389:, I have a slight quibble. The usual statement of the second incompleteness theorem includes as a hypothesis that the theory is able to prove the Hilbert–Bernays conditions (so "contains enough arithmetic" in the statement of the 2nd theorem is a stronger assumption than "contains enough arithmetic" in the statement of the 1st theorem). Therefore, as it is usually stated, the second incompleteness theorem does not apply to Q, because Q doesn't meet the hypotheses of the theorem. In my opinion the paragraph in the article beginning "Gödel's theorems only apply to axiomatic systems defining sufficient arithmetic to carry out the coding constructions " doesn't clearly distinguish between the hypotheses of the first and second theorems. I will try to edit the article to take this into account. — Carl
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1319:. The opening sentences of this article may not provide sufficient context for the statement in question, as the notion of a fragment of first-order arithmetic is not defined within Knowledge. Buss defines a fragment of first-order arithmetic as an axiomatizable subtheory of first-order arithmetic in his chapter of the
1301:: For each statement in Presburger arithmetic, either it is possible to deduce it from the axioms or it is possible to deduce its negation." It is true that the languages of these theories differ. My point is that the precise sense of 'weaker' intended here should be stated at the beginning of this article.
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Thank you, Emil, I understand. And one more: Could you say something about what the axiom (3) is necessary for? If I understand it right, the axiom (3) is a "weaker" replacement of induction scheme. But it doesn't make possible inductive inference. It excludes such models as nonnegative reals, but in
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I don't think the analogy with
Urysohn's metrization theorem works. What is called "second incompleteness theorem" is in any case different from Gödel's formulation, the statement has been generalized by relaxing the assumptions, so it's a kind of double standard to disallow another relaxation of the
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for the second incompleteness theorem isn't Q, it's something stronger than Q. By analogy, if I state a theorem in analysis with "any sufficiently smooth function has property X" and then in the proof I use the third derivative, what I really assumed in the hypotheses was that the function is 3 times
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The common belief among experts is that Q does not prove the
Hilbert–Bernays derivability conditions that are needed for the standard proof of the second incompleteness theorem. However, there is no published proof that any particular derivability condition is not provable in Q. This was discussed on
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The sentence does not imply that the theory minus each one of the axioms should have uninteresting models, only that it does not satisfy Gödel’s theorems, and anyway the word is used here informally. However, for more clarity I changed extension to end-extension (one of the uninteresting models of Q
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My opinion is that the standard, textbook statement of the second incompleteness theorem is the one that assumes the derivability conditions. So the proof that Pudlak gives is not what is usually called the "second incompleteness theorem" – it's a distinct, stronger theorem. This is parallel to how
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Nevertheless, there is an alternate proof that Q does not, in fact, prove Con(Q). In this limited sense, the second incompleteness theorem holds for Q. I believe that the first proof that Q does not prove Con(Q) is due to A. Bezboruah and J. C. Shepherdson, "Gödel's Second
Incompleteness Theorem for
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I changed a sentence saying something like "Q, like the Peano axioms, has non-standard models of all infinite cardinalities" to say Peano arithmetic instead of Peano axioms. I usually think of "Peano axioms" as meaning the 19th century version which has a single induction axiom in full second-order
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that doesn't really cover identity in the sense of 'first order logic with identity' (versus without identity). I think we either need to add a section to that article, or retarget this link to a more relevant location (possibly a new as-yet unwritten article?) Does anyone more familiar than I know
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In particular, I think that when we say "second incompleteness theorem" in an article we should assume readers will think of the theorem that assumes the derivability conditions. I think that this confusion is what led Grudeu to make the edits that he or she make to the article. I'm not saying that
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with any of the axioms removed is not essentially undecidable (any extension in the same language is also undecidable), this is quite true, but then goes on to state that some of its subtheories obtained by removing some of the states axioms are decidable - in particular, that one such subtheory is
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I went ahead and removed it. I don't have the edition of the reference that was cited (the 4th edition), but I was able to see the cited page on google books and it doesn't include the fact claimed. The 5th version, which I do have, also says nothing. So I can't say if this was a misreading of the
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seems to go to an unintended target, I do not see how it is relevant to
Robinson's arithmetic. The given definition of + in terms of S and · relies on basic properties like commutativity and associativity of + and ·, distributivity, and most importantly, cancelativity of + and ·. None of these are
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Emil, maybe I'm going beyond the scope of the article, but it's interesting subject to be noted somewhere: If I'm not mistaken, the existence of decidable extention of the theory means inapplicability of the 1st Goedel's incompleteness theorem to the theory, doesn't it? And it's interesting where
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to both decidable and undecidable fragments of undecidable theories. The
Knowledge article on First-Order Logic contains examples of fragments in which the language is restricted: "For instance, first-order logic is undecidable, meaning a sound, complete and terminating decision algorithm is
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and
Presburger arithmetic are different (Presburger arithmetic doesn't have multiplication), so I don't think it makes sense to talk about them in this way. Should this be removed? (This is my first time doing this sort of thing, so I feel a little uneasy about altering things myself.)
