Talk:Peano axioms/Archive 1

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Older posts

I deleted this:

This was a proof using logic alone, but of course infinite. It gives an algorithm for simplifying a :possible proof of contradiction by a series of simple transformations, until it becomes very short. :To see that it becomes very short we need transfinite induction. Gentzen's proof has been :humorously called "assuming the dubious to prove the obvious".

since (a) it was not a proof using 'infinite logic' but a straightforward mathematical proof; (b) the procedure makes the proofs longer, not shorter; (c) whoever called the proof that doesn't understand it.

Someone was getting confused about Peano's axioms vs. first order Peano Arithmetic. I(different I than the above: this I's a PhD student who's area is models of PA) changed

Although natural numbers satisfy these axioms, there are other, nonstandard models of arbitrary large cardinality - by Compactness theorem the existence of infinite natural numbers cannot be excluded in any axiomatization.


Dedekind proved, in his 1888 book Was sind und was sollen die Zahlen, that any model of the second order Peano axioms is isomorphic to the natural numbers. On the other hand, the last axiom listed above, the mathematical induction axiom, is not itself expressible in the first order language of arithmetic.

If one replaces the last axiom with the schema:

  1. If P(0) is true and for all n P(x) implies P(x+1)

for each first order property P(x) (an infinite number of axioms) then although natural numbers satisfy these axioms, there are other, nonstandard models of arbitrary large cardinality - by the Compactness theorem the existence of infinite natural numbers cannot be excluded in any axiomatization; by the Lowenheim-Skolem theorem there exist models of all cardinalities.

Thanks for the clear-up and help! Revolver 07:04, 30 Sep 2004 (UTC)

Presumably "For all n, P(x) implies P(x+1)" really means:

"For all x, P(x) implies P(x + 1)."

Michael Hardy 00:07, 25 Oct 2003 (UTC)

Peano fails on 0 (zero) not being a natural number and zero (the absolute absence of anything) exists only in mathematics. Zero is a natural number by definition alone and not by any construct that is derived from the science; it is an add-on and leads to "axiomatic" for that construct alone, and nowhere else. Like all attempts to by-pass absolutes, short-cuts are always based on some form of assumption that is passed off as knowledge by some one exercising status or authority. For every natural number, Peano is correct, but for the unnatural numbers of zero and infinite, he also has no way of inclusion. Both are outside the science, at all times. From nowhere to somewhere but not at either end. Oct 29, 2005

Peano system

Why is the name Peano system used in the article? The criteria are due to Dedekind. ---- Charles Stewart 11:30, 29 Sep 2004 (UTC)

This is the term that I seem to remember hearing/reading most often. You can certainly check the literature and see if you find Dedekind system to be more standard. Whether the criteria are due to Dedekind is worth noting for historical reasons but not really related to current standard usage — we all know how many things in math aren't named after the proper person. Revolver 07:04, 30 Sep 2004 (UTC)
No doubt you are right, I think that the structure is normally given an ad hoc name such as "numerical structure"; I don't think there is a standard name for it, but I think the name Peano system is doubly unfortunate, because, besides the historical point, one usually uses the term "structure" to talk about these things and not system.
FWIW, Google tells me that "Peano system" is not much used, 180 hits, the most common being for a Perl ORB on freshmeat, and most of the relevant links referrring to Peano'x axioms and not the number structure. ---- Charles Stewart 12:14, 30 Sep 2004 (UTC)
I can do a check of several set theory books over the next week or so, just as an initial look. I'm not sure how to interpret the google quote you give (180 hits) without comparison to an alternative (how many hits does "___" get?) Revolver 08:25, 1 Oct 2004 (UTC)
"Numerical structure" got about 1600 hits, but the very FIRST hit is some religious numerology babble, so I take these google hit-numbers with a few grains of salt. Revolver 08:31, 1 Oct 2004 (UTC)
`Meaning' of 180 hits: the structure is obviously very fundamental, so 180 hits is low, especially since Wikipedia has quite a few syndicates. By comparison "Hopf algebra" gets 12700 hits, despite being an obviously less fundamental concept. I think there is no standard way of talking about this structure; given this, it looks to me that we should call the structure something non-misleading. Dedekind structure, gets only one relevant hit, but it is for exactly this item, and it is what I would call it. ---- Charles Stewart 12:09, 1 Oct 2004 (UTC)
It may be more "fundamental" in the foundational sense, but my guess is that more people study Hopf algebras than Dedekind structures. There does seem to be no standard terminology. I googled "Peano structure" and found at least half a dozen relevant hits. I admit I have no real knowledge on the history of the situation. Since both "Dedekind" and "Peano" seem to be in use, why not use "Dedekind-Peano structure" (which I have seen), and make some comments about other terminology? Note: I have also seen "Dedekind-Peano axioms" for "Peano axioms". Revolver 01:30, 3 Oct 2004 (UTC)
I'd guess the structure of positive integers sees as much study and more application than Hopf Algebras; however everytime you see a Hopf algebra applied you will see the fact advertised using the standard name. I have no objection to Dedekind-Peano structure. I think the name Dedekind-Peano axioms is generally used for second-order Peano arithmetic, but saying "second-order Peano axioms" I think is more suggestive. Should we add a terminology section? ---- Charles Stewart 09:27, 3 Oct 2004 (UTC)

first order?

The article claims that "Peano axioms (or Peano postulates) are a set of first-order axioms" can someone please tell how is it possible to postulate: "if a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers." in a first order logic? i think you will have two use second order for that claim.(it seems to quantify over properties(definition of second order logic))

(post moved to bottom of page)
As an axiom, induction is second-order. However it can be formulated in a first-order manner by adding not a single axiom, but an axiom schema with infinitely many instances. Since the schema defines a(primitive) recursive set, from the point of view of proof theory, it is acceptable. ---- Charles Stewart 07:46, 26 Jan 2005 (UTC)
I see, for example we can get around induction by using no-cycles claim, (no size 1 cycle, no size 2 cycle..etc up to infinity).Still, In this case the text of the artcile should be changed to this particular(infinitary) axiomatization, because as it stands it is confusing as to why is it first-order.--Hq3473 08:02, 26 Jan 2005 (UTC)
If I'm not mistakened, forbidding cycles is both too weak and can't be captured by a first-order axiomatisation. I agree the text needs (much) improvement, but I guess the issue won't be clear until we give an actual axiomatisation in a Hilbert-style proof theory - then we can talk about this point properly. I'll put it on my overburdened to do list ---- Charles Stewart 10:22, 26 Jan 2005 (UTC)
i see, cycles will not rule out higher orders. Any way, can you maybe give me link or a refernce to riogorous first order peano axiomatization? i want to see how is the deduction overcome.--Hq3473 21:44, 26 Jan 2005 (UTC)
I agree with the original objection that the induction axiom is second order. The fact that it can be replaced by an infinite number of first order formulae does not change this statement. Having arbitrarily large sets of formulae is not usually meant when speaking of a "first order theory" and it is commonly said that the natural numbers are not first order axiomatizable (meaning "by a single formula (or a finite set of formulae)"). Allowing for infinite sets of first-order axioms gives you quite a lot of expressive power, e.g. you can describe the natural numbers and any subset thereof. I think it is misleading to speak of first-order in the intro, especially since the section on Peano arithmetic also speaks of "the restriction of Peano axioms to a first-order theory". --Markus Krötzsch 20:19, 15 November 2005 (UTC)
Peano arithmetic is definitely a first-order theory; this is completely standard usage. The fact that it's not finitely axiomatizable makes no difference. (Actually, "true arithmetic", the set of all true first-order statements about the natural numbers, is also a first-order theory; the difference is that it doesn't have a computable or even computably enumerable set of axioms.). The full Peano axioms are second order because you quantify over properties, not just over individual natural numbers. --Trovatore 20:28, 15 November 2005 (UTC)

Discardng induction?

