1406:" ....an infinite strictly decreasing sequence cannot exist, or equivalently, every strictly decreasing sequence of ordinals do terminate (and cannot be infinite)." The grammar seems to have gone awry here. Most of us use a singular verb after "every". "All things come to an end" but "Every man comes to an end". Apart from that, the statement seems (to the uninitiated) incorrect. The sequence ω, ω - 1, ω - 2 .... looks strictly decreasing to me and it seems to go for ever. I think non-specialists would appreciate either a bit more explanation here or a reference to an article which explains the meaning of the statement about strictly decreasing sequences and proves that is true.
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sequences are a worldwide standard after all these years. Now that you have done that, what is the purpose in raising numbers up to a superscript, or bringing them down to a subscript. Are you suggesting they have exponential effects of some kind? In any case, don't pepper your examples up with 'C" source code, that's downright inscrutable. Just frame your argument in terms of ordinary lines of BASIC code, if you will, and I, like many others who come by this area, will probably understand what you are talking about.
1279:(as at the second link). The relationship to Goodstein's theorem is exactly the same for both representations of the Hydra game, so I suggest a more evenhanded treatment. The fact that the second link presents the game as the execution of a "program" composed of trees, and also explains a more general form of the game, would hardly seem to matter in this regard. On the other hand, if the concern is really about academic or other credentials (rather than relevancy), then I don't wish to pursue the matter. Discussion? —
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problem is that "it cannot be expressed in PA language"?!! That's actually nice and reassuring because on a first glance it seems that one needs not to know the theory of infinite ordinal numbers, but that it should be sufficient (since actually equivalent) to "code" the numbers using "vectors" (nested lists) of finite length, together with sufficient description of how the "bump to base b+1" and "minus 1" is done with these vectors (i.e., keeping track of length and "height" of these vectors). —
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It seems that original
Goodstein theorem (as written in his work) was a bit more strong. He does not restrict changing the base from 2 to 3, from 3 to 4 and so on. He says that if we have _any_ permutation of N and we will change bases accorgind to this permutation, anyway finally we will finish with
984:
notation make a lot more sense? Isn't Base 2 the same thing as binary? I think a lot more people would understand 35 in binary to be %00100011, and not 2 ^ 2 ^ 2 + 1. I guess all I am saying, is that the example given in the main page, seemed/seems to proceed from the assumption that binary was
515:
What about CON(PA), certainly a natural statement since the whole
Hilbert school was chasing after a proof of it before Gödel showed that was impossible? I guess both CON(PA) and Gentzen's induction principle would count as "metamathematical", though I personally don't see why that should matter.
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of the text counts as superscript, and how much farther it has to go, to be super-superscript, and conversely, how far below the baseline you should go, to make it a subscript, and so on. Naturally, I will assume you are already familiar with
Hewlett-Packard Laserjet escape sequences since those
988:
Articles written for
Knowledge should not be dumbed down, but they should be organized in such a way that ordinary people can make heads or tails out of what you are saying. If you want to use 35 as an example for some strange superscript/subscript notation system, you might want to start out by
328:
Yes, Goodstein states the theorem that way in his 1944 paper, which I just double-checked. But he isn't proving the independence from PA; he is just showing the the sequence must terminate. When stated in this general form, the theorem is not expressible in the language of Peano arithmetic. To
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f(G(m)(n+1),n+2), as G'(m)(n)=G(m)(n+1)+1. is not an ordinal.” is hard to understand. Could please someone rewrite it starting with simple steps and ending with conclusions? Already the second sentence is not understandable, and the whole para (in fact, the whole claim) needs a complete clean
374:
So actually it's well possible that this version of the theorem is possible to (state and) prove in the PA, and only the one depending on some permutation β of the naturals is not...?! And actually one cannot even say that the "full" version cannot be proved in PA, but actually the main (only?)
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One has to be careful when saying things like "replace every instance of n with the first infinite ordinal number ω", as in all the definitions and examples above the coefficients appear on the left, and ordinal multiplication isn't commutative. You don't want to have, e.g., 3·ω = ω, but rather
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it's a choice between the two, I recommend the first link, since pictorial graphs are the "standard" and are probably visually preferable. Aside from some generalisations, the second link primarily shows how these games can be played using bracket expressions rather than graphs. —
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Unless I'm failing to see something, this expression is not a base-2 expression on any level, let alone on all levels, since it contains the digit 2. A base-n expression cannot contain a digit representing n, just like base-10 doesn't have a single digit representing the number 10.
