View source for Talk:Hilbert 2nd problem

I created this article (obviously still under construction) partly in pursuit of my lifelong interest in the history and philosophy of mathematics.

I have only recently discovered the work of Gregory Chaitin. He and I are similar in age and, in addition, studied only minutes from one another along the Pennsylvania Railroad: he mathematics at CCNY in New York City, I philosophy at Rutgers University in New Brunswick, New Jersey.

Chaitin's interest in Hilbert's promotion of mathematical formalism and my own interest in Carnap's et. al. promotion of logical positivism are, I feel, mirror images of a single mind.

The subject has engaged the comments of many mathematicians and philosophers of mathematics since Hilbert's 1900 lecture. The title that Chaitin gave to his own 2000 lecture allows us to see his work as a homage to Hilbert's vision.

== Feasible and infeasible ==

Encouraged by you, I've read that text by Chaitin, and like to share some impression. Here is ''my'' outrageous claim. Results of Goedel, Turing, Chaitin and probably some others are a wonderful proof of an ''intuitively evident'' point that a mathematical theory cannot prove "the truth, the whole truth, and nothing but the truth" about natural numbers. I do not know whether or not this was Hilbert's program, but if it really was, then it is a wonder that for Hilbert, the opposite was ''intuitively evident''; and then we observe a dramatic change of our intuition, somewhat similar to the change from "evidently, a continuous function must be differentiable in most of the points" to "evidently, a generic continuous function is nowhere differentiable".

Once upon a time I was thinking hard about probabilistic cellular automata (there was a wonderful problem attacked by me, among others, but ultimately solved in an exciting work by Peter Gacs). And I got very pessimistic, for the following reason. Imagine a deterministic cellular automation powerful enough for having "computers" and "robots" among possible finite configurations. Introduce small randomness; being small it does not prevent these computers and robots from successful functioning during reasonably long time; being non-zero, the randomness introduces "mutations", it ensures that every possible computer/robot will emerge somewhere, sooner or later. Now ask a clever question about the asymptotic behavior of this automation on large time.

In order to answer such a question, you probably need to understand what is "the optimal civilization", a finite but growing combination of computers/robots most succesfully enlarging itself and fighting all other civilizations! Or alternatively you need to understand a sequence of civilizations that tends to optimality...

Why hope that such knowledge follows from reasonable axioms?

Yes, we mathematicians have an encouraging experience: when we want to prove or disprove something, sooner or later we can. But this only shows that our intuition is able to choose only feasible tasks.

Go to a wild, find a tiger that looks at a zebra, and ask a physicist to predict the result: who will succeed this time, the tiger or the sebra? No, the physicist cannot. He is successful in answering properly chosen questions. And the same applies to us mathematicians.

And now I wonder, which one of the following two scenarios is closer to the historical truth.

The first scenario. Hilbert believed that every question formulated in the formal arithmetic is "mathematically feasible". Thus, his intuition looks quite naive nowadays.

The second scenario. Hilbert's intention was more modest: to create a formal arithmetics that proves "the truth, and nothing but the truth" but not quite "the whole truth" about natural numbers. It was enough for him, if the theory solves all "mathematically feasible" questions.

[[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 11:03, 6 June 2015 (CEST)

::Well, from the article (as of now) I understand that the answer to my question is "the first scenario". [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 18:49, 9 June 2015 (CEST)

::::Yes, as you say, Hilbert, a mathematician of great renown, nevertheless entertained notions that we nowadays regard as naive.

::::Yet, as we say this, we smile and recall that few things in mathematics (even fewer in everyday life?) come into being fully formed and wholly mature -- as did Athena, who is said to have sprung full-blown from the head of Zeus!

::::It is enough to note that the subtitle of the ‘’E of M’’ article [[Set theory]] is ''naive'' to remind ourselves of the difficult journey from yesterday’s ‘’Unknown Unknowns’’ to tomorrow’s ‘’Known Knowns’’ -- I'm thinking here of U.S. Vice President Dick Cheney's famous remarks about these two notions. With respect to perplexing matters in mathematics (and in everyday life!), we repeatedly find ourselves somewhere in the midst of such a journey.