I asked a vague question at Talk:Preintuitionism ... are there any examples of systems in which the first four axioms are kept, the fifth (induction) is intentionally broken or discarded, and yet are not finite, and somehow manage to achieve some sort of "infinity"? (By "intentionally broken", I mean, are there different set of axioms, in which induction is a theorem that may be proven false?) (By requiring "infinity", I want to exclude "obvious" systems like finite groups and fields (and finite state machines?) which seem like they might not need induction as an axiom). linas 05:50, 7 September 2005 (UTC)

Yes, this is an active area of study. If you limit the induction schema to formulas all of whose quantifiers are bounded, you get a theory called 0, which is insufficient to prove that the exponential function (the function that sends n to 2n) is total. (No warranties on the exact definition of 0). There's a whole hierarchy of theories weaker than PA. --Trovatore 20:49, 16 October 2005 (UTC)
Looks like I didn't read your question carefully enough; this isn't what you were after. But it still might be of interest to you.... --Trovatore 23:24, 16 October 2005 (UTC)

definition of PA

The most glaring deficiency of this article in its current form is that it does not give any explicit definition of first-order PA, which makes the introduction to the article seem almost a misrepresentation. PA is alluded to in the third paragraph of the "Metamathematical discussion" section (a section which has serious POV flaws as well), but an actual precise characterization of PA is never given. It should be mentioned that the language needs to include multiplication--this isn't necessary for the second-order Peano axioms, but it is for first-order PA; otherwise you get something much weaker, and I believe actually decidable. --Trovatore 20:56, 16 October 2005 (UTC)

We could borrow the treatment from PlanetMath. Presburger arithmetic is sort of PA without multiplication, and is indeed decidable. --- Charles Stewart 01:37, 17 October 2005 (UTC)

Hmmm... I think the Peano Induction axiom is necessarily a second order axiom, and there is a similar first order axiom schema, but which is slightly weaker. Perhaps the claim in the opening sentence that Peano's axioms are first order should be removed? -Lethe | [[User talk:Lethe|Talk]] 19:26, 15 November 2005 (UTC)

User:Jeekc has copied the PlanetMath material, which I've touched up a little bit. I think it's OK now. It's important that we get this right, because the initialism PA refers to the first-order theory and we probably talk about it all over the place in Wikipedia. --Trovatore 19:32, 15 November 2005 (UTC)
I suppose there should be an article Peano arithmetic and that article should use the first-order schema. -Lethe | [[User talk:Lethe|Talk]] 19:41, 15 November 2005 (UTC)
I want to look it up in a book, but I think the term "Peano's axioms" should and does normally include the stronger second order axiom. If that's the case, the recent changes here are not correct. -Lethe | [[User talk:Lethe|Talk]] 19:43, 15 November 2005 (UTC)
What do you mean? The recent changes are about "Peano arithmetic", not "Peano's axioms". --Trovatore 19:49, 15 November 2005 (UTC)
The article now claims that "Peano axioms are a set of first order axioms". I suggest that this is wrong: Peano axioms are a set of second order axioms. Peano arithmetic uses first order axioms, and therefore Peano arithmetic is not defined by the Peano axioms but rather by some other set of weaker axioms. I'm not positive about this, so I'd like to go check perhaps in EDM2. -Lethe | [[User talk:Lethe|Talk]] 20:26, 15 November 2005 (UTC)
PS what is going on with my signature?
Ah, right you are, that's bad. The article certainly shouldn't say that the Peano axioms are first order, and then list them as a second-order axiomatization. However it's not a recent change. It's been that way for at least a year. What we now have, that we didn't have before, is at least a list of the first-order axioms. --Trovatore 20:34, 15 November 2005 (UTC)
Oh, I see. JeekC added a new section on PA, and you were talking about that. Right then, I guess we're in agreement. -Lethe | [[User talk:Lethe|Talk]] 20:38, 15 November 2005 (UTC)

I've fixed the claims in the preamble, and added refs for Peano and Dedekind. --- Charles Stewart 17:24, 16 November 2005 (UTC)

there seems to be much confusion here about first order PA and second order peano axioms. The second order induction axiom allows for ARBITRARY properties, and thus it is NOT equivalent to a set of first order formulas. There are properties of natural numbers which are not expressable in terms of first order formulas, and there are nonstandard models of Peano arithmetics, while there are no nonstandard models for the second order axioms. Please do not mess up in the article and obscure this distinction. —Preceding unsigned comment added by (talkcontribs) 01:46, 19 November 2005
No one has been "messing up the article". It used to be much worse. Take a glance through the history. --Trovatore 01:57, 19 November 2005 (UTC)


The following part of the article badly needs to be reworded and cleared up:

But in 1931, Kurt Gödel in his celebrated second incompleteness theorem showed such a proof cannot exist. It is even impossible to prove consistency of Peano arithmetic while assuming the axioms themselves. Furthermore, we can never prove that any axiom system is consistent within the system itself, if it is at least as strong as Peano's axioms. In 1936, Gerhard Gentzen proved the consistency of Peano's axioms, using transfinite induction.
Most mathematicians assume that Peano arithmetic is consistent, although this relies on intuition only.

For a layman without a mathematical background, this basically reads "It's impossible to prove the consistency of PA. Gentzen proved the consistency of PA in 1936. It's impossible to prove the consistency of PA, but we assume it anyway", which of course is highly problematic. -- Schnee (cheeks clone) 21:55, 26 October 2005 (UTC)

0 and 1 do not need to be posited separately

To the anonymous user who tried to put this in, it is not needed. You can write to me about why this is true, but don't try to edit it this way again. The form of Peano's axioms is a matter of history and mathematical practice.

1 is defined as S(0) (S is a primitive notion; it is not defined as "adding 1".) Your counterexample is not a counterexample: the structure 0, pi, 2*pi, etc (non-negative integer multiples of pi in the usual real number system) is a perfectly good Peano-Dedekind structure: the "1" of this structure is the usual pi, and "adding 1" in this structure is adding pi in the subset of the real numbers which is its domain.

Randall Holmes 23:01, 25 December 2005 (UTC)

historical question about 0 vs. 1, quite distinct from the previous issue

Did Peano actually have 0 as the first natural number? The axioms are also sometimes presented with 1 as first, and I suspect this may be historically the original form. Of course, both are fine mathematically -- but one defines addition and multiplication differently in each case. I'm not going to edit the article on this point without historical sources. Randall Holmes 05:58, 27 December 2005 (UTC)

I suggest the article on Zermelo set theory as a model: someone has taken the trouble there to go back to the original text, and what he finds is rather interesting. I'm pretty sure that here the original text will bear out Lethe's claim that the original axiom set was second-order, but who knows what else we might find? Randall Holmes 06:00, 27 December 2005 (UTC)

undid incorrect changes to PA axiom set.

I reversed the changes to the PA axiom set, which were incoherent, and took the opportunity to eliminate the (older) references to <, which is not a primitive notion of PA (it is definable). Randall Holmes 06:29, 16 January 2006 (UTC)

You say the changes were inconsitent, however, the five axiom system descibed is the original one, from which other theorems can be derived. Also note that the layout of the theorems IS important in Peano arithmatic, despite many peoples' objections to it. —Preceding unsigned comment added by Evildictaitor (talkcontribs) 18:00, January 16, 2006 (UTC)
The list of axioms you give is not the original one. Moreover, I'm a professional mathematical logician; you aren't going to get anywhere lecturing me on what is correct and what isn't, because I know... Hofstadter is not an Authority, merely a popularizer. Randall Holmes 17:30, 17 January 2006 (UTC)
Forall a and b, it must be written seperately, and cannot be combined. Forall cannot be moved within the string, and parenthesis are not transmutable, and are implicitly required.
is a badly formed string, whereas is a valid one.
Also note that some "axioms" have been removed because they are not axioms at all, but theorems. The "axiom" is proovable within the system, and therefore can be used as one would use an axiom, but because it is a theorem of the system, not because it is an axiom. The five axioms are as follows:
Let us firstly state that all numbers are positive integers, such that
Let us next state that addition of zero to a number is equal to that first number, such that
Let us also say that succession of one number is tranmutable between elements being suceeded, such that,
Let us now also say any number multiplied by zero (including itself) is equal to zero.
Let us also say that multiplication is cummulative addition, such that
—Preceding unsigned comment added by Evildictaitor (talkcontribs) 18:00, January 16, 2006 (UTC)

The typographical details do not matter (many different conventions on notation are possible). This said, I have no particular objections to your notational changes. The axiom is essential: you cannot prove it from the other axioms. The notation is generally understood and perfectly correct, by the way. Randall Holmes 17:30, 17 January 2006 (UTC)
I think the colons are a bit strange; I don't recall seeing those in any standard text. Usually one uses parentheses, either around each quantifier or around the matrix (or both, but I think that's usually excessive). I seem to recall that Hofstatder uses the colons. --Trovatore 19:33, 17 January 2006 (UTC)
Right, something like for is common. The notation changes suggested are useless and confusing. Gew75 03:19, 25 January 2006 (UTC)
No, I didn't mean the (x) notation, which looks more like Russell-Frege era stuff; it's not used much these days, at least in mathematics (I think some philosophers may still use it). I meant you can write either
--Trovatore 03:25, 25 January 2006 (UTC)

Reverted all changes by Evildictaitor; will continue to do this as necessary

It's all in the title, really. Randall Holmes 17:45, 17 January 2006 (UTC)

Hofstadter is not the source for Peano arithmetic, and his choices of notation are somewhat unusual. Your modifications to the informal description of the axioms were entirely inappropriate. You do not understand what the usual sets of axioms are (hint: Hofstadter is a popular writer, not an authority). I will revert your edits as necessary. I always watch this page. Randall Holmes 17:45, 17 January 2006 (UTC)

I'm perfectly happy to talk here about the reasons for all this, but don't make ill-informed changes to the main article (or at least, don't expect them to stay there). Randall Holmes 17:52, 17 January 2006 (UTC)

Circular Reasoning?