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This theorem is more or less the only place that hereditary base-n notation is used. So having an extra article on that would probably give it more weight than it deserves. When people learn the theorem, they are not likely to be familiar with hereditary base-n notation
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0 (starting with any natural number). And (as far as I understand) Goodstein theorem in this (strong) meaning cannot be proved from Peano axioms. It is not clear whether
Goodstein theorem in the written in the article (weak) meaning cannot be proved from Peano axioms.
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was sufficient to prove the consistency of PA; (c) Gödel's second incompleteness theorem says that it's impossible to prove that a system is consistent using methods available in the system itself; so therefore, PA is not capable of doing transfinite induction up to
361:. I don't think that the independence result would be sensitive to which increasing function you use, but checking it would be a lot of work. In any event, the theorem that is now known as Goodstein's theorem is the one mentioned in this article.
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The natural theorem expressible but not provable in PA is
Gentzen's induction principle. He maks this very clear in 1943 and it has been noted many times, e.g., in Schwichtenberg's paper in the Handbook of Mathematical Logic, ed. Barwise. 1977.
1378:
I corrected the proof since there were a misunderstanding of the argument (G(m) does note converge, G(m) terminates, the fact that P(m) dominates G(m) plays no role at all). Please carefully read it before blind cancellation of my edit... ;-)
1217:
It is a base-2 expansion, so a sum of powers of 2, not a base-2 expression. You start by expanding 35 as such a sum, viz. 35 = 2^5 + 2^1 + 2^0. Then you do the same for all of the exponents you used in that base-2 expansion: replace them with
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as an example of an undecidable but 'natural' statement of number theory. Its one and only interesting property is its undecidability. Since the undecidability is the most interesting thing about the theorem, it should be mentioned first.
329:
make the theorem expressible in the language, you have to choose some fixed increasing function to use for the base changes. The independence result proved by Kirby and Paris was for the version of the theorem using the increasing function
1299:. I think relevancy may have been the wrong reason for deletion, and I apologize for not checking more carefully. I'll to consider this more closely, and determine whether your page is the better one; if so, I'll restore it. —
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The lede is carefully worded to avoid giving the impression that the result can be "proved" in PA but as a result the lede never actually states that
Goodstein actually proved anything, only "stated". Should this be clarified?
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Smorynski, Craig. Some rapidly growing functions, Math. Intell., 2 1980, 149-154. / The
Varieties of Arboreal Experience, Math. Intell., 4 1982, 182-188. / "Big" News from Archimedes to Friedman, Notices AMS, 30 1983, 251-256.
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On your other point — you can't get an ordinal smaller than ω by subtracting 1. Every ordinal less than ω is finite. The expression "ω−1" either does not make sense at all, or else evaluates to ω, depending on your notational
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I lean toward removing both, actually, but you really shouldn't add a link to your own research, even though your academic credentials are not being questioned—it's not a matter of academic credentials, but of
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Thanks for clarifying that ordianls are not the same a surreal numbers. I get it now ω is not replaced by ω - 1 but by some number which is always finite. But I still think the articel could be clearer.
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I actually found the theorem quite interesting even without the reference to its undecidability, but I'm not an expert at writing encyclopedias, so I'll not start an edit war today. :) --
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I think you're absolutely right on the grammar. (There's a tiny question in my mind whether this could be a Yank–Brit difference but I don't think so.) I'll fix it if no one else does.
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be given an article all unto itself? The definition given so far, may make sense to a professional mathematician, but somehow the underlying gist escapes me, a fairly ordinary person.
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Yeah, that's silly. The easiest solution is just to say "proved", and then explain further down how much strength you need to prove it, and that that's more than PA. --
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100011. This doesn't quite agree with the way the article writes it, though I don't think it makes a difference to the construction since b^0=1 for any b anyway.
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I wanted to put that in, but it's complicated, and I wasn't personally certain of the details. I believe that one proof is that (a) Trnasfinite induction up to ε
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As to how to explain this to non-specialists, open to suggestions. Is a detailed explication of the notion of ordinal really on-topic here? Maybe it actually
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be OK to say a little about it. Usually I'm skeptical about that sort of digression, but this is a topic where you might expect to attract readers who
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A user removed the second link (but left the first one), commenting that it was "not directly related". However, trees can be represented using either
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L. A. S. Kirby and J. B. Paris. Accessible
Independence Results for Peano Arithmetic, Bulletin of the London Mathematical Society, 14, 1982, 285-293.
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You are right that "hereditary base 2 notation" is not the same as "base 2 notation", but they are ver similar. In base 2, 35 is %100011 which means
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Why is this notation system called "hereditary"? This basically looks like a way of avoiding superscripts (upward departures from the baseline).
