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Axiom ax-groth 6712
Description: The Tarksi-Grothendieck Axiom. For every set there is an inaccessible cardinal such that is not in . The addition of this axiom to ZFC set theory provides a framework for category theory, thus for all practical purposes giving us a complete foundation for "all of mathematics." This version of the axiom is used by the Mizar project (http://www.mizar.org/JFM/Axiomatics/tarski.html). Unlike the ZFC axioms, this axiom is very long when expressed in terms of primitive symbols - see grothprim 6723. An open problem is finding a shorter equivalent.
Assertion
Ref Expression
ax-groth
Distinct variable group:   ,,,,

Detailed syntax breakdown of Axiom ax-groth
StepHypRef Expression
1 vx . . . 4
2 vy . . . 4
31, 2wel 1428 . . 3
4 vw . . . . . . . . 9
54cv 1424 . . . . . . . 8
6 vz . . . . . . . . 9
76cv 1424 . . . . . . . 8
85, 7wss 2634 . . . . . . 7
94, 2wel 1428 . . . . . . 7
108, 9wi 4 . . . . . 6
1110, 4wal 1341 . . . . 5
12 vv . . . . . . . . . 10
1312cv 1424 . . . . . . . . 9
1413, 7wss 2634 . . . . . . . 8
1512, 4wel 1428 . . . . . . . 8
1614, 15wi 4 . . . . . . 7
1716, 12wal 1341 . . . . . 6
182cv 1424 . . . . . 6
1917, 4, 18wrex 2139 . . . . 5
2011, 19wa 360 . . . 4
2120, 6, 18wral 2138 . . 3
227, 18wss 2634 . . . . 5
23 cen 5678 . . . . . . 7
247, 18, 23wbr 3358 . . . . . 6
256, 2wel 1428 . . . . . 6
2624, 25wo 359 . . . . 5
2722, 26wi 4 . . . 4
2827, 6wal 1341 . . 3
293, 21, 28w3a 912 . 2
3029, 2wex 1346 1
Colors of variables: wff set class
This axiom is referenced by:  axgroth5 6713  axgroth2 6714  grothpw 6715
Copyright terms: Public domain