-
Notifications
You must be signed in to change notification settings - Fork 108
Expand file tree
/
Copy pathmmil.raw.html
More file actions
20357 lines (16994 loc) · 412 KB
/
Copy pathmmil.raw.html
File metadata and controls
20357 lines (16994 loc) · 412 KB
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
"https://www.w3.org/TR/html4/loose.dtd">
<!-- The file mmil.html is generated from mmil.raw.html -
see the regen-from-raw script for details -->
<HTML LANG="EN-US">
<HEAD>
<!-- improve mobile display -->
<META NAME="viewport" CONTENT="width=device-width, initial-scale=1.0">
<META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1">
<TITLE>Home - Intuitionistic Logic Explorer</TITLE>
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<LINK REL="shortcut icon" HREF="favicon.ico" TYPE="image/x-icon">
<STYLE TYPE="text/css">
<!--
/* Math symbol image will be shifted down 4 pixels to align with normal
text for compatibility with various browsers. The old ALIGN=TOP for
math symbol images did not align in all browsers and should be deleted.
All other images must override this shift with STYLE="margin-bottom:0px".
(2-Oct-2015 nm) */
img { margin-bottom: -4px }
-->
</STYLE>
</HEAD>
<!-- <BODY BGCOLOR="#EEFFFA"> -->
<BODY BGCOLOR="#FFFFFF" STYLE="padding: 0px 8px">
<TABLE BORDER=0 CELLSPACING=0 CELLPADDING=0 WIDTH="100%">
<TR>
<TD ALIGN=LEFT VALIGN=TOP><A HREF="../index.html"><IMG SRC="_icon-il.gif"
BORDER=0
ALT="Metamath Home"
TITLE="Metamath Home"
HEIGHT=32 WIDTH=32 ALIGN=TOP STYLE="margin-bottom:0px"></A>
</TD>
<TD ALIGN=CENTER VALIGN=TOP><FONT SIZE="+3"
COLOR="#006633"><B>Intuitionistic Logic Explorer Home Page</B></FONT>
</TD>
<TD NOWRAP ALIGN=RIGHT VALIGN=TOP><FONT SIZE=-2
FACE=sans-serif> <A HREF="wn.html">First ></A><BR><A
HREF="ax1hfs.html">Last ></A></FONT>
</TD>
</TR>
<TR>
<TD COLSPAN=3 ALIGN=LEFT VALIGN=TOP><FONT SIZE=-2
FACE=sans-serif>
<A HREF="../mm.html">Mirrors</A> >
<A HREF="../index.html">Home</A> >
ILE Home >
<A HREF="mmtheorems.html">Th. List</A> >
<A HREF="mmrecent.html">Recent</A>
</FONT>
</TD>
</TR>
</TABLE>
<HR NOSHADE SIZE=1>
<CENTER>
<B><FONT COLOR="#006633">Created by Mario Carneiro</FONT></B>
</CENTER>
<HR NOSHADE SIZE=1>
<B><FONT COLOR="#006633">Intuitionistic Logic Proof Explorer</FONT></B>
<!--
<P><I>Intuitionistic Logic</I> (<A
HREF="https://en.wikipedia.org/wiki/Intuitionistic_logic">Wikipedia</A>
[accessed 19-Jul-2015], <A
HREF="http://plato.stanford.edu/entries/logic-intuitionistic/"> Stanford
Encyclopedia of Philosophy</A> [accessed 19-Jul-2015]) is a logic weaker
than classical logic that can be thought of as a weakening of classical
logic such that the law of excluded middle, (<FONT
COLOR="#0000FF"><I>φ</I></FONT> <FONT FACE=sans-serif> ∨</FONT>
¬ <FONT COLOR="#0000FF"><I>φ</I></FONT>), doesn't always hold.
Specifically, it holds if we have a proof for <FONT
COLOR="#0000FF"><I>φ</I></FONT> or we have a proof for ¬ <FONT
COLOR="#0000FF"><I>φ</I></FONT>, but it doesn't necessarily hold if
we don't have a proof of either one. Intuitionistic logic can be
thought of as a constructive logic in which we must build and exhibit
concrete examples of objects before we can accept their existence. A
proof by contradiction, where denial of an assertion to be proved leads
to asserting a falsehood, will generally not be valid in intuitionistic
logic.
-->
<!-- the above was replaced by David A. Wheeler: -->
<P><I>Intuitionistic Logic</I> (<A
HREF="https://en.wikipedia.org/wiki/Intuitionistic_logic">Wikipedia</A>
[accessed 19-Jul-2015], <A
HREF="http://plato.stanford.edu/entries/logic-intuitionistic/"> Stanford
Encyclopedia of Philosophy</A> [accessed 19-Jul-2015]) can be
thought of as a constructive logic in which we must build and exhibit
concrete examples of objects before we can accept their existence.
Unproved statements in intuitionistic logic are not given an intermediate truth value,
instead, they remain of unknown truth value until they are either proved or disproved.
Intuitionist logic can also be thought of as a weakening of classical
logic such that the law of excluded middle (LEM), (<FONT
COLOR="#0000FF"><I>φ</I></FONT> <FONT FACE=sans-serif> ∨</FONT>
¬ <FONT COLOR="#0000FF"><I>φ</I></FONT>), doesn't always hold.
Specifically, it holds if we have a proof for <FONT
COLOR="#0000FF"><I>φ</I></FONT> or we have a proof for ¬ <FONT
COLOR="#0000FF"><I>φ</I></FONT>, but it doesn't necessarily hold if
we don't have a proof of either one.
There is also no rule for double negation elimination.
Brouwer observed in 1908 that LEM was abstracted from finite situations,
then extended without justification to statements about infinite collections.