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not the same thing as base 2, and base 2 was not binary, even though all these years (over 25 years) I have assumed the two were one and the same.
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The
Smorynski papers reprinted in: Harrington, L.A. et.al. (editors) Harvey Friedman's Research on the Foundations of Mathematics, Elsevier 1985.
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claim to be the first natural theorems not provable by PA. I don't know which is true, but I suspect they can't both be true at the same time.
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I cannot find any result in the Kirby/Paris paper concerning the derivation length of the Goodstein sequence. Where should this be stated?
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439:'s definition is the correct one, so where does this claim come from? Can 3 · 2 − 1 be written differently? If so it should be altered. --
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Perhaps someone with access to a good library could find an accessible independence proof, write it up, and add it to the article.
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The article had contained the following two external links (the second one, recently added, being to a page of my own creation):
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rewrite from scratch. A so-called “example” is not helpful the way it is written. Even if it were, an example is not a proof.
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Perhaps it would be best to have a WP page on Hydra games; but, lacking that, I hope at least one of these links is retained.
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I don't think it's that bad. Apparently the first sentence is considered self-evident since it just says replacing the base
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base-2 expansions, so 5 = 2^2 + 2^0, 1 = 2^0, and then 35 = 2^(2^2+2^0) + 2^(2^0) + 2^0. That corresponds to the base-2
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tags with no more than one level of super/subscript to inline HTML, and corrected the italicization of such phrases as
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The whole para “Then f(G(m)(n), n+1) = f(G'(m)(n), n+2). Now we apply the minus 1 operation, and f(G'(m)(n),n+2) : -->
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because only the 5s are replaced by ωs. If this is not what you are asking, could you rephrase your question? — Carl
1641:(which is maybe not that obvious, but at least somewhat intuitive), and the part after "as" is just the observation
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This is inconsistent as 3 ≠ 402653211. All references (and they are scarce) that I can find on the Internet suggest
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Is there some analysis of the ordinal strength of this theorem? It looks to me like it follows from induction on
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Note that the exponent "5" is not itself in base 2 notation. So in hereditary base 2 notation, we would write
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Also perhaps: Spencer, Joel. Large numbers and unprovable theorems, Amer. Math. Monthly, Dec 1983, 669-675.
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The undecidability of this theorem is notable, but let's define and prove the theorem first, shall we ;-)
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Simpson, Stephen G. Unprovable theorems and fast-growing functions, Contemporary Math. 65 1987, 359-394.
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Sorry to be dim but if ω - 1 equates to ω how can we say that P(m) decreases? Surely P(m + 1)= P(m)
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The value 0 is reached at base 3 · 2 − 1, which, curiously, is a generalized
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does not see the difference.) If anyone objects, please let me know. —
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Just move 2 from the right factor into the left. 3 · 2 = 402653184 · 2.
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How does one prove that this theorem is independent of Peano's axioms?
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No, it does not avoid superscripts; 2^(2^2 + 2^0) + 2^1 + 2^0 means
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35 = 1*2^5 + 0*2^4 + 0*2^3 + 0*2^2 + 1*2^1 + 1*2^0 = 2^5 + 2^1 + 2^0
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completely unimportant? Thats an extremely closeminded attitude.