::::Hilbert was not the first mathematician to insist that some branch or other of mathematics needed to be placed on a firm(er) foundation, that mathematical concepts needed to be defined (more) rigourously, that mathematical assumptions needed to be stated (more) explicitly, and that more formal methods needed to replace intuition.

::::Hilbert must have known in 1900 what we know today, namely, that in all such efforts, the formalisms (e.g. definitions and axioms) that we develop '''inside mathematics''' are a way of expressing our regard for and paying our respect to the intuitive notions '''outside mathematics''' from which those formalisms have spring and upon which they are ultimately based.

::::[[User:Whayes43|William Hayes]] ([[User talk:Whayes43|talk]]) 17:22, 11 June 2015 (CEST) Best regards. It's always a pleasure to read and reflect on your comments.

==The difference between mathematics and philosophy?==

Thank you for your candid and very interesting reflection. Quite obviously you, yourself, have also been a student of philosophy!

::<small>No, never!  :-)  Though, one of the two my most influential teachers was a logician. [http://www.tau.ac.il/~tsirel/Research/myspace/remins.html] [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 18:21, 8 June 2015 (CEST)</small>

Thinking about this subject (Foundations of Math -- Indeed, are there any?) brought to mind a long-ago event in which one of my math profs at Rutgers, in a brief conversation, worked some mentoring magic in my undergraduate life.

Prof Dekker was, as I recall, a logician, though my studies with him were of algebra and topology. One afternoon, just before or just after a class (interesting that very important events happen on the periphery of and almost in spite of the plans we make for ourselves) I mentioned to him Nagel and Newman’s book on Godel’s incompleteness theorems, through which I was working on my own -- there was, at the time, no one in the philosophy department capable of helping me with that material. Prof Dekker made a bit of a face and then shared an anecdote from his own student past:

::As an undergraduate, I had wrestled with the issue of academic direction: would I pursue philosophy or mathematics. The answer came to me as a revelation during a scholarly conference to which one one of my philosophy professors had kindly invited me.

::A respected scholar rose and delivered a keynote address on a subject that was of great interest to me. Immediately afterwards, another scholar, equally respected, rose as a devil’s advocate to deliver some comments to the contrary of the keynote address. Finally, the keynote speaker rose once more and made a brief reply, which he began with the words, “But you have completely misunderstood me!”

Dekker paused for a moment, then looked at me and said the words that caused him to choose mathematics rather than philosophy:

::You know, those words made the decision (between math and philosophy) for me. They would never have been said by a mathematician. We don’t ever misunderstand one another!

A bit of an exaggeration (?) and yet his words helped me with my own academic indecision, although in my case, the choice I made (for mathematics) was rendered moot by other, overarching events in my life. C’est la vie!

[[User:Whayes43|William Hayes]] ([[User talk:Whayes43|talk]]) 16:06, 8 June 2015 (CEST)

Feeling this article was growing unmanageably large, I today determined that it should deal only with the mathematics that ''preceded'' Hilbert's Problems Address, including (of course) his statement of the 2nd problem itself. Accordingly, I have revised the introductory remarks and excised headings for the sections no longer needed. I will deal with the subsequent history of the development of Hilbert's program itself in another article, [[Hilbert program]], which I have also created today. Trust that this seems reasonable. Thanks for your occasional, watchful, and helpful edits. --[[User:Whayes43|William Hayes]] ([[User talk:Whayes43|talk]]) 17:43, 25 July 2015 (CEST)

== Hilbert's Grundlagen ==

"6 undefined relations: being on, being in, being between, being congruent, being parallel, & being continuous" — I got puzzled with the last (being continuous). "Being on" is a relation between a point and a line, or a point and a plane, or a line and a plane, I guess. "Being parallel" - between two lines. "Being continuous" - between what? According to the text it should be some ''n''−place relation; for which ''n''?
[[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 19:00, 19 August 2015 (CEST)