<HR NOSHADE SIZE=1>
<TABLE WIDTH="100%"><TR>
<TD VALIGN=top>
<B><FONT COLOR="#006633">Contents of this page</FONT></B>
<MENU>
<LI> <A HREF="#overview">Overview of intuitionistic logic</A></LI>
<LI> <A HREF="#overview2">Overview of this work</A></LI>
<LI> <A HREF="#axioms">The axioms</A></LI>
<LI> <A HREF="#theorems">Some theorems</A></LI>
<li><a href="#flavors">Different flavors of constructive mathematics</a></li>
<li><a href="#existence">A note on existence</a></li>
<li><a href="#set-size">The size of sets</a></li>
<LI> <A HREF="#intuitionize">How to intuitionize classical proofs</A></LI>
<LI> <A HREF="#setmm">Metamath Proof Explorer cross reference</A></LI>
<LI> <A HREF="#bib">Bibliography</A></LI>
</MENU></TD>
<TD VALIGN=top>
<B><FONT COLOR="#006633">Related pages</FONT></B>
<MENU>
<LI> <A HREF="mmtheorems.html">Table of Contents and Theorem List</A></LI>
<LI>
<A HREF="mmrecent.html">Most Recent Proofs
(this mirror)</A>
(<A HREF="https://us.metamath.org/ileuni/mmrecent.html">latest</A>)
</LI>
<LI> <A HREF="mmbiblio.html">Bibliographic Cross-Reference</A></LI>
<LI> <A HREF="mmdefinitions.html">Definition List</A></LI>
<LI> <A HREF="mmascii.html">ASCII Equivalents for Text-Only Browsers</A></LI>
<LI>
<A HREF="../metamath/iset.mm">Metamath database iset.mm (ASCII file)</A>
</LI>
</MENU>
<B><FONT COLOR="#006633">External links</FONT></B>
<MENU>
<LI>
<A HREF=" https://github.com/metamath/set.mm">GitHub repository</A>
[accessed 06-Jan-2018]
</LI>
</MENU>
</TD>
</TR></TABLE>
<HR NOSHADE SIZE=1><A NAME="overview"></A><B><FONT COLOR="#006633">
Overview of intuitionistic logic</FONT></B>
<P>(Placeholder for future use)
<HR NOSHADE SIZE=1><A NAME="overview2"></A><B><FONT COLOR="#006633">
Overview of this work</FONT></B>
<P>(By Gérard Lang, 7-May-2018)
<P>Mario Carneiro's work (Metamath database) "iset.mm" provides in Metamath a
development of "set.mm" whose eventual
aim is to show how many of the theorems of set theory and
mathematics that can be derived from classical first-order logic can
also be derived from a weaker system called "intuitionistic logic." To
achieve this task, iset.mm adds (or substitutes) intuitionistic
axioms for a number of the classical logical axioms of set.mm.
<P>Among these new axioms, the first six
( ~ ax-ia1 ,
~ ax-ia2 ,
~ ax-ia3 ,
~ ax-io ,
~ ax-in1 ,
and
~ ax-in2 ), when added to
~ ax-1 ,
~ ax-2 ,
and
~ ax-mp ,
allow for the development of intuitionistic propositional logic.
We omit the classical axiom
<SPAN CLASS=math>((¬
<SPAN CLASS=wff STYLE="color:blue">𝜑</SPAN> → ¬ <SPAN
CLASS=wff STYLE="color:blue">𝜓</SPAN>) → (<SPAN CLASS=wff
STYLE="color:blue">𝜓</SPAN> → <SPAN CLASS=wff
STYLE="color:blue">𝜑</SPAN>))</SPAN> (which is ax-3 in
set.mm). Each of our new axioms is a theorem of classical
propositional logic, but ax-3 cannot be derived from them. Similarly,
other basic classical theorems, like the third middle excluded or the
equivalence of a proposition with its double negation, cannot be derived
in intuitionistic propositional calculus. Glivenko showed that a
proposition <FONT COLOR="#0000FF"><I>φ</I></FONT>
is a theorem of classical propositional calculus if and only
if ¬¬<FONT COLOR="#0000FF"><I>φ</I></FONT>
is a theorem of intuitionistic propositional calculus.
<P>The next four new axioms
( ~ ax-ial ,
~ ax-i5r ,
~ ax-ie1 ,
and
~ ax-ie2 )
together with the set.mm axioms
~ ax-4 ,
~ ax-5 ,
~ ax-7 ,
and
~ ax-gen
do not mention equality or distinct variables.
<P>The ~ ax-i9 axiom is just a slight variation of the classical ~ ax-9 .
The classical axiom ~ ax12 is strengthened into first ~ ax-i12 and then
~ ax-bndl (two results which would be fairly readily equivalent to ~ ax12
classically but which do not follow from ~ ax12 , at least not in an obvious
way, in intuitionistic logic).
The substitution of ~ ax-i9 , ~ ax-i12 , and ~ ax-bndl for ~ ax-9 and ~ ax12
and the inclusion of
~ ax-8 ,
~ ax-10 ,
~ ax-11 ,
~ ax-13 ,
~ ax-14 ,
and
~ ax-17
allow for the development of the intuitionistic predicate calculus.
<P>Each of the new axioms is a theorem of classical first-order
logic with equality. But some axioms of classical first-order logic
with equality, like ax-3, cannot be derived in the intuitionistic
predicate calculus.</P>
<P>One of the major interests of the intuitionistic predicate calculus
is that its use can be considered as a realization of the program of the
constructivist philosophical view of mathematics.
<HR NOSHADE SIZE=1><A NAME="axioms"></A><B><FONT COLOR="#006633">
The axioms</FONT></B>
<P>As with the <A HREF="mmset.html#axioms">classical axioms</A>
we have propositional logic and predicate logic.
<P>The axioms of intuitionistic propositional logic consist of some of the axioms from
classical propositional logic, plus additional axioms for the operation of the 'and',
'or' and 'not' connectives.
