such that 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01a68b412c4282555f15546cf6e1fc42893b7e07f271557ceb021821098dd66c1b associates an element of 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01108c995b953c8a35561103e2014cf828eb654a99e310f87fab94c2f4b7d2a04f to at most one element of 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c013ada92f28b4ceda38562ebf047c6ff05400d4c572352a1142eedfef67d21e662. That is, if 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c0149d180ecf56132819571bf39d9b7b342522a2ac6d23c1418d3338251bfe469c8 and 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01a21855da08cb102d1d217c53dc5824a3a795c1c1a44e971bf01ab9da3a2acbbf then necessarily 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01c75cb66ae28d8ebc6eded002c28a8ba0d06d3a78c6b5cbf9b2ade051f0775ac4. When 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01ff5a1ae012afa5d4c889c50ad427aaf545d31a4fac04ffc1c4d03d403ba4250a defines a function, we use the notation 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c017f2253d7e228b22a08bda1f09c516f6fead81df6536eb02fa991a34bb38d9be8.

Properties of functions

A function is injective if different inputs give different outputs. That is, if 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c018722616204217eddb39e7df969e0698aed8e599ba62ed2de1ce49b03ade0fede and 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c0196061e92f58e4bdcdee73df36183fe3ac64747c81c26f6c83aada8d2aabb1864 then 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01eb624dbe56eb6620ae62080c10a273cab73ae8eca98ab17b731446a31c79393a.

A function is surjective if its codomain is equal to its range. That is, if 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01f369cb89fc627e668987007d121ed1eacdc01db9e28f8bb26f358b7d8c4f08ac then for all 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01f74efabef12ea619e30b79bddef89cffa9dda494761681ca862cff2871a85980 there is 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01a88a7902cb4ef697ba0b6759c50e8c10297ff58f942243de19b984841bfe1f73 with 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01349c41201b62db851192665c504b350ff98c6b45fb62a8a2161f78b6534d8de9.

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

Ordinals

We can construct the natural numbers by identifying 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c0198a3ab7c340e8a033e7b37b6ef9428751581760af67bbab2b9e05d4964a8874a with the empty set 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c0148449a14a4ff7d79bb7a1b6f3d488eba397c36ef25634c111b49baf362511afc, and the proceeding to define the successor 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c015316ca1c5ddca8e6ceccfce58f3b8540e540ee22f6180fb89492904051b3d531. This way we get an ordinal for each natural number 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01a46e37632fa6ca51a13fe39a567b3c23b28c2f47d8af6be9bd63e030e214ba38. These are the finite ordinals. We can take the union of all finite ordinals to get the first infinite ordinal, which we denote 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01bbb965ab0c80d6538cf2184babad2a564a010376712012bd07b0af92dcd3097d.

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 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c0144c8031cb036a7350d8b9b8603af662a4b9cdbd2f96e8d5de5af435c9c35da69 and an injection 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01b4944c6ff08dc6f43da2e9c824669b7d927dd1fa976fadc7b456881f51bf5ccc then there exists a bijection 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01434c9b5ae514646bbd91b50032ca579efec8f22bf0b4aac12e65997c418e0dd6.

We denote the cardinality of a set 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01bdd2d3af3a5a1213497d4f1f7bfcda898274fe9cb5401bbc0190885664708fc2 by 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c018b940be7fb78aaa6b6567dd7a3987996947460df1c668e698eb92ca77e425349, and we identify 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c01cd70bea023f752a0564abb6ed08d42c1440f2e33e29914e55e0be1595e24f45a with the smallest ordinal 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c0169f59c273b6e669ac32a6dd5e1b2cb63333d8b004f9696447aee2d422ce63763 such that 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c011da51b8d8ff98f6a48f80ae79fe3ca6c26e1abb7b7d125259255d6d2b875ea08 is equinumerous with 5529d549a95e11ba1f09a439801893e632e428aed0fbca3b4d13878cc11c0c018241649609f88ccd2a0a5b233a07a538ec313ff6adf695aa44a969dbca39f67d.

If we assume the Axiom of Choice, or equivalently, the well-ordering principle,