These axioms seem (to me) to have a chicken-and-egg problem, so to speak. You can't tick off how many times you applied recursion [ f(f(f(..))) ] until you have integers to do it with. Seems to me you have to go back to pebbles or the like and construct examples first. 21:40, 6 February 2006 (UTC) sorry I was logged out - the foregoing on circular reasoning was from Carrionluggage 21:45, 6 February 2006 (UTC)

So it might help if you tried to elucidate just what proposition it is that you think is being demonstrated circularly. If you simply mean that the presentation of the Peano axioms is more about showing you how your intuitive idea of natural numbers is formalized than it is about telling you what natural numbers "really are", well, of course you're right. Hope this isn't bad news, but you understood what a natural number is as well when you were ten years old, as you're ever going to. Answering that question for you is not the job of a formalization.
Still, the section you're referring to is undoubtedly written in a confusing fashion at best; unless someone wants to sepcify a little better what's meant by saying that the Peano axioms are "summed up" in the diagram with the f's, then that passage should be deleted. --Trovatore 22:20, 6 February 2006 (UTC)

Thanks for the reply. What I meant, in more detal is illustrated by two examples: It is written : "where each of the iterates f(x), f(f(x)), f(f(f(x))), ... of x under f are distinct. "

Now, that looks clever, but if you do not have a way to count up the nesting that was just presented, you have accomplished nothing (as I see it). I suggest you might tell the reader just to put little pebbles or beans in a container instead. Of course you can't get to infinity, but people can get the idea that if they could sit there forever with an unlimited supply of beans, their container could eventually exceed any "target" weight. In other words, what is the value added of that expression above with all the parentheses, and how does it compete with counting beans?

Next case:

It is written: "so that

  • 0 := {}
  • 1 := S(0) = {0}
  • 2 := S(1) = {0,1} = {0, {0}}
  • 3 := S(2) = {0,1,2} = {0, {0}, {0, {0}}}

and so on. This construction is due to John von Neumann."

Once again, it seems to me that the question is begged (chicken-egg problem), because you can't count up the lines displayed (or, again, the curly brackets) unless you know what the integers are. Is this a nutty complaint? I always had a similar problem with group isomorphisms - so it is well that I did not continue in math past a course using van der Waerden and some real and complex variables. In the latter case, when you have a group with a lot of isomorphic subgroups (as I recall) you can (usually?) indice a permutation among them - say a cyclic permutation. But after this permutation, the subgroups look the same as before, so how does anyone know you permuted them amongst each other? Well, not to bother if this is not interesting. Thanks Carrionluggage 19:08, 7 February 2006 (UTC)

It's not a nutty complaint at all, just an insoluble one. What it indicates is that you want more from formal theories than they're able to deliver. The point of the von Neumann construction is not to tell you what the naturals are, just how to code them into set theory, so that the methods of set theory can then be applied to them. As I said, you already know what the naturals are, as well as you're ever going to (for that matter, as well as anyone else is ever going to). --Trovatore 19:17, 7 February 2006 (UTC)

Thanks for explaining. Carrionluggage 05:33, 10 February 2006 (UTC)

Dead Link to Gentzen's Proof

This link in the bottom group on the article page: [1] would not work for me at this time - perhaps a bad omen :-) Carrionluggage 21:38, 10 February 2006 (UTC)


Perhaps commutativity, associativity and distributivity should be proved here? This would show the axioms in action as it were.

Formalization of induction

The article uses , but this doesn't seem a correct formulation. For example, let denote the statement . Then consider the case of . , so the statement is false.

I would formalize induction instead as . 06:05, 16 April 2006 (UTC)

No history here? No zero?

I was hoping to find something about any relationship between Dedekind and Peano, and Peano's "use" of Dedekind's postulates/axioms, along this line:

"Peano acknowledges (1891b, p.93) that his axioms came from Dedekind (1888, art. 71, definition of a simply infinite system; see also below, pp. 100-101)" (van Heijenoort, p. 84)

(Page 100-101 contains Dedekind's expression of his axioms in a defense against the criticisms of Keferstein.) Maybe there's nothing more to be said ... was there any consequential rancor? Why are they called "the Peano axioms? rather than the "Dedekind axioms?"

Also for me it was an eye-opener when I read the axioms in van Heijenoort, that the "real", "true" Peano axioms do not define "zero" except as #8 which I read as "no number exists for which the unit is its successor". Am I reading this right? Someone else above raised this point. Thanks, wvbaileyWvbailey 14:40, 11 September 2006 (UTC)

By the way, I found this to be is a nicely-written, erudite article.wvbaileyWvbailey 14:43, 11 September 2006 (UTC)

I am not familiar with the history, except that many older works began the natural numbers with one rather than with zero. But zero is the standard starting point today. This makes sense logically because if a natural number is the number of elements which can be in a finite set (or pebbles in a basket), then zero is needed to describe an empty set (or empty basket). Back in the old days, an empty basket would not even be considered as having a number of pebbles. It just did not occur to them to ask the question. JRSpriggs 04:22, 12 September 2006 (UTC)
Anybody out there know how/when "zero" crept into the axioms? Was this von Neumann up to his tricks? (I'm familiar with his set-theoretic notion of "an empty box as 'the unit'", at least indirectly through Halmos, Naive Set Theory.) Lemme know, 'twould be interesting to add a small history section to this article. Thanks, wvbaileyWvbailey 13:48, 12 September 2006 (UTC)
I just peeked into van Heijenoort and discovered von Nemann (1923) there on p. 346 ff. He defines "the null set" (how on earth can something have a name "the null basket" -- the non-basket -- and at the same time truly represent/be nothingness?) as O and the unit as (O). As I thumb backwards I see in Lowenheim's The Calculus of Relatives what looks a bit like von Neumann's 0 (Lowenheim's ordered pair 1ij = (i, j), 0 = ~1. And he is fiddling around with Schröder and he prints out Müller's axioms that look suspiciously like here is where the 0 got into the game (cf page 240 in van Heijenoort). Anybody know any details? Thanks, wvbaileyWvbailey 14:05, 12 September 2006 (UTC)
The link to natural number was a good idea, but it looks like Ernst Schroder and his works that Eugen Muller codified (1909, 1910) precedes Whitehead and Russell's P.M. (cf commentary in van Heijenoort p. 231).wvbaileyWvbailey 14:29, 12 September 2006 (UTC)

The historical sources

We have a pretty good department library; it is cleary better than my understanding of Latin and Italian is. Within our library I found three volumes of Opere Scelte of Peano, in an edition from the 1950:s; and therin, in vol. II, pp. 20-55, the full Arithmetices principia nova methodo exposita (in Latin). This is not the 1889 printing; but I assume that there are no essential changes. From this text I deduce the following consequences:

  1. In the second paragraph, where Peano's original paper and nine axioms are briefly discussed, we have to state that he starts from one (not, as it now is written, zero).
  2. At the beginning of the 'informal' exposition of the five axioms, we should mention in so many words that there is some modernisation in notation and by starting from zero, but this in no way changes the essential properties of his approach.
  3. It would be reasonable to insert a sentence somewhere, stating that his work both was influenced by and influenced his contemporaries, and that he especially mentions a book by Grassmann and the paper by Dedekind (vide infra).JoergenB

I'll try to give anyone with a deviating opinion an as fair chance as possible to decide it for h*rself, by quoting the Latin text and giving some of my interpretation of it. Real Latin experts are welcome to give better translations. If you want me to quote some other part, I'll gladly do so. Unhappily, while the original text from 1889 should be free now, I' don't think this holds for this Italian edition; otherwise, scanning it into Wikisource would be an option, if there were sufficient interest.

