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isn't 'intuitive'), justification of its inclusion is omitted here. So, it seems to me that we have the "inside baseball" people and the peanut gallery (general public). It's the latter that I believe would greatly benefit from some mention of the "with or without 0" issue. (other than just an aside). Oh, speaking of asides, it's claimed here that N is necessarily infinite. OK, but that should be changed to "countably infinite", imho.
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877:. While they seem to have abandoned the process of assigning A-class ratings more than a decade ago, it is nevertheless a WikiProject-based rating that was never rescinded, so it's up to them to do any reassessment or retraction. You and the other editors here can certainly chime in there to explain why you feel the rating is not merited at this time. Best of luck.
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theorem is to have a system where provability is recursively enumerable. Strong enough FOL theory with decidable axiom schemas is one typical way to get recursively enumerable definition of provability. Second order logic with full semantics is not such an effective definition: there is no way to enumerate all true facts according to such semantics.
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I went ahead and split the article into the 1) historical+2nd-order part and 2) FOL part. I feel that this was the right thing to do because the historical+2nd-order part one is meant to describe natural numbers in a fairly general sense as witnessed by categoricity, whereas the second part has clear
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I suggest as the main way towards improving the article to have more balance between 1) the historical second-order formulation that seems to be the way the article started and 2) more modern and more widely used first-order variant (called Peano arithmetic) here. Many people would be more than happy
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article, for example, second-order logic with "full semantics", plus the original Peano axioms, is enough to make the theory categorical â there is only one model up to isomorphism. "Full semantics" means that you have first-order variables that range over natural numbers and second-order variables
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reasons (though historians may still be interested). And yes, I suppose it's probably good to know about if you want to read Gödel's original paper, but the same remarks apply â Gödel was doing something for the first time, and putting it in the context of the day, but there are much more efficient
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At least when only 0 and S are considered as function symbols, substitutivity follows from the "only if" part of axiom 7. When also + and * are admitted, and they are defined recursively as usual, substitutivity might be provable by induction (I'm not sure about this). On the other hand, e.g. Hermes
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include 0. Even Peano did it initially, (before he understood its value in
Abstract Algebra and how it allowed N to have lots of connections that an N-without-0 would lack.) Note that other than the silly ahistorical claim that 0 is intuitive (for certain values of 'intuitive', there's nothing that
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ISO 80000-2:2009 apparently requires 0 to be an element of N. The obvious problem with that is that zero can't be put into a one-to-one correspondence with anything. There is also the just plain false claim that zero is intuitive. Well, no: it must be taught; it is not a natural concept. (Everyone
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I think that textual rendering would make the proofs more accessible. If this is to be merged, it should go into the
Arithmetic section because they are one part of the proof of the claim that the axioms imply the properties of an ordered semiring. The proofs are easy exercises in induction, so I
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Trovatore: Henking semantics for second order (and higher-order) logic could be applied to the question of models of Peano arithmetic and provability, but do we actually have interesting facts to mention in that direction (other than it is a valid thing to consider)? If yes, this line of thinking
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While there are nontrivial points to be made about the conception of zero and the mental steps needed to make it intuitive, I respectfully suggest that this is not the right article to go into detail on that. This is an article about a particular axiomatic framework, not about zero and not even
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Jochen: Thank you for pointing out to this historical development, of which I was not aware of! I was merely following the expositions in several textbooks, including
Mendelson and the handbook by Samuel Buss. These all refer to FOL formulation of the problem. What matters for the incompleteness
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It is true that the older notion of natural number did not include zero. That was a historical mistake, which has largely been corrected. The more modern and useful conception of natural number does in fact include zero. However terminology has not converged to a standard; there are still
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I agree with you that the A-rating seems a bit too high. I'm not an expert in logic, but the article seems to have many deeper flaws. For instance, the induction is expressed in second-order logic, but then the addition (defined immediately after the axioms) is given like the induction is a
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could be used to expand and clarify analogously the paragraph on categoricity proof carried out in first-order axiomatization of ZFC. I thought how to clarify that paragraph, but I am not entirely sure about the intention was with it originally, so I left it as it is.
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65.36.124.105: Your link above is to a discussion that I understood as being about including a version of the axioms that axiomatized exponentiation. I was against that â exponentiation, unlike multiplication, does not need to be axiomatized in PA. You can simply
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probably standard these days, and I would look favorably on a presentation incorporating that idea, assuming it can be found in good sources. The axioms that deal purely with equality can be presented for historical interest in the "history" section, perhaps.
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the first-order and second-order variables range over. That is, you limit the second-order variables to ranging over a pre-specified class of sets of naturals, rather than all of them. Now the theory is no longer categorical. It's actually a theory of
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a predicate variable, ".", "Î ", and "â" meaning "and", "for all", and "implies", respectively). I guess I'm unaware of some modern treatment of Gödel's result which transcribed it to FOL. Maybe you could even devise a clarifying remark for the page
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I'm not an expert on this topic, but the presentation of the induction schema in the "Peano arithmetic as first-order theory" section seems unnecessarily complicated (though I'm not claiming it's wrong). Instead of an axiom for each formula
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Eh? Many natural formulas become awkward if you let zero be a natural number. They have to be marked up by changing x to x+1 which often makes the formula twice as large to display. That, or requires some qualification that all of the
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by all choices of the variables (i.e. 0, S(0), S(S(0)), etc for each y_i). This seems a lot simpler, though maybe there is some subtle difference that I am not appreciating, which is why I am posting this here rather than editing.
