### Tarski Grothendieck Set Theory

by
Andrzej Trybulec

Copyright (c) 1989 Association of Mizar Users

MML identifier: TARSKI
[ MML identifier index ]

```environ

vocabulary TARSKI;

begin

reserve x,y,z,u,N,M,X,Y,Z for set;

:: theorem  TARSKI:1       ---- everything is a set
canceled; :: as obvious

::      ---- extensionality

theorem
(for x holds x in X iff x in Y) implies X = Y;

:: singletons & unordered pairs

definition let y; func { y } means
x in it iff x = y;
correctness;
let z; func { y, z } means
x in it iff x = y or x = z;
correctness;
commutativity;
end;

canceled 2;

definition let X,Y;
pred X c= Y means
x in X implies x in Y;
reflexivity;
end;

definition let X;
func union X means
x in it iff ex Y st x in Y & Y in X;
correctness;
end;

canceled 2;

::        ---- regularity

theorem
x in X implies ex Y st Y in X & not ex x st x in X & x in Y;

scheme Fraenkel { A()-> set, P[set, set] }:
ex X st for x holds x in X iff ex y st y in A() & P[y,x]
provided for x,y,z st P[x,y] & P[x,z] holds y = z;

:: ordered pairs

definition let x,y;
func [x,y] equals :Def5:
{ { x,y }, { x } };
coherence;
end;

canceled;

definition let X,Y;
pred X,Y are_equipotent means
ex Z st
(for x st x in X ex y st y in Y & [x,y] in Z) &
(for y st y in Y ex x st x in X & [x,y] in Z) &
for x,y,z,u st [x,y] in Z & [z,u] in Z holds x = z iff y = u;
end;

:: Alfred Tarski
:: Ueber unerreichbare Kardinalzahlen,
:: Fundamenta Mathematicae, vol.30 (1938), pp.68-69

:: Axiom A. (Axiom der unerreichbaren Mengen). Zu jeder Menge N gibt es
:: eine Menge M mit folgenden Eigenschaften :
:: A1. N in M;
:: A2. ist X in M und Y c= X, so ist Y in M;
:: A3. ist X in M und ist Z die Menge, die alle Mengen Y c= X und keine
::     andere Dinge als Element enthaelt, so, ist z in M;
:: A4. ist X c= M und sind dabei die Menge X und M nicht gleichmaechtig,
::     so ist X in M.

:: also

:: Alfred Tarski
:: On Well-ordered Subsets of any Set,
:: Fundamenta Mathematicae, vol.32 (1939), pp.176-183

:: A. For every set N there exists a system M of sets which satisfies
::    the following conditions :
:: (i)    N in M
:: (ii)   if X in M and Y c= X, then Y in M
:: (iii)  if X in M and Z is the system of all subsets of X, then Z in M
:: (iv)   if X c= M and X and M do not have the same potency, then X in M.

theorem
ex M st N in M &
(for X,Y holds X in M & Y c= X implies Y in M) &
(for X st X in M ex Z st Z in M & for Y st Y c= X holds Y in Z) &
(for X holds X c= M implies X,M are_equipotent or X in M);
```