:I have made (some) sense of Boyer's mention of the relation of "being continuous" as follows:
::Consider Hilbert's 5 groups of axioms, grouped as follows: Incidence, Order, Congruence, Parallelism, and Continuity. Euclid ''Elements'' treats these as follows:
::* Congruence and Parallelism are dealt with to some extent in axioms.
::* Incidence is referred to (and therefore relied upon) in Euclid's 1st postulate that has to do with drawing a line between 2 points.
::* Order and Continuity are not dealt with at all.
::Hilbert's axioms of Order draw on Pasch's work in 1880s, to the extent that the 4th of Hilbert's axioms of Order is commonly called Pasch's Axiom.
::In the various editions of the Grundlagen, Hilbert used different axioms for Continuity. In modern presentations (re-statements) of "Hilbert's Axioms," authors choose one or another of Hilbert's axiom sets for Continuity based on ... what ... personal preference? whim? ouija board consultation? I have no idea! Many authors have used two axioms (e.g. Archimedes and Line Completeness); at least one author has used a single, strong axiom of completeness (Dedekind's Axiom -- which implies Archimedes and Line Completeness).
:In any case, the matter in Euclid's ''Elements'' that the axioms of Continuity seek to address is the ''intersection'' of lines and circles. It is this matter of two lines or two circles or one of each intersecting one another that makes sense of Boyer's talk of the relation of "being continuous".

- - - - -

:You ask: some ''n''−place relation; for which ''n''?
:I took the "n" to be, very specifically, the $n$ (geometric) terms of Hilbert's axiom set $AX$.
:Blanchette provides this more general example:
::If $AX$ is the set $\{$There are at least two $points$; Every $point$ lies on at least two $lines$$\}$, then $R_{AX}$ is the relation that holds of any triple <$P$, $LO$, $L$> such that $P$ has at least two members, $L$ has at least two members, and $LO$ is a relation that holds between each member of $P$ and at least two members of $L$.
:I am not persuaded that this example captures all of what Hilbert meant when he wrote the following in a letter to Frege:
::If in speaking of my points I think of some system of things, e.g. the system: love, law, chimney-sweep … and then assume all my axioms as relations between these things, then my propositions, e.g. Pythagoras' theorem, are also valid for these things. In other words: any theory can always be applied to infinitely many systems of basic elements.

:Hilbert's choices of "love, law, chimney-sweep" reminded me of Rudolph Carnap's own life-choices of "love, music, and logic" mentioned in a mathematical memoir.
- - - - -

::Whenever I read the phrase "undefined relations" I think of the many modern families (of which mine is one) that are the product of multiple (re)marriages yielding step-grandkids (whom one of my friends dubs "gift-grandkids") and ever more compound-complex relations and sort-of descendants.

--[[User:Whayes43|William Hayes]] ([[User talk:Whayes43|talk]]) 18:43, 20 August 2015 (CEST)

::All that is nice, about ''axiom(s)'' of continuity. No doubt, there are such axioms. I ask, what could be ''relation'' of continuity? Beween how many objects, and which kind of objects?
::On [https://en.wikipedia.org/wiki/Hilbert%27s_axioms WP] I see 1+3+2=6 primitive relations, one ternary and 5 binary, and two axioms of continuity; and that looks logical (though I did not read Hilbert). [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 22:31, 20 August 2015 (CEST)
::If at all a continuity relation can be imagined, it is probably of [http://mathoverflow.net/questions/56301/relations-of-infinite-arity infinite arity]... [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 12:50, 21 August 2015 (CEST)

:::As I understand them, the Axioms of Continuity (Axiom of Line Completeness and Axiom of Archimedes) are intended to address the matter of '''intersection''' that arises  whenever Euclid deals with either of the following processes:
:::# the ''actual intersection'' of 2 lines or 2 circles or one line and one circle
:::# the ''implied intersection'' involved in the process of using one line segment to ''measure'' another line segment 
:::Hilbert's Axioms of Continuity establish that lines and circles and line segments are sufficiently dense with points to establish that these processes do indeed determine points of intersection that both exist and also are unique. Euclid doesn't consider this matter at all.
:::In this sense, the ''property of continuity'' substantiates our speaking of both ''a relation of intersection'' (involving two lines or two circles or a line and a circle) and ''a relation of measurement'' (involving two line segments).
:::Either that or I am completely in the dark about what Boyer meant. :-))  

::::As far as I know, Hilbert does not introduce new undefined relations if not quite necessary. Intersection of two lines is, by definition, a point that belongs to both lines. Only one undefined relation (one of the three containment relations) is used in this definition. No need to invent a new one.