<CENTER>
<TABLE BORDER CELLSPACING=0 BGCOLOR="#EEFFFA"
SUMMARY="Axioms of intuitionistic propositional calculus">
<CAPTION><B>Axioms of intuitionistic propositional calculus</B></CAPTION>
<TR ALIGN=LEFT><TD> <A HREF="ax-1.html"> Axiom <I>Simp</I></A></TD>
<TD><FONT COLOR="#006633"><B>ax-1</B></FONT></TD>
<TD><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
COLOR="#0000FF"><I>φ</I></FONT> → (<FONT
COLOR="#0000FF"><I>ψ</I></FONT> → <FONT
COLOR="#0000FF"><I>φ</I></FONT>))</TD>
</TR>
<TR ALIGN=LEFT><TD><A HREF="ax-2.html">Axiom <I>Frege</I></A></TD> <TD
NOWRAP><FONT COLOR="#006633"><B>ax-2</B></FONT></TD>
<TD>
<FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>((<FONT
COLOR="#0000FF"><I>φ</I></FONT> → (<FONT
COLOR="#0000FF"><I>ψ</I></FONT> → <FONT
COLOR="#0000FF"><I>χ</I></FONT>)) → ((<FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
COLOR="#0000FF"><I>ψ</I></FONT>) → (<FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
COLOR="#0000FF"><I>χ</I></FONT>)))</TD>
</TR>
<TR ALIGN=LEFT><TD><A
HREF="ax-mp.html">Rule of Modus Ponens</A></TD> <TD NOWRAP><FONT
COLOR="#006633"><B>ax-mp</B></FONT></TD>
<TD>` |- ph ` &
` |- ph -> ps ` ⇒ ` |- ps `</TD>
</TR>
<TR ALIGN=LEFT><TD><A
HREF="ax-ia1.html">Left 'and' elimination</A></TD><TD><FONT
COLOR="#006633"><B>ax-ia1</B></FONT></TD><TD>
` |- ( ( ph /\ ps ) -> ph ) `</TD></TR>
<TR ALIGN=LEFT><TD><A
HREF="ax-ia2.html">Right 'and' elimination</A></TD><TD><FONT
COLOR="#006633"><B>ax-ia2</B></FONT></TD><TD>
<FONT COLOR="#808080" FACE="sans-serif">⊢ </FONT>((<FONT COLOR="#0000FF"><I>φ</I></FONT>
<FONT FACE="sans-serif">∧</FONT> <FONT COLOR="#0000FF"><I>ψ</I></FONT>) →
<FONT COLOR="#0000FF"><I>ψ</I></FONT>)</TD></TR>
<TR ALIGN=LEFT><TD><A
HREF="ax-ia3.html">'And' introduction</A></TD><TD><FONT
COLOR="#006633"><B>ax-ia3</B></FONT></TD><TD>
<FONT COLOR="#808080" FACE="sans-serif">⊢ </FONT>(<FONT COLOR="#0000FF"><I>φ</I></FONT>
→ (<FONT COLOR="#0000FF"><I>ψ</I></FONT> → (<FONT COLOR="#0000FF"><I>φ</I></FONT>
<FONT FACE="sans-serif">∧</FONT> <FONT COLOR="#0000FF"><I>ψ</I></FONT>)))</TD></TR>
<TR ALIGN="LEFT"><TD><A
HREF="ax-io.html">Definition of 'or'</A></TD><TD><FONT COLOR="#006633"><B>ax-io</B></FONT></TD><TD>
<FONT COLOR="#808080" FACE="sans-serif">⊢ </FONT>(((<FONT COLOR="#0000FF"><I>φ</I></FONT>
<FONT FACE="sans-serif"> ∨</FONT> <FONT COLOR="#0000FF"><I>χ</I></FONT>)
→ <FONT COLOR="#0000FF"><I>ψ</I></FONT>)
↔ ((<FONT COLOR="#0000FF"><I>φ</I></FONT> →
<FONT COLOR="#0000FF"><I>ψ</I></FONT>) <FONT FACE="sans-serif">∧</FONT>
(<FONT COLOR="#0000FF"><I>χ</I></FONT> → <FONT COLOR="#0000FF"><I>ψ</I></FONT>)))
<TR ALIGN="LEFT"><TD><A
HREF="ax-in1.html">'Not' introduction</A></TD><TD><FONT COLOR="#006633"><b>ax-in1</B></FONT></TD><TD>
<FONT COLOR="#808080" FACE="sans-serif">⊢
</FONT>((<FONT COLOR="#0000FF"><I>φ</I></FONT> → ¬ <FONT COLOR="#0000FF"><I>φ</I></FONT>)
→ ¬ <FONT COLOR="#0000FF"><I>φ</I></FONT>)</TD></TR>
<TR ALIGN="LEFT"><TD><A
HREF="ax-in2.html">'Not' elimination</A></TD><TD><FONT COLOR="#006633"><B>ax-in2</B></FONT></TD><TD>
<FONT COLOR="#808080" FACE="sans-serif">⊢ </FONT>(¬ <FONT COLOR="#0000FF"><I>φ</I></FONT>
→ (<FONT COLOR="#0000FF"><I>φ</I></FONT> → <FONT COLOR="#0000FF"><I>ψ</I></FONT>))</TD></TR>
</TABLE>
</CENTER>
<P>Unlike in classical propositional logic, 'and' and 'or' are not readily defined in terms of
implication and 'not'. In particular, <FONT COLOR="#0000FF"><I>φ</I></FONT> ∨
<FONT COLOR="#0000FF"><I>ψ</I></FONT> is not equivalent to ¬ <FONT COLOR="#0000FF"><I>φ</I></FONT> →
<FONT COLOR="#0000FF"><I>ψ</I></FONT>,
nor is <FONT COLOR="#0000FF"><I>φ</I></FONT> →
<FONT COLOR="#0000FF"><I>ψ</I></FONT> equivalent to ¬ <FONT COLOR="#0000FF"><I>φ</I></FONT> ∨
<FONT COLOR="#0000FF"><I>ψ</I></FONT>, nor is <FONT COLOR="#0000FF"><I>φ</I></FONT> ∧
<FONT COLOR="#0000FF"><I>ψ</I></FONT> equivalent to ¬ (<FONT COLOR="#0000FF"><I>φ</I></FONT> → ¬
<FONT COLOR="#0000FF"><I>ψ</I></FONT>).</P>
<P>The ax-in1 axiom is a form of proof by contradiction which does hold intuitionistically. That is, if
<FONT COLOR="#0000FF"><I>φ</I></FONT> implies a contradiction (such as its own negation),
then one can conclude ¬ <FONT COLOR="#0000FF"><I>φ</I></FONT>. By contrast, assuming
¬ <FONT COLOR="#0000FF"><I>φ</I></FONT>
and then deriving a contradiction only serves to prove ¬ ¬ <FONT COLOR="#0000FF"><I>φ</I></FONT>,
which in intuitionistic logic is not the same as <FONT COLOR="#0000FF"><I>φ</I></FONT>.</P>
<P>The biconditional can be defined as the conjunction of two implications, as in