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My personal preference would be only to add the single proof of associativity. I think it aids the understanding of how the Peano axioms can be used to infer well-known properties of arithmetic, and reflects on the axioms themselves.
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everywhere, which also adds text and lengthens formulas. The term "non-negative integer" works, if you absolutely have to have zero. Stick to the conventional, historical usage taught in grammar school. It works cleanly.
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of getting rid of that, but it didnÊŒt go anywhere. However, if we do include the axioms for equality, weÊŒre missing the one for substitution. We use this implicitly in the examples of addition and in the proof that
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first-order schema. No examples of interesting predicates are talked about which differentiates this system from simpler axiomatic schemes (e.g., primality, divisibility). Decidability isn't discussed at any length.
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about the natural numbers. The "zero" issue should be touched on in the history section (I assume it already is; haven't checked) but beyond that it's a distraction from the main concerns of the article. --
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Anyway this is not the place to discuss it. I unwisely responded in that key half a year ago, but the right answer is the one I gave in the previous section, at 19:27, 19 August 2022 (UTC). --
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who knows its history knows this.) I don't argue for a revolution in the
Abstract Algebra main-stream here. I DO argue that more discussion should be included with an (alternative) N that does
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is the multiplicative left identity; we should either include it in the main list of nine axioms, or do as we do elsewhere in the article and refer to âthe usual axioms of equalityâ. --
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I and some of the other editors believe that the entry does not meet the GA criteria and that the A rating for the item is already inactive. Can we downgrade it to a B rating?--
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I believe that including an example of how the Peano axioms prove associativity of addition, say, would be helpful for an understanding of the Peano axioms. On the other hand,
998:-order logic (in sect.2, p.176, variable symbols of arbitrary high order are introduced before fn.17; on p.177, axiom I.3 is the classical 2nd-order induction axiom, with
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question than a general math-logic question. The notation and terminology used by
Russell turned out to be pretty much of a dead end; hardly anyone learns it anymore for
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However, the project is no longer processing the revocation of A-level entries. This seems to have resulted in how we cannot revoke the A-level status of the entry.--
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Since no one so far seems to come forward to defend the A-Rating, I would suggest downgrading it and seeing if anyone reinstates it. Then we could take it to
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Honestly I'm not sure the "proofs" article belongs in
Knowledge at all, but that's not really a road I want to go down; it isn't hurting anyone where it is.
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the statements you wanted to add as axioms. Probably more to the point, I don't think it's standard to include exponentiation as part of the language of PA.
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Aside: I don't think the article should be nominated for either a GA or FA until the significance of the Peano axioms is addressed (in the article).
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article to this one. This can be done through normal editing channels, no complete merger is deemed necessary (and in fact holds at least one full
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Does that help? It might be a nice thing to bring up at the refdesk if you want more details. We could use some good questions like that. --
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Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.
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Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.
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I'm not competent to answer the question directly because I've never bothered to put in the effort needed to understand
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recursive axiomatization(s), exhibits non-standard models, and was the subject of Goedel's incompleteness theorem.
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is of little use as an independent page without the context from here. Therefore, I suggest selectively to merge
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As regards considering equality to be part of the logic rather than the non-logical axioms, I think that
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Could this article be nominated for GA? Or maybe even FA? You can do discussion on the talk page.
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seems like it can be realized by (infinitely many) instances of the above given by replacing
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File:Inductive proofs of properties of add, mult from recursive definitions svg.svg
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anyone can change the assessment, and disputes should be settled on the talk page.
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On the other hand, you can take exactly the same theory but interpret it with
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that range over sets of natural numbers, and the latter really range over
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Talk:Gödel's_incompleteness_theorems#First-order_vs._higher-order_version
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observed that any other natural number may be given as a number with no
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would say that one of them is likely sufficient to state as an example?
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to take FOL variant as the starting point for an entire article.
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The article currently defines equality; there was discussion
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Accordingly, this article needs to be edited and revised.
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comment) from the conversation participants. Proposer,
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1408:â Preceding
1405:
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1293:Oppose merge
1292:
1239:
1217:Peano axioms
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1182:
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51:WikiProjects
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674:BlueMoonset
659:BlueMoonset
123:Mathematics
114:mathematics
70:Mathematics
1545:Categories
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899:TheZuza777
801:3540058192
341:, namely
1508:Trovatore
1431:Trovatore
1373:is not a
1351:Trovatore
1301:Trovatore
1160:CONSENSUS
1074:Trovatore
837:Trovatore
39:is rated
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1397:variable
1276:Felix QW
1221:Felix QW
1172:Felix QW
924:Felix QW
789:Comment:
622:Rhkramer
584:unsigned
226:Archives
205:Approved
1261:Vkuncak
1116:Vkuncak
1095:Vkuncak
955:Vkuncak
938:Vkuncak
183:Process
150:on the
41:C-class
1240:biased
1168:Oppose
1164:Proofs
1153:CLOSED
996:higher
821:define
186:Result
47:scale.
1467:: -->
1215:into
1067:first
825:prove
736:above
28:This
1526:talk
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1371:Zero
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1323:Zero
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180:Date
1379:one
1330:not
1138:.)
1051:all
911:By
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