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Generalised Hydra game as a bracket-expression rewriting system
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1496:, ω - 1 does exist, but has nothing to do with this topic. —
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really seen the concept, and it's central to the argument. --
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Now everything is in base 2, including the exponents. — Carl
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694:{\displaystyle \omega ^{\omega ^{\omega }}+\omega }
614:{\displaystyle \omega ^{\omega ^{\omega }}+\omega }
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http://www.u.arizona.edu/~miller/thesis/thesis.html
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654:{\displaystyle \omega ^{\omega ^{\omega }}+4}
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1047:
1046:
1029:
1028:
1027:
1021:
1020:
1019:
1012:
1011:
969:
964:
961:
933:Sprouts (game)
928:
925:
924:
923:
897:
894:
887:
883:
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857:
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838:
834:
831:
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581:"for example,
578:
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564:67.122.209.126
549:
545:
532:
529:
518:67.122.209.126
484:
481:
470:165.125.144.16
461:
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437:Woodall number
433:
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406:Woodall number
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390:
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1022:
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1007:
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1001:
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986:
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629:
621:decreases to
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315:Shlomi Hillel
309:
308:
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167:
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142:
138:
132:
129:
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108:
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91:
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50:
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41:
35:
27:
23:
18:
17:
1669:
1663:
1659:
1655:
1651:
1647:
1643:
1638:
1632:
1628:
1624:
1619:
1615:
1611:
1607:
1606:+2 and then
1603:
1599:
1574:— Preceding
1569:
1555:Arthur Rubin
1546:
1542:
1538:
1534:
1531:
1528:
1525:Reformatting
1508:
1498:Arthur Rubin
1452:
1435:
1431:
1408:
1405:
1382:— Preceding
1377:
1341:
1316:
1301:Arthur Rubin
1276:
1272:
1270:
1255:
1223:
1219:
1146:
987:
979:
972:
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930:
580:
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514:
486:
463:
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429:
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322:
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291:
288:
285:
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278:
275:
257:
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202:
196:
194:
182:
164:— Preceding
161:
137:Low-priority
136:
96:
62:Low‑priority
40:WikiProjects
1474:subtraction
1252:Hydra games
1185:Denis Kasak
938:—Preceding
491:—Preceding
441:Alex Watson
424:− 1, where
112:Mathematics
103:mathematics
59:Mathematics
1691:Categories
1674:Bbbbbbbbba
1224:expression
973:Shouldn't
497:SternJacob
460:Two Firsts
1566:Bad proof
1478:ω - 1 = ω
1440:Trovatore
1360:Trovatore
1297:WP:EL#ADV
483:One First
428:+ 2 : -->
247:AxelBoldt
1588:contribs
1576:unsigned
1512:Bukovets
1472:, using
1455:Bukovets
1427:choices.
1410:Bukovets
1396:contribs
1384:unsigned
991:baseline
952:contribs
940:unsigned
703:Hirak 99
505:contribs
493:unsigned
363:CMummert
166:unsigned
1535:G(m)(n)
1436:haven't
313:ω·3...
294:Dominus
204:Dominus
139:on the
30:C-class
1618:+1 by
1610:+2 by
1602:+1 by
1558:(talk)
1501:(talk)
1468:As an
1388:Gasole
1345:Tkuvho
1322:r.e.s.
1304:(talk)
1281:r.e.s.
1273:graphs
1010:first.
982:binary
451:Goplat
197:except
36:scale.
1672:bad.
1486:ω - 1
1432:would
1220:their
404:From
238:Timwi
185:Timwi
183:Said
1678:talk
1670:that
1654:+1)=
1584:talk
1516:talk
1459:talk
1444:talk
1414:talk
1392:talk
1364:talk
1349:talk
1326:talk
1285:talk
1232:talk
1189:talk
1157:talk
1060:talk
1039:talk
1000:talk
948:talk
916:talk
707:talk
701:? --
568:talk
522:talk
501:talk
474:talk
381:Talk
222:talk
218:Rich
174:talk
1666:)-1
1537:to
1476:,
1328:)
1287:)
1153:CBM
1035:CBM
935:.
912:CBM
377:MFH
131:Low
1693::
1680:)
1662:)(
1656:G'
1650:)(
1631:,
1590:)
1586:•
1545:)(
1518:)
1461:)
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1416:)
1398:)
1394:•
1366:)
1351:)
1317:If
1234:)
1191:)
1155:·
1062:)
1037:·
1002:)
954:)
950:•
914:·
886:ω
882:ω
877:ω
827:−
784:ω
774:ω
770:ω
765:ω
709:)
689:ω
679:ω
675:ω
670:ω
639:ω
635:ω
630:ω
609:ω
599:ω
595:ω
590:ω
570:)
544:ϵ
524:)
507:)
503:•
476:)
420:·
412:A
408::
340:↦
224:)
187::
176:)
1676:(
1664:n
1660:m
1658:(
1652:n
1648:m
1646:(
1644:G
1639:u
1635:)
1633:k
1629:u
1627:(
1625:f
1620:ω
1616:n
1612:ω
1608:n
1604:n
1600:n
1582:(
1547:n
1543:m
1541:(
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1187:(
1159:)
1151:(
1147:n
1133:1
1130:+
1125:1
1121:2
1117:+
1112:)
1109:1
1106:+
1101:2
1097:2
1093:(
1089:2
1058:(
1041:)
1033:(
998:(
946:(
918:)
910:(
896:4
893:+
856:4
853:+
846:5
842:5
837:5
833:=
830:1
824:5
821:+
814:5
810:5
805:5
781:+
744:4
741:+
734:4
730:4
725:4
705:(
686:+
649:4
646:+
606:+
566:(
548:0
520:(
499:(
472:(
430:b
426:n
422:b
418:n
379::
349:1
346:+
343:n
337:n
271:0
269:ε
266:0
262:0
258:n
254:0
220:(
172:(
143:.
42::
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