<A HREF="dfbi2.html">dfbi2</a> and <A HREF="df-bi.html">df-bi</a>.</P>
<P><A NAME="pcaxioms"></A><B><FONT COLOR="#006633">Predicate
logic</FONT></B> adds set variables (individual variables) and the ability to quantify
them with ∀ (for-all) and ∃ (there-exists). Unlike in classical logic, ∃
cannot be defined in terms of ∀. As in classical logic, we also add = for equality
(which is key to how we handle substitution in metamath) and ∈ (which for current
purposes can just be thought of as an arbitrary predicate, but which will later come to
mean set membership).</P>
<p>Our axioms are based on the classical set.mm predicate logic axioms, but adapted for
intuitionistic logic, chiefly by adding additional axioms for ∃ and also changing
some aspects of how we handle negations.</p>
<CENTER>
<TABLE BORDER CELLSPACING=0 BGCOLOR="#EEFFFA"
SUMMARY="Axioms of intuitionistic predicate logic">
<CAPTION><B>Axioms of intuitionistic predicate logic</B></CAPTION>
<TR ALIGN=LEFT><TD><A HREF="ax-4.html">Axiom of Specialization</A></TD>
<TD><FONT COLOR="#006633"><B>ax-4</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
COLOR="#0000FF"><I>φ</I></FONT>)</SPAN></TD></TR>
<TR ALIGN=LEFT><TD> <A HREF="ax-5.html">Axiom of Quantified Implication</A></TD>
<TD><FONT COLOR="#006633"><B>ax-5</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I>(<FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
COLOR="#0000FF"><I>ψ</I></FONT>) → (<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>ψ</I></FONT>))</SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-i5r.html">The converse of ax-5o</A></TD>
<TD><FONT COLOR="#006633"><B>ax-i5r</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>((<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>ψ</I></FONT>) → <FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
COLOR="#0000FF"><I>ψ</I></FONT>))</SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-7.html">Axiom of Quantifier Commutation</A></TD>
<TD><FONT COLOR="#006633"><B>ax-7</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">y</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">y</FONT></I><FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT>)</SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-gen.html">Rule of Generalization</A></TD>
<TD><FONT COLOR="#006633"><B>ax-gen</B></FONT></TD><TD>
<FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT><FONT
COLOR="#0000FF"><I>φ</I></FONT> => <SPAN >
<FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT><FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT></SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-ial.html"><SPAN ><I><FONT
COLOR="#FF0000">x</FONT></I></SPAN> is bound in <SPAN ><FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT></SPAN></A></TD>
<TD><FONT COLOR="#006633"><B>ax-ial</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT>)</SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-ie1.html"><SPAN ><I><FONT
COLOR="#FF0000">x</FONT></I></SPAN> is bound in <SPAN ><FONT
FACE=sans-serif>∃</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT></SPAN></A></TD>
<TD><FONT COLOR="#006633"><B>ax-ie1</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
FACE=sans-serif>∃</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
FACE=sans-serif>∃</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT>)</SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-ie2.html">Define existential
quantification</A></TD>
<TD><FONT COLOR="#006633"><B>ax-ie2</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I>(<FONT
COLOR="#0000FF"><I>ψ</I></FONT> → <FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>ψ</I></FONT>) → (<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I>(<FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
COLOR="#0000FF"><I>ψ</I></FONT>) ↔ (<FONT
FACE=sans-serif>∃</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
COLOR="#0000FF"><I>ψ</I></FONT>)))</SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-8.html">Axiom of Equality</A></TD>
<TD><FONT COLOR="#006633"><B>ax-8</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<I><FONT
COLOR="#FF0000">x</FONT></I> = <I><FONT COLOR="#FF0000">y</FONT></I> →
(<I><FONT COLOR="#FF0000">x</FONT></I> = <I><FONT COLOR="#FF0000">z</FONT></I>
→ <I><FONT COLOR="#FF0000">y</FONT></I> = <I><FONT
COLOR="#FF0000">z</FONT></I>))</SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-i9.html">Axiom of Existence</A></TD>