:::::I've revised Boyer's summary description of Hilbert's theory and added a comment to the footnote -- all for the better, I think. Thank you for your clarifying comments, which are both persistent and supportive.

::::::Ah, yes, I see it written in Boyer: [[Media:Boyer-AHistoryOfMathematics-658.pdf|PDF]]. Now I should think more. [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 22:19, 22 August 2015 (CEST)
::::::And here are the relevant pages of "Grundlagen": [[File:Hilbert-2.png]] [[File:Hilbert-16.png]] [[File:Hilbert-17.png]] English translation: [[Media:Hilbert-2-15-16.pdf|PDF]].

::::::Here is what I observe in Hilbert's text.

:::::::We think of these points, straight lines, and planes as having certain mutual relations, which we indicate by means of such words as “are situated,” “between,” “parallel,” “congruent,” “continuous,” etc. The complete and exact description of these relations follows as a consequence of the axioms of geometry.

::::::However, the word "continuous" does not occur in the two "axioms of continuity" (Archimedean axiom and axiom of completeness), nor in any other axiom. Thus, the "complete and exact description" of the continuity relation is void. It means, no such relation at all. [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 08:41, 23 August 2015 (CEST)

::::::It is natural to ask, why did Hilbert mention it in the list. Let me note that Hilbert pays great attention to the list of axioms, and very little attention to the list of undefined relations. Just note his "such words as" rather than "the following undefined relations".

::::::Also, his "Verzeichnis der Begriffsnamen" (index) contains "Seite eines Punktes auf einer Geraden" (belonging of a point to a line) and more like that; "Zwischen" (between); "Kongruenz..."; but nothing about "stetig" (continuous) except for axiom of continuity.

::::::It is a pity that Boyer interprets 'such words as “are situated,” “between,” “parallel,” “congruent,” “continuous,” etc'as 'six undefined relations—being on, being in, being between, being congruent, being parallel, and being continuous'. (By the way, his ''two'' items "being on" and "being in" mean really ''three'' incidence relations: a point on a line, a point on a plane, and a line on a plane.) [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 09:36, 23 August 2015 (CEST)

::::::No wonder that we have no idea even of the arity of "the continuity relation". This "relation" just '''was not introduced''' by Hilbert. Nor by anyone else (I am pretty sure). If I say "abracadabra", it does not make it an existent notion, and we still know nothing about "it". [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 12:05, 23 August 2015 (CEST)

::::::For some reason, I left off consulting Hilbert's own text until yesterday, when I looked into Townsend's 1902 translation into English of Hilbert's first edition (1899) of the ''Grundlagen''. We may need to cut Boyer some slack. In using the term 'relations,' Boyer may simply have followed Townsend's translation -- apparently authorized by Hilbert. Here is how Townsend renders Hilbert's first two paragraphs ('''emphasis''' added):
:::::::Let us consider three distinct systems of things. The things composing the first system, we will call points and designate them by the letters A, B, C,. . . ; those of the second, we will call straight lines and designate them by the letters a, b, c,. . . ; and those of the third system, we will call planes and designate them by the Greek letters α, β, γ,. . . The points are called the elements of linear geometry; the points and straight lines, the elements of plane geometry; and the points, lines, and planes, the elements of the geometry of space or the elements of space.