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How can it be true? Let us drop the axiom (3). However, every term in form of "1+ ... +1" represents a natural number, and it's provable, that different such terms are not equal. That is, all standard natural numbers are contained in any given model of the theory, aren't they?
487:: PRA, PA, and ZFC. When discussing the requirements of the second incompleteness theorem on p. 839, he emphasizes that for the second theorem "mere binumerability of the primitive recursive relations" is not sufficient. That "mere binumerability" is exactly what we get with Q.
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differentiable. If I later prove that some particular function that is only once differentiable has property X, I cannot say that this new result was a consequence of my original theorem, since the original theorem only applied to functions that were 3 times differentiable.
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to be a "syntactically restricted subset of formulae of the theory." In this sense, both
Robinson's arithmetic and Presberger's arithmetic are fragments of Peano Arithmetic. This may not be what experts in the proof theory of arithmetic intend; nevertheless, there are
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my opinion such models are not worse than models with nonstandard numbers. Perhaps, not all computable functions are representable in the theory without the axiom (3), unlike Q? Is there an example of computable function, which is not representable in such theory?
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our articles should take a position on which theorem is "really" the incompleteness theorem, just that our articles should be written in a way that accommodates readers who are only familiar with the theorem stated in terms of the derivability conditions. — Carl
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Before I respond too much, I want to say I think we can find a wording in the article that is acceptable without resolving this disagreement, which is linguistic and not mathematical. I was satisfied with the wording you added after I made just a few cosmetic
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true with Q as the base theory. I don't know what exactly is the claim in
Bezboruah and Shepherdson, but the Pudlák article I linked does state it in that form (well, he actually states it for finitely axiomatized theories, but there is no deep reason behind
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equivalent to
Presburger arithmetic and so is decidable. This is simply incorrect, since any finitely axiomatizable, essentially undecidable theory is also hereditarily undecidable (any subtheory in the same language is also undecidable). The languages of
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the Nagata–Smirnov metrization theorem is distinct from
Urysohn's metrization theorem. Unfortunately, the situation we have in logic is like a world where Urysohn's theorem is commonly stated as "Any topological space with enough properties is metrizable".
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What you write is correct. However, I would say that in the nowadays common proof of Gödel’s first incompleteness theorem using representability of computable functions or r.e. sets, the most fundamental property that fails without (3) is
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logic, and which has no nonstandard models, while "Peano arithmetic" is the first-order axiomitization an infinite schema of induction axioms. I hope this change was appropriate -- please feel free to comment, I'm not any kind of expert.
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There is an issue in the given interpretation of +. Specifically, since * is not associative in some models of Q, the parentheses should be included, or there it should be verified that they are irrelevant in this case. Thank you.
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This section is based wholly on Boolos and Jeffrey. This definability of addition in terms of product and successor was discovered by Tarski. It is my understanding that product associates, given the intended interpretation of
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exactly Goedel's proof fails: To implement Goedel's numbering and formula of provability we should express factorization by formula of arithmetic. And it's very strange why this expression doesn't work without the axiom (3).
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on a cut, prove Hilbert–Bernays conditions for this fragment, and then use a variant of the standard diagonalization argument), and because there is a popular misconception that it can't be used. However, the statement
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Also, you are looking at the wrong chapter in Hájek and Pudlák. The second incompleteness theorem for extensions of Q (or rather, its strengthening in terms of restricted provability) is Thm. 5.28 (ii) on page
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I understand! Without the axiom (3) we cannot prove, that prime numbers exist: So, the number 3 has dividers other than 1 and 3 - for example, 2 and 1.5 (in the model of nonnegative reals). Am I right?