From p. 22, at the end of the section titled PRAEFATIO, Peano mentions Cantor, Boole, H. Grassmann, and R. Dedekind. The third and second last paragraphs read:

In arithmetica demonstrationibus usus sum libro: H. Grassmann, Lehrbuch der Arithmetik, Berlin 1861.
Utilius quoque mihi fuit recens scriptum: R. Dedekind, Was sind und was sollen die Zahhlen, Braunschweig, 1888, in quo quaestiones, quae ad numerorum fundamenta pertinent, acute examinantur.

I suppose this means approximately (I really don't understand e.g. that form 'usus'):

In proofs of arithmetic I employ the book: H. Grassmann <title> . Also useful for me was the recent paper: R. Dedekind <title>, in which questions, which fundamentally concern numbers, are sharpely investigated.

On p. 23, under the heading SIGNORUM TABULA, Peano inter alia gives the 'Signum' the 'Significatio' non, and writes

Signa 1 , 2 , ... , = , > , <, +, -, vulgarent vulgarem habent significationem.

i.e., 'The signs... have their common meanings'. Under the subheading 'Signa composita' he writes e.g.

< non est minor ('is not less than')

On the next pages he (inter alia) explains that points are used for grouping, in a similar manner (?) as parentheses in algebra; and that a reverted capital C signifies deducitur, essentially 'implies'. Since I think I lack a more appropriate symbol, I represent it by ⊃.

On p. 34, under the main heading ARITHMETICES PRINCIPIA and subheading § 1. De numeris et de additione. he gives the 9 axioms, preceeded by:

Signo N significatur numerus (integer positivus).
Signo 1 significatur unitas.
Signo a + 1 significatur sequens a , sive a plus 1.
Signo = significatur est aequalis. Hoc ut novum signum considerandum est, etsi logicae signi figuram habeat.

As axiom 8 he states:

a N . ⊃ . a+1 = 1 .

This really should mean: For any natural number, its successor is different from 1 (one). Moreover, axiom 9 is the axion of induction, starting the induction at 1. Actually, he continues by constructing positive rationals (which he denotes R) and positive reals (denoted Q; sic); the latter construction seems related to Dedekind's.--JoergenB 17:31, 18 January 2007 (UTC)

usus sum means "I have used" or "I used". (perfect tense of "utor". Remember utor fruor fungor potior vescor?).
I hope you don't mind that I have replaced your symbol Ə by ⊃ in your text, since this is the usual (old-fashioned) way of writing "implies".
quae ad numerorum fundamenta pertinent means "that concern (or pertain to) the fundaments/foundations/fundamentals of numbers".
acute is probably "precisely" rather than "sharply".
I am not sure about vulgarent, perhaps this should be vulgare? --Aleph4 18:27, 18 January 2007 (UTC)
Thanks for the improvements; and you are oviously more than good enough to spot my typos :-) Peano wrote vulgarem, which I interpreted as a congruent attribute to significationem. (In this instance, since I had the correct text to look at, I had an unfair advantage:-) And yes, ⊃ is not exactly the symbol in this Italian edition; but it is much closer. I simply looked at the listed symbols near the editing window, since I do not know all HTML escape sequences.
Do you essentially agree with my three conclusions (vide supra)?--JoergenB 18:46, 18 January 2007 (UTC)
Concerning utor fruor fungor potior vescor: I've never had the pleasure or pain of studying Latin in school; i've gone to some low intensive evening Latin courses, and I've tried to learn it myself. Thus, I never had any mnemotechnical verses to learn (or forget). Are you saying that the enumerated verbs are deponent (i.e., active to content but passive in form)? (Speaking of Latin in school, did you incidently ever see Life of Bryan?)--JoergenB 18:58, 18 January 2007 (UTC)
yes, "vulgarem" is probably an attribute to "significationem". Yes, I essentially agree (though I think the article needs some rewriting). No, there are many more deponentia than only "utor fruor etc"; these are the verbs (or most common ones) that take the ablative (utor libro, not librum) where you would expect the accusative. Of course, Romanes eunt domus. --Aleph4 17:23, 20 January 2007 (UTC)
I am the person who felt obliged to add the text:
Peano's original axioms (1889) are preceded with the definitions:
  • "The sign N means number (positive integer).
  • "The sign 1 means unity" (italics in original, van Heijenoort (1976) p. 94)

His first axiom is "1 ε N" (ibid), the ε signifying "is an element/member of". No mention is made of the sign "0" (also cf commentary by van Heijenoort p. 83, and Dedekind's Letter to Keferstein" (1890) p.100).

My suggestion is edit the article as you see fit. If "the academy" disagrees you can be sure it will let you know. What disturbs me a bit is that we do not know why -- and so we loose the narrative thread to explain why -- Dedekind and Peano chose to start their axioms from the sign "1", nor do we know how and why the sign "0" aka zero crept in. A side-note: we can build a counter-machine (register machine) per the Peano axioms, without "zero", because "equality" between counts in registers is used as the conditional operation. What to do about "zero?" We do need "an origin". At the outset we define one register -- I always call it register "0" -- to contain ... well, no counts. It may be better to think of this per the Minsky (ca 1958) convention (aka Minsky machine). We think of "the continuum" beginning at a single mark at the left end and every positive integer a square to its right; Minsky considered a register a left-ended Turing tape with a single mark " | " in the square to the immediate right of the left end. The machine need never print thereafter -- movement left (considered decrement) must be preceded by a test to see if the mark is there; movement right is considered increment. To turn this "Minsky-convention machine" into a "Peano-machine" we dispense with left motion and allow the machine to print a mark to act as a new origin (aka "zero"). Thus we only have "increment register" (i.e. move one square right), "test two registers for equality", and "set new origin for register" (i.e. print a mark). wvbaileyWvbailey 20:23, 18 January 2007 (UTC)
All right, I'll try an edit.
As to the reasons for the original formulation: Both Dedekind and Peano continue by constructing positive rational numbers by means of pairs of positive integers as the next step. By not allowing zero, they need not exclude it as demoninator. This could well be their reason, or at least part of it. One reason for the later inclusion of zero could be the set-theoretical approach, where 'natural numbers' may be identified as the finite cardinal numbers. The cardinal number of the empty set is zero. (Note that I know the mathematical facts; but I'm just guessing how they may have influenced the choice to include or exclude zero.)
Actually, Peano's definition of the ε sign doesn't seem to be exactly one of membership. I suspect that he does not clearly distinguish what we now would call 'sets' and 'predicates'; but this is probably not important. He writes that ε stands for est (i.e., is); but he uses the sign in a very 'modern' manner.--JoergenB 21:56, 19 January 2007 (UTC)

Formalized atoms?

The subsection Formalized atoms appeared recently, with no prose and an unclear comment in the page source code. I don't think it belongs here at all, but if it does it ought to be expanded somehow (the induction axiom is particularly tricky to formalize...). The entire point of the "Peano axioms" is that they are informally stated; the formalized version is Peano arithmetic.

In fact, the entire section The axioms seems to repeat itself three times. And it is certainly false that Dedekind defined a "Dedekind-Peano structure" in 1888. Any objections to a rewrite of that section? CMummert 03:26, 3 January 2007 (UTC)

No objection here. I TeX'd it up, but anything you can do to improve it would be welcome. CRGreathouse (t | c) 05:01, 4 January 2007 (UTC)

References Question

Why are the references indented as they are. As a non-mathematician, not familiar with this topic, it appears to me that something is mixed up with this indentation. I believe that Wikipedia works best when it is clear to the non-specialist. Thanks. Nwbeeson 15:40, 3 January 2007 (UTC)

If you read the wording of the references, you'll see the reason for the indentation. However, the vertical spacing was inconsistent, which made the lay-out somewhat unclear. I've fixed it now. --Zundark 16:01, 3 January 2007 (UTC)

First section

I see that several people are editing this article, so let me point out the issues I see with the first section. I have been meaning to fix it, but I have to stop by the library first.