<TD><FONT COLOR="#006633"><B>ax-i9</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT><FONT
FACE=sans-serif>∃</FONT><I><FONT COLOR="#FF0000">x</FONT></I> <I><FONT
COLOR="#FF0000">x</FONT></I> = <I><FONT
COLOR="#FF0000">y</FONT></I></SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-10.html">Axiom of Quantifier
Substitution</A></TD>
<TD><FONT COLOR="#006633"><B>ax-10</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I> <I><FONT
COLOR="#FF0000">x</FONT></I> = <I><FONT COLOR="#FF0000">y</FONT></I> →
<FONT FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">y</FONT></I>
<I><FONT COLOR="#FF0000">y</FONT></I> = <I><FONT
COLOR="#FF0000">x</FONT></I>)</SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-11.html">Axiom of Variable
Substitution</A></TD>
<TD><FONT COLOR="#006633"><B>ax-11</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<I><FONT
COLOR="#FF0000">x</FONT></I> = <I><FONT COLOR="#FF0000">y</FONT></I> →
(<FONT FACE=sans-serif>∀</FONT><I><FONT
COLOR="#FF0000">y</FONT></I><FONT COLOR="#0000FF"><I>φ</I></FONT> →
<FONT FACE=sans-serif>∀</FONT><I><FONT
COLOR="#FF0000">x</FONT></I>(<I><FONT COLOR="#FF0000">x</FONT></I> = <I><FONT
COLOR="#FF0000">y</FONT></I> → <FONT
COLOR="#0000FF"><I>φ</I></FONT>)))</SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-i12.html">Axiom of Quantifier
Introduction</A></TD>
<TD><FONT COLOR="#006633"><B>ax-i12</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">z</FONT></I> <I><FONT
COLOR="#FF0000">z</FONT></I> = <I><FONT COLOR="#FF0000">x</FONT></I> <FONT
FACE=sans-serif> ∨</FONT> (<FONT FACE=sans-serif>∀</FONT><I><FONT
COLOR="#FF0000">z</FONT></I> <I><FONT COLOR="#FF0000">z</FONT></I> = <I><FONT
COLOR="#FF0000">y</FONT></I> <FONT FACE=sans-serif> ∨</FONT> <FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">z</FONT></I>(<I><FONT
COLOR="#FF0000">x</FONT></I> = <I><FONT COLOR="#FF0000">y</FONT></I> →
<FONT FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">z</FONT></I>
<I><FONT COLOR="#FF0000">x</FONT></I> = <I><FONT
COLOR="#FF0000">y</FONT></I>)))</SPAN></TD></TR>
<TR ALIGN=LEFT>
<TD><A HREF="ax-bndl.html">Axiom of Bundling</A></TD>
<TD><FONT COLOR="#006633"><B>ax-bndl</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">z</FONT></I> <I><FONT
COLOR="#FF0000">z</FONT></I> = <I><FONT COLOR="#FF0000">x</FONT></I> <FONT
FACE=sans-serif> ∨</FONT> (<FONT FACE=sans-serif>∀</FONT><I><FONT
COLOR="#FF0000">z</FONT></I> <I><FONT COLOR="#FF0000">z</FONT></I> = <I><FONT
COLOR="#FF0000">y</FONT></I> <FONT FACE=sans-serif> ∨</FONT> <FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">z</FONT></I>(<I><FONT
COLOR="#FF0000">x</FONT></I> = <I><FONT COLOR="#FF0000">y</FONT></I> →
<FONT FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">z</FONT></I>
<I><FONT COLOR="#FF0000">x</FONT></I> = <I><FONT
COLOR="#FF0000">y</FONT></I>)))</SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-13.html">Left Membership Equality</A></TD>
<TD><FONT COLOR="#006633"><B>ax-13</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<I><FONT
COLOR="#FF0000">x</FONT></I> = <I><FONT COLOR="#FF0000">y</FONT></I> →
(<I><FONT COLOR="#FF0000">x</FONT></I> <FONT FACE=sans-serif>∈</FONT>
<I><FONT COLOR="#FF0000">z</FONT></I> → <I><FONT
COLOR="#FF0000">y</FONT></I> <FONT FACE=sans-serif>∈</FONT> <I><FONT
COLOR="#FF0000">z</FONT></I>))</SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-14.html">Right Membership Equality</A></TD>
<TD><FONT COLOR="#006633"><B>ax-14</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<I><FONT
COLOR="#FF0000">x</FONT></I> = <I><FONT COLOR="#FF0000">y</FONT></I> →
(<I><FONT COLOR="#FF0000">z</FONT></I> <FONT FACE=sans-serif>∈</FONT>
<I><FONT COLOR="#FF0000">x</FONT></I> → <I><FONT
COLOR="#FF0000">z</FONT></I> <FONT FACE=sans-serif>∈</FONT> <I><FONT
COLOR="#FF0000">y</FONT></I>))</SPAN></TD></TR>
<TR ALIGN=LEFT><TD><A HREF="ax-17.html">Distinctness</A></TD>
<TD><FONT COLOR="#006633"><B>ax-17</B></FONT></TD><TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I><FONT
COLOR="#0000FF"><I>φ</I></FONT>)</SPAN>,
where <FONT COLOR="#FF0000">x</FONT> does not occur in <FONT
COLOR="#0000FF"><I>φ</I></FONT></TD></TR>
</TABLE>
</CENTER>
<P><A NAME="staxioms"></A><B><FONT COLOR="#006633">Set theory</FONT></B>
uses the formalism of propositional and predicate calculus to assert
properties of arbitrary mathematical objects called "sets." A set can
be an element of another set, and this relationship is indicated by the
<IMG SRC='in.gif' WIDTH=10 HEIGHT=19 ALT='e.'> symbol.