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Any serious university library should have a copy of Tarski et al (1953). Check it out and see if the entry does justice to what Tarski wrote. Incidentally, Pressburger arithmetic is not a subtheory of
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The author wrote: "... when any one of the seven axioms above is dropped. These fragments of Q ... have ... uninteresting models (i.e., models which do not extend the standard natural numbers)".
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What do you mean by the "usual statement of the second incompleteness theorem"? As far as I am aware, the usual statement of the second incompleteness theorem is that for any consistent theory
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615:(Edit conflict) Exactly – my point is that the theorem that they call the "Goedel's second incompleteness theorem" is the one on p.164. I wrote the following while you were writing.
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I for one think we should find a way to resolve these issues - but without too much verbiage. They do present a possibility for confusing someone who browses these articles. — Carl
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I noticed that Robinson arithmetic was the subject of some edits on the article. I was interested in that subject earlier this year, and when I looked into it this is what I found:
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is because it requires a more complicated proof than for a stronger base theory (basically, you need to interpret in the theory a decent fragment of bounded arithmetic like
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That’s the wrong way to think about it. Immediately from the definition, any subtheory of an incomplete theory that has the same language is also incomplete.
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I assume the IP simply didn't appreciate the difference between definability in the standard model of arithmetic and definability in a weak theory like Q.—
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I for one am certainly confused by the statement about weakness. Usually I see incompleteness described as a property of theories that are sufficiently
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lacking the recursive definition of addition are uninteresting, given that addition is definable. Tarski et al (1953) asserted that all fragments of
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I don't know of one on Knowledge, but I am planning to rework the first order logic article, so I will try to keep this in mind when I do it. — Carl
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There is more than one reference to Mendelson's textbook, but that deals with a theory stronger than Q (though still weaker than PA).
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Of course Q is strong enough for the standard proof of the first incompleteness theorem. This is what it was designed for, after all.
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incompleteness (i.e., having no recursively axiomatizable complete extension) is preserved upwards, but that’s a different matter.—
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I'm not sure I fully understood your above posting. Do you have particular suggestions what to improve in the article and how? -
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In the clause "even if we additionally restrict Gödel numbers of proofs to a definable cut" what does "definable cut" mean?
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There is an ambiguity concerning the notion of weakness at the beginning of the article. An opening sentence reads, "Since
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I don't follow what you are trying to say. That Q is usually not employed as the <insert-your-favourite-theory-here: -->
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But with this statement it is proveble, two symbols are not one symbol just as x is not on the position of y.......
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reference or what, but in any case I agree with removing the section. For reference the text was added here
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The theory is often left unspecified in the statement, but the <insert-your-favourite-theory-here: -->
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assumptions in the same spirit. But I can see what you mean. I am happy with the current formulation.—
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The reals are decidable, and the only functions representable in them are piecewise algebraic. See
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In Pudlák and Hájek's book on p. 164 their statement of the second incompleteness theorem begins:
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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That statement doesn't apply to Q because Q does not verify the derivablity conditions for any Σ