  1. It is worth listing Peano's original 5 axioms explicity, numbered as he did (which is why I have to stop at the library).
  2. The part about a "Dedekind-Peano structure" attributes modern concepts to Dedekind (1888), which is certainly inaccurate. Although it is accurate mathematically, it is not accurate historically and does not match the actual practice in contemporary mathematical logic, which treats PA as a first or second order theory in the usual way.
  3. The part about "Formalized axioms" belongs further down in the discussion of Peano arithmetic.
  4. The section on "Existence and uniqueness" should incorporate the standard second-order categoricity proof due to Dedekind.
  5. The part about the categorical interpreation should find a suitable book on topoi to list as a reference. Probably "Sheaves in Geometry and Logic" would work.
  6. The article on Hilbert's second problem is much improved, and the discussion of consistency should point to it. The entire metamatheamtics section will need to be cleaned up once the statement of the axioms is cleaned up.

So the general outline would then be:

  1. Peano's original axioms
  2. Binary operations and ordering
  3. Existence and uniqueness of the natural numbers
  4. Peano arithmetic
  5. Categorical interpretation
  6. Metamathematical discussion
  7. References

I'm planning to make these changes some time next week. CMummert · talk 17:40, 20 January 2007 (UTC)

Note that you cannot give Peanos original 5 axiom; originally, they were 9 :-)--JoergenB 22:39, 20 January 2007 (UTC)
I haven't looked at Peano's article yet. This article as it stands claims there are 4 axioms for equality and 5 for arithmetic. Presenting all 9 would be great by me. Right now there are 5 bullets, which I assume roughly correspond to Peano's original five axioms for arithmetic. But I am only familiar with Peano Arithmetic, so I have to look at the original source to confirm. CMummert · talk 23:39, 20 January 2007 (UTC)
I hope you are not going to take out the modernized view altogether. Where does it fit in your outline? JRSpriggs 13:15, 21 January 2007 (UTC)
I promise that the contemporary viewpoit will be reflected in whatever I do. CMummert · talk 13:17, 21 January 2007 (UTC)

The following from van Heijenoort pp. 83 ff. demonstrates the difficulty of the presentation of "the axioms" from a historical point of view. Whitehead and Russell evolved the symbols into PM, and we all remember how nasty that was to read as a first-time or casual reader. This is just as bad or even worse ( symbol Э should be a backwards C; observe that he uses it in two senses, just to add to the confusion).

"The sign &sup means one deduces [deducitur]2; [2Peano reads a &sup b "ab a deducitur b". Translated word for word, this either would be awkward ("from a is deduced b" or would reverse the relative positions of a and b ... Peano himself uses "on deduit" for "deducitur when writing in French... and this led to the translation adopted here.]" (p. 87).

"The sign K means class, or aggregate of objects.

The sign ε means is. Thus a ε b is read a is b; a ε K means a is a class; a ε P means a is a proposition.

... the sign -ε means is not.

The sign &sup means is contained in. Thus a ⊃ b means class a is contained in class b." (p. 89)


The sign N means number (positive integer).

The sign 1 means unity.

The sign a + 1 means the successor of a, or a plus 1.

The sign = means is equal to. We consider this sign as new, although it has the form of a sign of logic.

1. 1 ε N.

2. a ε N .⊃. a = a.

3. a, b ε N .⊃: a = b .=. b = a.

4. a, b, c ε N .⊃:. a = b.b. = c :⊃. a = c.

5. a = b.b ε N :⊃. a ε N.

6. a ε N .⊃. a + 1 ε N.

7. a, b ε N .⊃: a = b .=. a + 1 = b + 1.

8. a ε N .⊃. a + 1 - = 1.

9. k ε K :. 1 ε k :. x ε N.x ε k :⊃x .x+1 ε k ::⊃.N ⊃ k." (p. 94)

[The dots are used as parenthesis, as are the colons, triple dots :. and quadruple dots. What is in a sense more interesting are the recursive definitions 10. for "the numbers", 18 for addition, then subtraction, multiplication, powers, and division e.g.

"10. 2 = 1 + 1; 3 = 2 + 1, 4 = 3 + 1; and so forth." (94)

"18. a, b ⊃ N .Э. a + (b + 1) = (a + b) + 1" (p. 95)

Relative to 0, there is no a x 0, and no a0. The recursion for powers starts at a1 = a. This business of why no 0 is still bugging me. As noted above, for sure it would have complicated division, and 0n and 00 consequently become problems. We really need an answer to this question to keep the narrative coherent (i.e. the casual reader will raise the question, and avoiding an answer makes the article looks like handwaving. Better to say ... "not known" ... rather than avoiding the issue).

One approach is take a photo of a page of the axioms and insert it as a picture, as someone did for PM... am not sure how to do that technically. Anybody out there know? (use a digital picture, I suppose ... maybe the scanner can "save as" a photo). Then add a caption (I'll experiment. I don't have a cc of the original but maybe the college up the street does). As suggested above scanning the whole book is a good idea. wvbaileyWvbailey 18:04, 21 January 2007 (UTC)

aleph4 replaced my original try to represent the 'backwards C' by ⊃; I think that was a good choice; and I recommend the same supra. Peano seems to use ⊃ both for 'includes as class' (i.e., set), and for 'implies'. I am not sure that he considers this as fundamentally different uses; as far (not very far) I understood the Latin text, he does not wish to make an essential difference between a predicate and the class of objects satisfying it. Thus, I do not think he even wants to distinguish the interpretations "1 has the property of being a natural number" and "1 belongs to the class of natural numbers" of axiom 1. But, as I said, he explains these things in Latin in the pages about logic that preceeds the 9 axioms; and my Latin is not very strong. [User:Wvbailey|Wvbailey]], does your English version contain a translation of the full text (including the preceeding pages on logic and functions)?
It is not the full text but does have the following, in particular the "Preface" where all the logic is: Preface, Logical Notations I. Punctuation, II. Propositions, III. Propositions of Logic, IV. Classes, V. Inversion, VI. Functions. § Numbers and Addition. § 2 Subraction, §3 [missing], §4 Multiplication, §5 Powers, §6 Division.
The following is van Heijenoort (3rd printing 1976) reporting exactly what is missing. He worked from the Italian (vol. 2):
"Arithmetices principia consists of a long explanatory preface and ten sections: §1 Number and addition, §2 Subtraction, §3 Maxima and minima, §4 Multiplication, §5 Powers, §6 Division, §7 Various theorems, §8 Ratios of numbers, §9 Systems of rationals, §10 Systems of quantities. Below we print the preface and §1 in extenso; from §§2, 4, 5, 6 we give the "explanations" and "definitions", omitting the theorems, and we leave out the other sections entirely. The omitted parts consist almost exclusively of formulas and are readily available in Peano's collected works (1958).
Peano, Giuseppe, Opere scelte (Ediziioni cremonese, Rome), vol 1. (1957), vol. 2. (1958), vol. 3. (1959).
If you'd like I could copy/convert it and e-mail to you as a .pdf. I have a copier right by my side. I will also check right now to see if Dartmouth has a copy in the library. wvbaileyWvbailey 16:09, 22 January 2007 (UTC)
A photocopy of the original printing (the 1889 Latin edition) of the page(s) with the 9 axioms would be a rather nice and appropriate illustration to the text; and there could be no copyright issue over that edition. We don't seem to have it at the University of Stockholm; but I might try the Mittag-Leffler Institute, if none of you others have it easily accessible.
Another 'advantage' or at least feature with starting from 1 is that "a + 1" may be used to represent the successor, without introducing a 'redundant' symbol for the successor function. Again, in some modern approaches, this would rather be seen as a disadvantage, since then the sign + formally is equipped with two different meanings. To judge from the whole spirit of Peano's article (as filtered through my difficulty with the language), Peano was aware of these things, but preferred to use 'overlayered meanings', to borrow terminology from C++. Indeed, it is in the same spirit that we identify the natural numbers as a subset of the positive or of the non-negative rational numbers, or the rational numbers as a subset of the real numbers, without making distinctions e.g. between the integer 5 and the rational number 5/1 (although they customarily first are formally defined as different entities).
Anyhow, this is still guesswork; I'll try asking experts and see if I can find someone actually has studied the historical development.--JoergenB 12:44, 22 January 2007 (UTC)
Wvbailey, I've now discussed '1' versus '0' with a colleague who does know about these things. He does not remember exactly when 1 was first replaced with 0, but strongly recommends checking the various versions in van Heijenoort; since as far as he remembered, Kleene or Gödel used 0; but he suspected that Hilbert and Poincaré used 1. If you still have the volume at hand, could you please check? He also had opinions on why the change was made; but it would be better to see if indeed we can pinpoint the first 'switcher'. That person may have motivated the change, at least in the preface.
My colleague confirmed that Boole consciously made his operations possible to interpret both for logical statements and for classes; and actually Peano mentioned Boole too, in his preface. This at least makes it worth while to check if Peano explicitly states something similar somewhere.
As to why Peano (and Dedekind) started with 1, one sufficient reason could be that in general only positive integers were considered as natural numbers in these days – as in fact still is the case for some mathematicians, especially perhaps in number theory or in the USA. I suspect that this is a smaller problen than finding out how and why 1 was replaced by 0. Anyhow, all this shouldn't merit more than a couple of lines in the article. However, it would be rather nice to include a sentence like 'The first one to let the axioms start with 0 instead of 1 was NNNN in AAAA (YEAR), who motivated the change by XXXX'. (If the first person did have a motivation and is found in From Frege to Gödel, the I do not think quoting this could be accused as being 'original research'.)--JoergenB 16:30, 22 January 2007 (UTC)