Starting with the simplest mathematical object, called the empty set,
set theory builds up more and more complex structures whose existence
follows from the axioms, eventually resulting in extremely complicated
sets that we identify with the real numbers and other familiar
mathematical objects. These axioms were developed in response to <A
HREF="ru.html">Russell's Paradox</A>, a discovery that revolutionized
the foundations of mathematics and logic.</P>
<P><A NAME="izfaxioms"></A> In the IZF axioms that follow, <I>all set
variables are assumed to be</I> <A HREF="#distinct">distinct</A>. If
you click on their links you will see the explicit distinct variable
conditions.</P>
<CENTER><TABLE BORDER CELLSPACING=0 BGCOLOR="#EEFFFA"
SUMMARY="Intuitionistic Zermelo-Fraenkel Set Theory (IZF)">
<CAPTION><B>Intuitionistic Zermelo-Fraenkel Set Theory (IZF)</B></CAPTION>
<TR ALIGN=LEFT><TD><A HREF="ax-ext.html">Axiom of Extensionality</A></TD>
<TD NOWRAP><FONT COLOR="#006633"><B>ax-ext</B></FONT></TD>
<TD>
<SPAN ><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">z</FONT></I>(<I><FONT
COLOR="#FF0000">z</FONT></I> <FONT FACE=sans-serif>∈</FONT> <I><FONT
COLOR="#FF0000">x</FONT></I> ↔ <I><FONT COLOR="#FF0000">z</FONT></I> <FONT
FACE=sans-serif>∈</FONT> <I><FONT COLOR="#FF0000">y</FONT></I>) →
<I><FONT COLOR="#FF0000">x</FONT></I> = <I><FONT
COLOR="#FF0000">y</FONT></I>)</SPAN></TD></TR>
<TR ALIGN=LEFT>
<TD><A HREF="ax-coll.html">Axiom of Collection</A></TD>
<TD><FONT COLOR="#006633"><B>ax-coll</B></FONT></TD><TD>
<SPAN CLASS=math><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I> <FONT
FACE=sans-serif>∈</FONT> <SPAN CLASS=set
STYLE="color:red">𝑎</SPAN> <FONT FACE=sans-serif>∃</FONT><I><FONT
COLOR="#FF0000">y</FONT></I><FONT COLOR="#0000FF"><I>φ</I></FONT> →
<FONT FACE=sans-serif>∃</FONT><SPAN CLASS=set
STYLE="color:red">𝑏</SPAN><FONT FACE=sans-serif>∀</FONT><I><FONT
COLOR="#FF0000">x</FONT></I> <FONT FACE=sans-serif>∈</FONT> <SPAN
CLASS=set STYLE="color:red">𝑎</SPAN> <FONT
FACE=sans-serif>∃</FONT><I><FONT COLOR="#FF0000">y</FONT></I> <FONT
FACE=sans-serif>∈</FONT> <SPAN CLASS=set
STYLE="color:red">𝑏</SPAN> <FONT
COLOR="#0000FF"><I>φ</I></FONT>)</SPAN></TD></TR>
<TR ALIGN=LEFT>
<TD><A HREF="ax-sep.html">Axiom of Separation</A></TD>
<TD><FONT COLOR="#006633"><B>ax-sep</B></FONT></TD><TD>
<SPAN CLASS=math><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT><FONT
FACE=sans-serif>∃</FONT><I><FONT COLOR="#FF0000">y</FONT></I><FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">x</FONT></I>(<I><FONT
COLOR="#FF0000">x</FONT></I> <FONT FACE=sans-serif>∈</FONT> <I><FONT
COLOR="#FF0000">y</FONT></I> ↔ (<I><FONT COLOR="#FF0000">x</FONT></I>
<FONT FACE=sans-serif>∈</FONT> <I><FONT COLOR="#FF0000">z</FONT></I> <FONT
FACE=sans-serif>∧</FONT> <FONT
COLOR="#0000FF"><I>φ</I></FONT>))</SPAN></TD></TR>
<TR ALIGN=LEFT>
<TD><A HREF="ax-pow.html">Axiom of Power Sets</A></TD>
<TD><FONT COLOR="#006633"><B>ax-pow</B></FONT></TD><TD>
<SPAN CLASS=math><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT><FONT
FACE=sans-serif>∃</FONT><I><FONT COLOR="#FF0000">y</FONT></I><FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">z</FONT></I>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">w</FONT></I>(<I><FONT
COLOR="#FF0000">w</FONT></I> <FONT FACE=sans-serif>∈</FONT> <I><FONT
COLOR="#FF0000">z</FONT></I> → <I><FONT COLOR="#FF0000">w</FONT></I> <FONT
FACE=sans-serif>∈</FONT> <I><FONT COLOR="#FF0000">x</FONT></I>) →
<I><FONT COLOR="#FF0000">z</FONT></I> <FONT FACE=sans-serif>∈</FONT>
<I><FONT COLOR="#FF0000">y</FONT></I>)</SPAN></TD></TR>
<TR ALIGN=LEFT>
<TD><A HREF="ax-pr.html">Axiom of Pairing</A></TD>
<TD><FONT COLOR="#006633"><B>ax-pr</B></FONT></TD><TD>
<SPAN CLASS=math><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT><FONT
FACE=sans-serif>∃</FONT><I><FONT COLOR="#FF0000">z</FONT></I><FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">w</FONT></I>((<I><FONT
COLOR="#FF0000">w</FONT></I> = <I><FONT COLOR="#FF0000">x</FONT></I> <FONT
FACE=sans-serif> ∨</FONT> <I><FONT COLOR="#FF0000">w</FONT></I> = <I><FONT
COLOR="#FF0000">y</FONT></I>) → <I><FONT COLOR="#FF0000">w</FONT></I>
<FONT FACE=sans-serif>∈</FONT> <I><FONT
COLOR="#FF0000">z</FONT></I>)</SPAN></TD></TR>
<TR ALIGN=LEFT>
<TD><A HREF="ax-un.html">Axiom of Union</A></TD>
<TD><FONT COLOR="#006633"><B>ax-un</B></FONT></TD><TD>
<SPAN CLASS=math><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT><FONT
FACE=sans-serif>∃</FONT><I><FONT COLOR="#FF0000">y</FONT></I><FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">z</FONT></I>(<FONT
FACE=sans-serif>∃</FONT><I><FONT COLOR="#FF0000">w</FONT></I>(<I><FONT
COLOR="#FF0000">z</FONT></I> <FONT FACE=sans-serif>∈</FONT> <I><FONT
COLOR="#FF0000">w</FONT></I> <FONT FACE=sans-serif>∧</FONT> <I><FONT
COLOR="#FF0000">w</FONT></I> <FONT FACE=sans-serif>∈</FONT> <I><FONT
COLOR="#FF0000">x</FONT></I>) → <I><FONT COLOR="#FF0000">z</FONT></I>
<FONT FACE=sans-serif>∈</FONT> <I><FONT
COLOR="#FF0000">y</FONT></I>)</SPAN></TD></TR>
<TR ALIGN=LEFT>
<TD><A HREF="ax-setind.html">Axiom of Set Induction</A></TD>
<TD><FONT COLOR="#006633"><B>ax-setind</B></FONT></TD><TD>
<SPAN CLASS=math><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT>(<FONT