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definition of its provable formulas (or, at least, everyone believes that it doesn't). — Carl
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but the general statement x + y = y + x is not. Similarly, one cannot prove that Sx ≠ x
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Q", The Journal of Symbolic Logic Vol. 41, No. 2 (Jun., 1976), pp. 503-512,
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594:
593:
592:
591:
590:
561:
556:
552:
521:
518:
517:
516:
508:
491:
490:
489:
488:
477:
476:
475:
474:
466:
465:
452:
426:
422:
384:
383:
376:
372:
364:
361:
344:
341:
340:
339:
316:
315:
286:
283:
282:
281:
268:
267:
218:
215:
214:
213:
212:
211:
159:
156:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
1565:
1554:
1551:
1549:
1546:
1545:
1543:
1534:
1530:
1526:
1522:
1521:
1520:
1519:
1515:
1511:
1503:
1501:
1500:
1496:
1492:
1478:
1475:
1472:
1468:
1464:
1463:
1462:
1458:
1454:
1450:
1446:
1445:
1442:
1437:
1433:
1427:
1426:
1425:
1424:
1420:
1416:
1412:
1408:
1404:
1400:
1382:
1379:
1352:
1349:
1331:
1326:
1322:
1318:
1313:
1312:
1308:
1304:
1300:
1296:
1292:
1288:
1284:
1280:
1276:
1268:
1264:
1261:
1258:
1234:
1229:
1226:
1223:
1220:
1217:
1209:
1204:
1201:
1198:
1190:
1185:
1182:
1171:
1166:
1163:
1155:
1144:
1123:
1101:
1093:
1088:
1085:
1060:
1055:
1041:
1040:
1039:
1038:
1034:
1030:
1026:
1022:
1021:
1017:
1013:
1009:
1003:
1000:
997:
993:
989:
988:
987:
986:
982:
978:
974:
968:
965:
962:
957:
956:
955:
954:
950:
946:
942:
938:
932:
928:
925:
922:
903:
897:
891:
888:
885:
882:
876:
870:
847:
844:
841:
835:
826:
820:
812:
808:
804:
800:
799:
798:
796:
792:
788:
784:
775:Definable cut
774:
768:
765:
762:
758:
757:
756:
751:
747:
741:
738:
734:
733:
732:
731:
728:
725:
721:
716:
708:
696:
693:
690:
685:
684:
683:
678:
674:
667:
666:
665:
664:
663:
662:
661:
660:
649:
648:
647:
646:
645:
644:
643:
642:
631:
630:
629:
628:
627:
626:
625:
624:
614:
613:
612:
609:
606:
601:
600:
599:
598:
597:
596:
589:
586:
583:
578:
559:
554:
550:
540:
539:
538:
533:
529:
519:
514:
506:
502:
498:
497:
495:
494:
493:
492:
485:
481:
480:
479:
478:
470:
469:
468:
467:
464:
461:
458:
450:
446:
442:
424:
410:
406:
405:
404:
403:
398:
394:
388:
382:
377:
373:
370:
369:
368:
362:
360:
359:
355:
351:
350:66.127.55.192
342:
338:
334:
330:
329:123.255.28.93
326:
322:
318:
317:
313:
309:
304:
303:
302:
301:
297:
293:
284:
280:
277:
274:
270:
269:
266:
262:
258:
257:123.255.28.93
254:
250:
245:
244:
243:
242:
238:
234:
229:
224:
216:
210:
206:
202:
198:
197:
196:
191:
187:
181:
180:
179:
178:
174:
170:
165:
157:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
1507:
1488:
1466:
1448:
1402:
1398:
1314:
1274:
1272:
1142:
1027:
1023:
1010:
1006:
975:
971:
943:
939:
936:
810:
806:
802:
781:— Preceding
778:
719:
712:
576:
512:
504:
500:
483:
448:
444:
440:
408:
385:
366:
346:
343:Peano axioms
324:
320:
311:
307:
288:
252:
248:
227:
222:
220:
161:
137:Mid-priority
136:
96:
62:Mid‑priority
40:WikiProjects
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
1542:Categories
1453:Crisperdue
1279:incomplete
801:A formula
412:and every
201:Zero sharp
169:Zero sharp
1510:Jonaw2024
1467:Essential
742:. — Carl
292:Hitorikii
1415:FLeℵgyel
1407:FLengye|
1303:FLengye|
1299:complete
1029:Eugepros
1012:Eugepros
977:Eugepros
945:Eugepros
783:unsigned
633:changes.
251:because
1297:is "...
580:that).—
139:on the
1449:strong
863:, and
720:fairly
36:scale.
813:(0),
603:387.—
381:JStor
1529:talk
1514:talk
1495:talk
1471:Emil
1457:talk
1451:. -
1436:talk
1419:ta|k
1411:ta|k
1367:and
1307:ta|k
1257:Emil
1033:talk
1016:talk
996:Emil
981:talk
961:Emil
949:talk
921:Emil
791:talk
761:Emil
750:talk
724:Emil
689:Emil
677:talk
605:Emil
582:Emil
532:talk
499:Let
457:Emil
397:talk
354:talk
333:talk
296:talk
273:Emil
261:talk
237:talk
205:talk
190:talk
173:talk
1432:CBM
746:CBM
673:CBM
528:CBM
443:in
393:CBM
186:CBM
131:Mid
1544::
1531:)
1516:)
1497:)
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1459:)
1434:·
1421:)
1380:≤
1376:∃
1350:≥
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1309:)
1260:J.
1238:¯
1224:∨
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1199:∨
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951:)
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889:≤
883:∧
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748:·
727:J.
692:J.
675:·
608:J.
585:J.
577:is
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460:J.
447:,
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263:)
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1527:(
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1243:)
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391:(
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331:(
325:Q
321:Q
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308:Q
294:(
259:(
253:Q
249:Q
235:(
228:Q
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203:(
192:)
184:(
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143:.
42::
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