Dartmouth does not have the cc of the original Peano. But it does show this in the on-line card catalog:

Selected works of Giuseppe Peano. Translated and edited, with a biographical sketch and bibliography, by Hubert C. Kennedy
Imprint [Toronto] University of Toronto Press [1973]
Description xi, 249 p. 24 cm

I have made a cc of the van Heinjenoort article in .pdf form. If you or CMummert or anyone would like it e-mailed lemme know.

I will check through van Heijenoort. I did this a while back and first encountered something in von Neumann's set theory. But I will check more carefully.

There is another place to look: Principia Mathematica. I do not have a cc of the entire set, only the first volume up to *56. Peano is mentioned after *50. Tracing backwards through my cc, I see that "0" and "2" are defined in *54 Cardinal Couples. And there is some very interesting stuff in *51 that looks just like the Peano axioms.

In a nutshell PM defines 1 in terms of the unit class first, hints at 0, defines 2. Then defines 0 as a different entity not derived from 1 or 2.

[In *24, V is defined as the universal class, Λ as the null class, "the class with no members".]

SECTION A UNIT CLASSES AND COUPLES: Summary of Section A (p. 329-330):

"We next introduce a very important notation, due to Peano [italics added], for the class whose only member is x.... Peano uses the notation "ιx" for the class whose only member is x; we shall alter this to "ι'x", following our general notation for descriptive functions.
"Classs of the form ι'x are called unit classes, and the class of all such classes is called 1. This is the cardinal number 1, according to the definition of cardinal numbers which will be given in *100. The properties of 1, so far as they do not depend upon other cardinals, or upon the fact that 1 is a cardinal, will be studied in *52.
"After a number (*53) containing propostions involving 1 or ι, we pass to the consideration of cardinal couples (*54) and ordinal couples (*55). A cardinal couple is a class ι'x U ι'y, where x ≠ y. The class of such couples is defined as 2, and will be shown at a later stage (*101) to be a cardinal number.
  • 51 Unit Classes:
"In this number we introduce a new descriptive function ι'x meaning "the class of terms which are identical with x,", which is the same thing as 'the class whose only member is x....
"the distinction between x and ι'x is one of the merits of Peano's symbolic logic, as well as of Frege's." [boldface added]
"*51.4 ├ .: ∃ ! α . α ⊃ ι'x . ≡ . α = ι'x
"I.e. an existent class contained in a unit class must be identical with the unit class. From this proposition it will follow that 0 is the only cardinal which is less than 1." (p. 339)
"*51.401 ├ .: α ⊃ ι'x . ≡ : α = Λ. v. α = ι'x
[A proof here...]
This proposition shows that unit classes are the smallest existent classes.
"*51.41 ├ : α ⊃ ι'x U ι'y = ι'x U ι'z . ≡ . y = z "
[Sure looks like a Peano axiom to me ... there are more: cf *51.42, *51.421, *51.43]
  • 52 The cardinal number 1:
"In this number, we introduce the cardinal number 1, defined as the class of all unit classes. The fact that 1 is so defined is a cardinal number is not relevant at present, and cannot of course be proved until "cardinal number" has been defined....
"The properties of 1 to be proved in the present number are what we may call logical as opposed to arithmetical properties, i.e. they are not concerned with the arithmetical operations (addition, etc) which can be performed with 1, but with the relations of 1 to unit classes. The arithmetical properties of 1 will be considered later, in Part III."
  • 52.16 ├ :. α ε 1 . ≡ : ∃ ! α : x, y = ε α. ⊃x,y . x = y
"I.e. α is a unit class if, and only if, it is not null, and all its members are identical.
  • 52.4 ├ :. α ε 1 U ι'Λ . ≡ : x, y = ε α. ⊃x,y . x = y
"We shall define 0 as ι'Λ. Thus the above proposition states that a class has one member or none when, and only when, all its members are identical."

There is no *53.

  • 54 Cardinal Couples:
"We introduce here the cardinal number 0, defined as ι'Λ. That 0 so defined is a cardinal number will be proved at a later stage; for the present, we postpone the proof that 0 so defined has the arithmetical properties of zero." (p. 357)
  • 54.01 0 = ι'Λ Df.
  • 55 Ordinal Couples:
"Ordinal couples... are much more important, even in cardinal arithmetic, than cardinal couples ...they are the smallest existent relations, just as unit classes are the smallest existent classes.
  • 56 The Ordinal Number 2r

So it looks to me like this: the entire body of PM up to *50 allows the definition of "the unit class". Then comes the number "2" built on 1. The sign "0" is considered a different beast altogether, and is "just defined" as *54.01 0 = ι'Λ , period. Then it is shown to have certain properties, later. This seems much different than von Neumann set theory, which starts from 0 = { }. So the matter of 0 and Peano axioms may be more complicated than first thought. wvbaileyWvbailey 18:58, 22 January 2007 (UTC)

The following is a survey of Peano Axiom information -- the papers from van Heijenoort -- to see how the idea of "0" came to find itself in the axioms. I have no beatiful quotes that can move this out of the realm of "OR". I'm sure there are some out there, somewhere. Incidentally I've quoted some rather elegant ways the "Peano axioms" have been put into words rather than arcane and off-putting mathematical symbols (hint, hint ...). What I discovered is:

  • (i) That there were two approaches to the "foundations of mathematics" development -- the logicists/formalists start from 1 (i.e. foundations developed from the logic point of view -- Dedekind and Peano, Russell, early Hilbert) while the set-theoretics start from 0 (e.g. Zermelo and Von Neumann) (and this is because of the next bullet point: Cantor):
"One response to the challenge [of the Burali-Forti and Russell paradoxes] was Russell's theory of types... Another coming at almost the same time, was Zermelo's axiomatization of set theory. The two responses are extremely different..." (p. 199, van heijenoort's comments)
"The most comprehensive formal systems that have been set up hitherto are the system of Principia Mathematica (PM) on the one had and the Zermelo-Fraenkel axiom of set theory (further developed by J. von Neumann) on the other." (Godel 1931, p. 596)
  • (ii) The considerations of Cantor etc. drove the set-theoretic point of view because they (Cantor, etc) needed in particular to investigate the possibilities of putting the real numbers from 0 to 1 in one-to-one correspondence with "the continuum" from 1 to infinity. Cantor wastes no time adding 0 to the continuum (see below);
  • (iii) Most of the "later" mathematicians start from 0 -- even Hilbert, starting in 1927, goes over to the "0-side". However, a few more historically-enlightened folk such as Hodges mentions that the axioms start from 1.

What surprises is how Cantor adds 0 to his notion of Ω to produce the notion Ω'.

Cantor (1899), Letter to Dedekind: He adds 0 to his Ω to form Ω'. But he seems to consider "0" not a number per se.

"I now consider the system of all numbers and denote it by Ω.... a simply ordered system ... Hence when the system Ω, when naturally ordered by magnitude, forms a sequence.
"If we then add 0 to this sequence as an element – putting it first, of course – we now have the sequence Ω’,
"0, 1, 2, 3, . . ., ω0, ω0 + 1, . . ., γ, . . .
"in which, as we can readily see, ‘’every’’ number is the ‘’type’’ of the ‘’sequence of all elements’’ preceding it (including 0)....
"Certain numbers of the system Ω form, each one by itself, a number class; they are the finite numbers, 1, 2, 3, .... v,..."(p. 115)

Hilbert (1904), On the foundations of logic and arithmetic:

”We take as a basis of our considerations a first thought-object, 1 (one)....
”We now add a second simple thought-object and denote it by the sign = (equals).” (p. 131)

He then uses these to state the 5 axioms (p. 132 - 133).