FACE=sans-serif>∀</FONT><SPAN CLASS=set
STYLE="color:red">𝑎</SPAN>(<FONT
FACE=sans-serif>∀</FONT><I><FONT COLOR="#FF0000">y</FONT></I> <FONT
FACE=sans-serif>∈</FONT> <SPAN CLASS=set
STYLE="color:red">𝑎</SPAN> [<I><FONT COLOR="#FF0000">y</FONT></I> /
<SPAN CLASS=set STYLE="color:red">𝑎</SPAN>]<FONT
COLOR="#0000FF"><I>φ</I></FONT> → <FONT
COLOR="#0000FF"><I>φ</I></FONT>) → <FONT
FACE=sans-serif>∀</FONT><SPAN CLASS=set
STYLE="color:red">𝑎</SPAN><FONT
COLOR="#0000FF"><I>φ</I></FONT>)</SPAN></TD>
</TR>
<TR ALIGN=LEFT>
<TD><A HREF="ax-iinf.html">Axiom of Infinity</A></TD>
<TD><FONT COLOR="#006633"><B>ax-iinf</B></FONT></TD><TD>
<SPAN CLASS=math><FONT COLOR="#808080" FACE=sans-serif>⊢ </FONT><FONT
FACE=sans-serif>∃</FONT><I><FONT COLOR="#FF0000">x</FONT></I>(∅
<FONT FACE=sans-serif>∈</FONT> <I><FONT COLOR="#FF0000">x</FONT></I> <FONT
FACE=sans-serif>∧</FONT> <FONT FACE=sans-serif>∀</FONT><I><FONT
COLOR="#FF0000">y</FONT></I>(<I><FONT COLOR="#FF0000">y</FONT></I> <FONT
FACE=sans-serif>∈</FONT> <I><FONT COLOR="#FF0000">x</FONT></I> → suc
<I><FONT COLOR="#FF0000">y</FONT></I> <FONT FACE=sans-serif>∈</FONT>
<I><FONT COLOR="#FF0000">x</FONT></I>))</SPAN>
</TD></TR>
</TABLE></CENTER>
<P>The above axioms fairly closely follow the Intuitionistic Zermelo-Fraenkel
(IZF) axioms as laid out in [Crosilla].</P>
<P></P><HR NOSHADE SIZE=1><A NAME="theorems"></A><B><FONT COLOR="#006633">A
Theorem Sampler</FONT></B>
<P></P><CENTER><FONT COLOR="#006633"><I>From a psychological point of view,
learning constructive mathematics is agonizing, for it requires one to
first unlearn certain deeply ingrained intuitions and
habits acquired during classical mathematical training.</I>
<BR> —Andrej Bauer</FONT></CENTER>
<P>Listed here are some examples of starting points for your journey
through the maze. Some are stated just as they would be in a
non-constructive context; others are here to highlight areas which
look different constructively.
You should study some simple proofs from
propositional calculus until you get the hang of it. Then try some
predicate calculus and finally set theory.</P>
<P>The <A HREF="mmtheorems.html">Theorem List</A> shows the complete set of
theorems in the database. You may also find the <A
HREF="mmbiblio.html">Bibliographic Cross-Reference</A> useful.</P>
<P><TABLE BORDER=0><TR><TD VALIGN=TOP WIDTH="50%"><TR><TD VALIGN=TOP WIDTH="50%">
<B>Propositional calculus</B>
<MENU>
<LI>Law of identity ~ idALT</LI>
<LI>Praeclarum theorema ~ anim12</LI>
<LI>Contraposition introduction ~ con3</LI>
<LI>Double negation introduction ~ notnot</LI>
<LI>Triple negation ~ notnotnot</LI>
<LI>Definition of exclusive or ~ df-xor</LI>
<LI>Negation and the false constant ~ dfnot</LI>
</MENU>
<B>Predicate calculus</B>
<MENU>
<LI>Existential and universal quantifier swap ~ 19.12</LI>
<LI>Existentially quantified implication ~ 19.35-1</LI>
<LI><I>x</I> = <I>x</I> ~ equid</LI>
<LI>Definition of proper substitution ~ df-sb</LI>
<LI>Double existential uniqueness ~ 2eu7</LI>
</MENU>
<B>Set theory</B>
<MENU>
<LI>Commutative law for union ~ uncom</LI>
<LI>A basic relationship between class and wff
variables ~ abeq2</LI>
<LI>Two ways of saying "is a set" ~ isset</LI>
<LI>The ZF axiom of foundation implies excluded middle ~ regexmid</LI>
<LI>Russell's paradox ~ ru</LI>
<LI>Ordinal trichotomy implies excluded middle ~ ordtriexmid</LI>
<LI>Mathematical (finite) induction ~ findes</LI>
<LI>Peano's postulates for arithmetic:
~ peano1 ~ peano2 ~ peano3 ~ peano4 ~ peano5</LI>
<LI>Two natural numbers are either equal or not equal ~ nndceq (a special case of the law of the excluded middle which we can prove).</LI>
<LI>A natural number is either zero or a successor ~ nn0suc</LI>
<LI>The axiom of choice implies excluded middle ~ acexmid</LI>
<LI>Burali-Forti paradox ~ onprc</LI>
<LI>Transfinite induction ~ tfis3</LI>
<LI>Closure law for ordinal addition ~ oacl</LI>
</MENU>
<B>Real and complex numbers</B>
<MENU>
<LI>Archimedean property of real numbers ~ arch</LI>
<LI>Properties of apartness:
~ apirr ~ apsym ~ apcotr ~ apti</LI>
<LI>The square root of 2 is irrational ~ sqrt2irrap (a
different statement than "The square root of 2
is not rational" ~ sqrt2irr )</LI>
<LI>Convergence of a sequence of complex
numbers ~ climcvg1n given a condition on the rate of convergence</LI>
<LI>Triangle inequality for absolute
value ~ abstrii</LI>
<LI>The maximum of two real numbers ~ maxleb</LI>
</MENU>
</TD></TR></TABLE>
<P></P><HR NOSHADE SIZE=1><A NAME="flavors"></A><B><FONT COLOR="#006633">Different
flavors of constructive mathematics</FONT></B>
<p>Most of our development is based on the Intuitionistic Zermelo-Fraenkel
(IZF) set theory axioms shown above.</p>
<p>We do have a few sections which develop set theory using the weaker
Constructive Zermelo-Fraenkel (CZF) system, also described
in Crosilla. These sections start at ~ wbd (including the section header right
before it) and the biggest complication is the machinery to classify formulas
as bounded formulas, for purposes of the Axiom of Restricted Separation
~ ax-bdsep .</p>
<p>There are also a variety of statements stronger than IZF, up to and
including the law of the excluded middle, which we have theorems about.