:::::::We think of these points, straight lines, and planes as having certain mutual '''relations''', which we indicate by means of such words as “are situated,” “between,” “parallel,” “congruent,” “congruent,” etc. The complete and exact description of these '''relations''' follows as a consequence of the axioms of geometry. These axioms may be arranged in five groups. Each of these groups expresses, by itself, certain related fundamental facts of our intuition. We will name these groups as follows:
::::::::I, 1–7. Axioms of connection.
::::::::II, 1–5. Axioms of order.
::::::::III. Axiom of parallels (Euclid’s axiom).
::::::::IV, 1–6. Axioms of congruence.
::::::::V. Axiom of continuity (Archimedes’s axiom).
::::::--[[User:Whayes43|William Hayes]] ([[User talk:Whayes43|talk]]) 17:00, 24 August 2015 (CEST)

:::::Well, so what? Anyway, Hilbert's text is not a text of an international treaty. After all, a man can err in a detail; but, understanding the whole, we do not faint from slips. We do understand the axioms. We see, what relations occur in the axioms (this is not clear from the text itself! one really needs to understand the meaning). These relations matter. Others do not. Even if on some line of Hilbert we'll see "2+2=5", we'll not change our mind (unless we'll see that he really mean it, and explains why). There is simply no "continuity relation". No matter which words did Hilbert use in the informal text outside the axioms. [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 17:35, 24 August 2015 (CEST)

:::::This is not a matter of textual criticism. This is a matter of mathematics. A person not understanding the meaning cannot claim anything reasonable. [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 17:51, 24 August 2015 (CEST)

:::::Mathematics is, of course, invariant under translation from one natural language to another. If a mathematician knows some natural language to the extent that he understands the meaning of a mathematical text on that language, then this meaning does not depend on the language.  "Being on" and "being in" are not ''two'' relations; they seem to be two in English, probably not in another language.

:::::If someone tells me about a relation but appears to be unable to answer my question, what is its arity, it is like someone tells me about inhabitants of his apartment but appears to be unable to answer my question, who of these are humans, who are dogs and who are cats. What should I think then about the story he tells me? [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 18:31, 24 August 2015 (CEST)

:::::You are making important points that need to be made (and can only be made) by mathematicians. Boyer is, after all, not a mathematician, but a historian -- albeit of mathematics. I recall criticisms of some presentation and analysis of Cauchy's work by Grabiner (also a historian) to the effect that her mathematical reach exceeded her historical grasp. Eternal vigilance is the price not only of liberty, but also of knowledge. --[[User:Whayes43|William Hayes]] ([[User talk:Whayes43|talk]]) 18:49, 24 August 2015 (CEST)

::::::Nicely put! Probably, each historian of mathematics should be vigilant to stop before claiming something that he doesn't understand. "Never invest in a business you can't understand" [http://www.minterest.org/best-warren-buffett-quotes-on-investing/ Warren Buffet]. I was sad to see this, since I understand only math... Such advices are easy to give and nearly impossible to follow. [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 19:14, 24 August 2015 (CEST)

:::Interesting discussion of in-finite arity relations. The notion of validating infinitely long sentences generated by axiom schemas of first-order logic using truth tables of infinite size is confounding (and amusing) enough for me. https://en.wikipedia.org/wiki/First-order_logic#Infinitary_logics ??  --[[User:Whayes43|William Hayes]] ([[User talk:Whayes43|talk]]) 20:27, 21 August 2015 (CEST)

::::Probably, Hilbert would be disturbed, too. Metamathematics was created as a finitary theory that avoids infinity in order to be more reliable than mathematics. But now other options are tried, too. Clearly, infinitary logic can exists only ''within'' the set theory (or another very strong, and therefore not so reliable) mathematical theory. Hardly related to Hilbert's ideas. [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 21:18, 21 August 2015 (CEST)

If all goes as planned, immediately after posting this note I shall revise (again) the ''Grundlagen'' section, with a view to relating it more closely to Hilbert's 2nd problem. After all, only someone of exceptional stature and great chutzpah could stand before a congregation of mathematicians (true believers, all) to lecture them about what they needed to be doing and to insist that they immediately set about doing it! And the ''Grundlagen'' not only contributed to his stature, but also served as an introduction to the 2nd problem. So ....  --[[User:Whayes43|William Hayes]] ([[User talk:Whayes43|talk]]) 04:28, 13 September 2015 (CEST)

:So..., revision installed, resulting in (I believe) an improved structure that permits (requires -- actually) further additions/adjustments, but with better coherence. In all of this, I remain an obedient servant to my most diligent reader, editor, and (may I say) mentor?! Best regards, --[[User:Whayes43|William Hayes]] ([[User talk:Whayes43|talk]]) 05:04, 13 September 2015 (CEST)