Russell (1908), Mathematical logic as based on the theory of types:

See the definition of 0 from PM; this has the same definition for 0 (cf p. 178).

Zermelo (1908), Investigations in the Foundations of set theory:

“An object b may be called a se’ if and – with a single exception (Axiom II) – only if it contains another object, a, as an element....
“AXIOM II. (Axiom of elementary sets)... There exists a (fictitious) set, the null set, 0, that contains no element at all....
”The set Z0 contains the elements 0, {0}, 0, and so forth, and it may be called the number sequence, because its elements can take the place of the numerals.

Skolem (1923), Foundations of Elementary Arithmetic -- No mention is made of 0 or of any specific number excepting “1” [e.g. “...the proposition holds for c = 1 ...”, “ 1.a = a” :

”The notion number” and “the number n + 1 following the number n” (thus, the descriptive function n + 1) as well as the recursive mode of thought are taken as basis.” (p. 305)

Von Neumann (1923), On the introduction of transfinite numbers -- He invokes Cantor’s name in the first sentence:

”the aim of the present paper is to give unequivocal and concrete form to Cantor’s notion of ordinal number.”

and then goes on to define the numbers:

" 0 = O
" 1 = (O)
" 2 = (O, (O))
"where O is the null set ...” (p. 347) and “Let O be the empty set ...” (p. 348).

Hilbert (1925), On the infinite – continues to begin at “1”:

"That we have when, for example, we consider the totality of the numbers 1, 2, 3, 4, . . . itself as a completed entity...” (p. 373)
"It was Cantor, however, who systematically developed the notion of the actual infinite. If we look at the two examples of the infinite that we have mentioned, (1) 1, 2, 3, 4, . . . and (2) the points of the line segment from 0, to 1, or, what is the same, the totality of real numbers between 0 and 1, then the idea that suggests itself most readily is to consider them purely from the point of view of cardinality, and when we do this we observe surprising facts that are familiar to every mathematician today.” (p. 374)

Hilbert (1927), The foundations of mathematics: By this time Hilbert has apparently moved away from the logicist start from 1 and accepted the set-theoretic start from 0:

”VI. Axioms of number
”16. a’ ≠ 0
”17. (A(0) & (a)(A(a)  A(a’)))  A(b) (principle of mathematical induction.
”Here a’ denotes the number following a, and the integers 1, 2, 3, . . . can be written in the form 0’, 0’’, 0’’’, . . .” (p. 467)

Weyl (1927), Comments on Hilbert’s second lecture on the foundations of mathematics:

“For after all Hilbert, too, is not merely concerned with, say, 0’ or 0’’’, but with any 0’’...’, with an ‘’arbitrary concretely given’’ numeral.” (p. 482-483).

Ackermann (1928), On Hilbert’s construction of the real numbers – van Heijenoort’s introduction begins as follows:

"The notion of primitive recursive functions appeared, in Dedekind 1888, as a natural generalization of the recursive definitions of addition and multiplication. Primitive recursive functions are obtained from 0, the successor function, and the identity function by composition and the following schema ... [the induction formula follows]." (p. 494)

Godel (1931), On formally undecidable propositions of Principia mathematica and related systems I – van Heijenoort observes, and Godel indeed admits, that Godel adjoined the Peano axioms to PM in order to simplify the presentation but the axioms could have been derived from PM (cf. p. 599). He begins with the sign “0” and applies the successor function to it (cf p. 599). He then maps the numeral 1 to the primitive sign “0” (p. 601), etc.

Reichenbach (1947), The Elements of Symbolic Logic -- His Section 44. The Definition of Number, follows Russell and PM; begins with the number 1. No mention of Peano axioms.

Rosenbloom (1950), The Elements of Mathematical Logic [This edition is marred by no references! The publisher left them out, I guess] – gives the Peano axioms starting from 1: “In words, P1 says that 1 is a positive integer, P2 that the successor of an integer is an integer, P3 that an integer can have at most one successor, ...[etc].” P1-P5 and P11 [induction] are essentially Peano’s postulates for arithmetic.

”It is interesting to contrast the theory of Quine [I]45 (no null class), that of Zermelo (Lz in III3) (no universal class), and that of Aristotle (neither null class nor universal class). In the von Neumann-Bernays version of Zermelo’s system (...) there is, indeed a universal class, but there is a distinction between classes and sets, and the universal class is not a set.” (p. 195).

Kleene (1952), Introduction to Metamathematics, presents the peano axioms as beginning from 0: “1. 0 is a natural number. 2. If n is a natural number, then n’ is a natural number. 3. The only natural numbers are those given by 1 and 2” [etc.] (p. 20).

Nagel and Newman (1958), Godel’s Proof:

"His axioms are five in number. They are formulated with the help of three undefined terms, acquaintance with the latter being assumed. The terms are: ‘number’, ‘zero’, and ‘immediate successor of’. Peano’s axioms can be stated as follows:
"1. Zero is a number
"2. The immediate successor of a number is a number
"3. Zero is not the immediate successor of a number
"4. No two numbers have the same immediate successor.
"5. Any porperty belonging to zero, and also to the immediate successor of every number that has the property, belongs to all numbers.
"The last axiom formulates what is often called the ‘principle of mathematical induction.’" (p. 103)

Suppes (1960), Axiomatic Set Theory, defines the 5 Peano axioms as beginning with 0:

"P1. 0 is a natural number.
"P2. If x is a natural number then x’ is a natural number.
"P3. There is no natural number x such that x’ = 0.
"P4. If x and y are natural numbers and x’ = y’ then x = y.
"p5. If φ(0) and for every natural number x if φ(x) then φ(x’), then for every natural number x, φ(x).” (p. 121)

Halmos (1970), Naïve Set Theory, begins the 5 Peano axioms as follows:

"To say that ω is a successor set means that
"(I) 0 ε ω
"(where of course , 0 = Ø), and that
"(II) if n ε ω, then n+ ε ω
"(where n” = n U { n }).
[etc.]” p. 46.

Hodges (1983), Allan Turing: The Enigma

”Dedekind . . . showed in 1888 that all arithmetic could be derived from three ideas: that there is a number 1, that every numbere has a successor, and that a principle of induction allows the formlation of statements about ‘’all’’numbers. ... in 1889, the Italian mathematician G. Peano gve the axioms in what became the standard form.
"In 1900 Hilbert greeted the new century by posing seventeen unsolved problems to the mathematical world. Of these, the second was that of proving the consistency of the ‘Peano axioms’ on which, as he had shown, the rigour of mathematics depended."(p. 83)

Boolos, Burgess, Jeffrey (2002), Computability and Logic: Fourth Edition, speaks of “the theory ‘’’P’’’ of Peano arithmetic’’”(p. 214) in the chapter on “Representability of Recursive Functions”. Their presentation uses the symbol “0” as a basis.

Dawson (1997), Logical Dilemmas, in an excellent chapter called Excursus presents the Peano axioms in their original form (i.e. from the unit 1), even using a similar (but not quite identical) symbolism (p. 45).

Davis (2000), Engines of Logic, in a long note 12 starting on page 225 Davis gives an example of encoding PA (Peano Arithmetic) for undecidable propositions. Rather than use 0 he uses the symbol "1" plus the others of set theory and Boolean algebra.

Franzen (2005), Godel's Theorem: An incomplete guide to its use and abuse, presents the axioms as beginning with 0 (Chapter 7.2 "PA as a First-Order Theory", p. 131 ff)

Goldstein (2005), The Proof and Paradox of Kurt Godel, begins with 0:

"Here are the first three: 0 is a number. The successor of any number is a number. No two numbers have the same successor." (p. 127)

wvbaileyWvbailey 22:32, 23 January 2007 (UTC)

Wvbailey, your survey is already very long, but just a few examples would be enough to show that both conventions are still alive, and that is all the article needs to say. Also, your book is biased towards introductory texts, which are more likely to stick to historical anachronisms in my experience. You leave out the many advanced books that study formalized arithmetic, which are more likely to contain modern treatments rather thanhistorical ones. CMummert · talk 00:02, 24 January 2007 (UTC)

edit 2007-1-23

I tried to follow as many previous conventions as I could when editing this evening; I'm sorry if I stepped on any toes. I was afraid of edit conflicts, but I'm done for a while so feel free to edit away.