We have notations for these statements, shown in the table below,
rather than including them as axioms.</p>
<p>Most of these statements are known as omniscience principles or
choice principles. In the case of the omniscience principles, most of
them also have an analytic form (stated in terms of real numbers),
which implies the corresponding non-analytic form. In reading the
following table ` RR ` is the real numbers and ` DECID ph ` is
` ph \/ -. ph ` (see ~ df-dc ).</p>
<table border cellspacing=0 bgcolor="#EEFFFA">
<tr>
<th>Name</th>
<th>our notation</th>
<th>analytic principle</th>
<th>notes and related theorems</th>
</tr>
<tr>
<td>Principle of Omniscience (law of excluded middle)</td>
<td>EXMID</td>
<td>-</td>
<td>~ df-exmid</td>
</tr>
<tr>
<td>Limited Principle of Omniscience (LPO)</td>
<td>` _om e. Omni `</td>
<td>` A. x e. RR A. y e. RR ( x < y \/ x = y \/ y < x ) `</td>
<td>~ trilpo</td>
</tr>
<tr>
<td>Weak Limited Principle of Omniscience (WLPO)</td>
<td>` _om e. WOmni `</td>
<td>` A. x e. RR A. y e. RR DECID x = y `</td>
<td>~ redcwlpo</td>
</tr>
<tr>
<td>Markov's Principle (MP)</td>
<td>` _om e. Markov `</td>
<td>` A. x e. RR A. y e. RR ( x =/= y -> x =//= y ) `</td>
<td>~ neapmkv</td>
</tr>
<tr>
<td>Lesser limited principle of omniscience (LLPO)</td>
<td><i>not yet defined, but the definition will be something like</i>
`A. f e. ( 2o ^m _om ) A. g e. ( 2o ^m _om ) ( -. 1o e. ( ran f i^i ran g ) ->
( -. 1o e. ran f \/ -. 1o e. ran g ) ) `</td>
<td>` A. x e. RR A. y e. RR ( x <_ y \/ y <_ x ) `</td>
<td>Not yet defined but <a href="https://github.com/metamath/set.mm/issues/4058"
>issue 4058</a> sketches out how we will be able to define it and prove relevant
theorems. Related theorems: ~ dichmul0or ~ ivthdich</td>
</tr>
<tr>
<td>Axiom of choice</td>
<td>CHOICE</td>
<td>-</td>
<td>implies excluded middle as shown at ~ exmidac</td>
</tr>
<tr>
<td>dependent choice</td>
<td><i>not yet defined</i></td>
<td>-</td>
<td></td>
</tr>
<tr>
<td>countable choice</td>
<td>CCHOICE</td>
<td>-</td>
<td>~ df-cc</td>
</tr>
<tr>
<td>Zorn's Lemma</td>
<td><i>not yet defined</i></td>
<td>-</td>
<td>not equivalent to the axiom of choice in the absence of
excluded middle; see <a href="https://github.com/metamath/set.mm/issues/4497"
>issue 4497</a> for discussion about adding it</td>
</tr>
</table>
<P></P><HR NOSHADE SIZE=1><A NAME="existence"></A><B><FONT COLOR="#006633">A
note on existence</FONT></B>
<p>When we mathematically say something exists, there are at least three
things that might mean:</p>
<ul>
<li><b>explicit existence</b>: we can demonstrate what the thing is.
We represent this by giving a class variable as in
the examples below.</li>
<li><b>truncated existence</b>: we can demonstrate there is such a thing,
but we do not expose what it is. We represent this by ` E. ` .</li>
<li><b>classical existence</b>: there is no way that such a thing can fail
to exist. The wording in the previous sentence may look like a double
negation, and indeed ` -. -. E.` is our notation for classical
existence. Example: ~ ismkvnex concerns the statement
that classical existence implies truncated existence in a particular
context.</li>
</ul>
<p>Examples:</p>
<ul>
<li>
If we have the coefficients of a polynomial, ~ elplyr gives explicit
existence (in the form of a function definition) for a polynomial
with those coefficients. But suppose we have a polynomial: can we
get its coefficients? In the case of truncated existence, yes:
~ elply2 does that. But explicit existence in this case would be
something like df-coe from set.mm.
</li>
<li>
Saying that a sequence converges using the notation ` F e. dom ~~> `
is making an assertion of truncated existence; we are saying that
there is a rate at which the sequence converges but we are providing
no way to get that rate. (Note the use of ` E. ` in ~ df-clim ).
On the other hand, the hypothesis in ~ climcvg1n gives the rate at
which the sequence converges, and often we will need that rate of
convergence rather than the truncated (mere) existence of such a rate.
</li>
<li>
In ~ ctssdccl we show a mapping ` G ` and specify explicitly,
via a hypothesis used as a local definition, what it is (contrast
with ~ ctssdc which is using truncated existence).
</li>
<li>
Because the existence for ` g ` in ~ omiunct is truncated,
we need countable choice to get explicit existence and thus apply
~ ctiunct which requires explicit existence.
</li>
</ul>
<p>For more on these topics, see [HoTT] especially section 3.10.
Section 11.2.2 discusses the issues around truncated existence
and sequence convergence.</p>
<P></P><HR NOSHADE SIZE=1><A NAME="set-size"></A><B><FONT COLOR="#006633"
>The size of sets</FONT></B>
<p>When working without excluded middle, familiar concepts such
as finiteness or countability spawn several concepts. Although
we need to list them in some order, beware of thinking of each
category as including all smaller sets (for example the proposition
that any subset of a finite set is finite is equivalent to excluded
middle by ~ exmidssfi ).</p>
<table border cellspacing=0 bgcolor="#EEFFFA">
<tr>
<th>name</th>
<th>our notation</th>
<th>notes and related theorems</th>
</tr>
<tr>
<td>finitely enumerable</td>