I do feel strongly that for axiom systems such as ZFC and PA that have many different but equivalent axiomatizations it is worthwhile to literally quote the axioms as one author presents them. This is already the way the ZFC article is arranged. I know the list of axioms of PA is long, but it isn't that long. CMummert · talk 03:59, 24 January 2007 (UTC)

Ack! edit conflict! I lost it all.... Not to worry, I'm not going to add any of the above as it appears -- it is just there for me and JoergenB to look at (he asked me to research van Heijenoort to find out if what he had been told is true re the appearance of zero (mostly true)). I could not find a quotable quote re zero and the axioms. I know one is out there somewhere... this is just what happened a year ago re the Davis/halting-problem question that an anonymous reader finally resolved by pointing me to a quote that I could confirm.

I am trying to get a historical perspective, too. Who, why, and where and when? My modus operandi, more or less. (I noticed that Martin Davis does the same in his writing -- he even uses the same sources!). I agree that the quotation(s) of the 5 axioms (not the nine) should be as one author presents them. But which author? The original is impossible to read. Also the Hilbert notion of "primitive idea" is important. wvbaileyWvbailey 04:35, 24 January 2007 (UTC)

You shouldn't lose information in an edit conflict - it is in the lower text box of the edit conflict page.
I looked, it was gone. Not to worry.
The strange notation in the original is why I rephrased the axioms in English. I don't mind including all nine of Peano's axioms; it gives insight into the issues he thought needed to be addressed. Its obvious to anyone who looks at the original that the English descriptions are correct. CMummert · talk 04:40, 24 January 2007 (UTC)

I like your presentation of the axioms in English -- that was to be one of my recommendations.

Concerning the "source" of the axioms, there are a couple interesing things in van Heijenoort:

"Peano acknowledges (1891b, p. 93) that his axioms come from Dedekind (1888, art. 71, definition of a simply infinite system [also in van Heijenoort Dedekind's letter of defense to Keferstein]. As for Frege, Peano learned of his work immediately after the publication of Arithmetices principia." (p. 81)
"Today we would consider that Axioms 2, 3, 4, and 5, which deal with identity, belong to the underlying logic. This leaves the five axioms that have become universally known as "the Peano axioms"." (p. 81)

Peano says he used Grassmann 1861 for the proofs (p. 86). whereas his parts II, III, IV:

"can be traced to the works of many writers, especially Boole." (p. 86). He goes on to cite Boole, Schroder, Peirce, Jevons , and McColl.

One thought about "the 5" versus "the nine" is, in a historical view, to present "the major five" as did Hilbert in his 1904 "Foundations of Logic and Arithmetic" -- this is where he calls "1" a "thought object". (This thereby opens the philosophic issues of what are "thought objects" ... but we put that aside for the time being ... there's the very interesting Padoa 1900 that seems to get into this side of it...). There are two other Hilbert papers where they appear prominently (but never with Peano's name attached) 1925 On the infinite and a sequel 1927 The Foundations of Mathematics. All three are excellent. But first I will have to study them carefully .

If you know of any "source books" etc. re the later developments re the Peano axioms lemme know, please. wvbaileyWvbailey 19:49, 24 January 2007 (UTC)

I left an almost empty section "Historical placement of the axioms" which I was thinking would contain information about the inspiration and sources for Peano's work, including Dedekind. You have read all about this, and probably know more about it than I do.
I don't see how it hurts to include all nine of Peano's axioms - it only adds a few lines. Although I didn't read van Heijenoort, I already said in the article that the equality axioms are taken for granted nowadays. So he and I agree on that point.
Godel's 1931 paper gives Peano's axioms, but in the context of finite-type arithmetic, and Godel already had eliminated the first 6 axioms. There are unlikely to be source papers after that time, because the material was classical enough to be covered in texts instead. So I bet Kleene's Intro to Metamathematics has something to say. I can count at least 6 modern texts on formalized arithmetic, and I am sure that is an underestimate. Kaye's book is a standard reference and is quite well written, but it is a graduate-level book. CMummert · talk 23:22, 24 January 2007 (UTC)
Thanks. I have made a cc of the van Heinjenoort Peano article in .pdf form. If you would like it e-mailed lemme know.
I find Kleene's presentation a bit odd:
"1. 0 is a natural number. 2. If n is a natural number, then n' is a natural number. 3. The only natural numbers are those given by 1 and 2.
"4. For any natural numbers m and n, m' = n' only if m = n. 5. For any natural number n, n' ≠ m' if m = n.
"Also it is understood that ' is a univalent operator or single-valued function, so that conversely to 4: For any natural numbers m and n, m' = n' if m = n.
"...Peano stated Proposition 3 instead as the principle of mathematical induction... [he defines this in the next secton: "If (1) P(0), and (2) for all n, if P(n) then P(n'), then, for all n, P(n)". ] wvbaileyWvbailey 00:39, 25 January 2007 (UTC)
Kleene's definition is not really very different than the one in Godel (1931). All that the induction axiom says is that the natural numbers are the smallest set containing 1 closed under successor. CMummert · talk 02:48, 25 January 2007 (UTC)

Last sentence

The following statement (very last in the article), is not "intuitively obvious" (and perhaps incorrect [?]):

"The small number of mathematicians who advocate finitism reject Peano's axioms because the axioms require an infinite set of natural numbers."

I do not believe this is correct -- the axioms do not provide for a completed infinity, or even "an infinity" whatever that means, but simply show how numbers get created. We need an axiom of infinity to "go there." I've not read that intuitionists, e.g. Brouwer's bunch, reject the Peano axioms. Maybe some do, but then they won't have induction available to them. They don't allow for Cantor's "completed infinity", but they do allow for a never-ending progression of "successor events," and in particular they are great fans of induction:

"A generality statement all natural numbers n have the property P, or briefly for all n, P(n), is understood by the intuitionist as an hypotheitical assertion to the effect that, if any particular natural number were given to us, we couldb e sure that that number n has the property P. This is a meaning which does not require us to take into view the classicaal completed infinity of the natural numbres.
"Mathematical induction is an example of an intuitionistic method for proving generality propostions abou the natural numbers. A proof by induction of the proposition for all n, P(n) shows that any given n would have to have the property P, by reasoning which uses only the number from 0 up to n ... Of course, for a particular proof by induction to be intuitionistic, also the reasonings used within its basis and induction step must be intuitionistic." (Kleene 1952, 1971 p. 49)

I suggest the line in question should go away or get an in-line reference, if someone knows of one. wvbaileyWvbailey 17:34, 25 January 2007 (UTC)

You are correct that ordinary intuitionism has no problem with the infinite set of natural numbers or with induction. But see for example this paper by Vladmir Sazonov, where he studies what he calls "feasible numbers". He does not accept the fact that 2^x is a total function on the natural numbers, and gives the specific value of 2^1024 as one that does not exist because it is too large to be physically computed. So not only does he reject a completed infinity of natural numbers, he rejects even a potential infinity of natural numbers, because he rejects (at least) the existence of any number bigger than 2^1024. This viewpoint is sometimes called "ultrafinitism". It is not the same meaning of the word finitism that is used in the context of Hilbert's program. The number of mathematicians who subscribe to this viewpoint is exceedingly small. CMummert · talk 17:55, 25 January 2007 (UTC)

Interesting. Maybe there is a way to rephrase the last sentence. I will mull it over for a while. [I am developing a certain sympathy with this notion of ultrafinitism: Lately I have been applying my feeble brain to the Collatz Conjecture (Ulam's problem). The notions of completed versus add-on infinity are beginning to intrude: I can give you an example of a (class of) numbers that will never reduce to 1 but they must be "infinite" in extent (i.e. an infinite number of digits of a certain type ωvile , e.g. 2infinity - 1 is the easiest example). But are these entities "numbers"? What are they exactly? And my poor Excel spreadsheet runs out after 15 decimal digits. So I am in the process of designing a "process" so I can go beyond the 15-digit accuracy and make my (finite) vile numbers even bigger.] wvbaileyWvbailey 19:39, 25 January 2007 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Peano axioms/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

See the A-Class review discussion. Geometry guy 15:47, 9 June 2007 (UTC)

Last edited at 21:45, 4 August 2010 (UTC). Substituted at 06:49, 7 May 2016 (UTC)