== Hilbert’s conception of consistency ==

"In the context of formal theories, Hilbert’s conception of consistency and his associated methodology for consistency-proofs are, for the most part standard today." — Yes, but how much is this methodology Hilbert's? According to [https://en.wikipedia.org/wiki/Non-Euclidean_geometry#Hyperbolic_geometry Wikipedia], Beltrami in 1868 "defined the Klein model which models the entirety of hyperbolic space, and used this to show that Euclidean geometry and hyperbolic geometry were equiconsistent so that hyperbolic geometry was logically consistent if and only if Euclidean geometry was." [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 09:23, 25 August 2015 (CEST)

:It is likely that Hilbert was aware of Beltrami's work, if not directly, then through Klein himself, with whom Hilbert carried on an extensive correspondence and (in 1897-98) taught seminars. Here's a link to a Math Stack Exchange question about Beltrami's consistency proof and a lively exchange in the answers:
::[http://math.stackexchange.com/questions/665981/how-did-beltrami-show-the-consistency-of-hyperbolic-geometry-in-his-1868-papers How did Beltrami show the consistency of hyperbolic geometry in his 1868 papers?]
:Who ever said mathematics is dull?! --[[User:Whayes43|William Hayes]] ([[User talk:Whayes43|talk]]) 16:37, 25 August 2015 (CEST)

::Well, Beltrami or not, in this or that book, anyway, the conception of consistency and the associated methodology for consistency-proofs probably existed before Hilbert. But, probably, some details were more vague before Hilbert, and less vague (but still not perfect) after. [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 17:08, 25 August 2015 (CEST)

== How many axioms of continuity? ==

Here on the talk page you write "As I understand them, the Axioms of Continuity (Axiom of Line Completeness and Axiom of Archimedes) are intended to address the matter of intersection".  Yes; axiom of completeness does.  But in the article you does not mention it, do you?  Only Archimedes.

I do not claim anything here; I feel I do not understand what happens; only guess.  Hilbert hesitates of the completeness axiom.  And no wonder.  Nowadays, it does not qualify for an axiom.  It is rather a property of a model (of this system of axioms).  It cannot be formulated in the given language (points, lines, betweenness... you know what).  "Nothing can be added" means really "for every other model that contains the given model..."; and this is terrible: not only the word "model" is beyond the language of this theory (geometry, not logic nor metamathematics), but worse, we need a quantifier over all possible models!  Nowadays, one would say instead: for every Cauchy sequence of points... or: for every subset of the line...  Still bad (not a first-order logic), but much better.

Well, I understand why Hilbert feels the need in this axiom but hesitates.  But why do you hesitate to mention it in the article? [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 18:13, 25 August 2015 (CEST)

:Changes (dare I say corrections) were proposed (and made) to Hilbert's original text, among which were the particular axioms used in the group named "Axioms of Continuity." Here is something ''a propos'' this matter of "completeness" from the 1902 English translation:
::Remark. (Added by Prof. Hilbert in the French translation -- and included in this English translation. WH.). To the preceeding five groups of axioms, we may add the following one, which, although not of a purely geometrical nature, merits particular attention from a theoretical point of view. It may be expressed in the following form:
:::Axiom of Completeness. (Vollst¨andigkeit): To a system of points, straight lines, and planes, it is impossible to add other elements in such a manner that the system thus generalized shall form a new geometry obeying all of the five groups of axioms. In other words, the elements of geometry form a system which is not susceptible of extension, if we regard the five groups of axioms as valid.
::This axiom gives us nothing directly concerning the existence of limiting points, or of the idea of convergence. Nevertheless, it enables us to demonstrate Bolzano’s theorem by virtue of which, for all sets of points situated upon a straight line between two definite points of the same line, there exists necessarily a point of condensation, that is to say, a limiting point. From a theoretical point of view, the value of this axiom is that it leads indirectly to the introduction of limiting points, and, hence, renders it possible to establish a one-to-one correspondence between the points of a segment and the system of real numbers. However, in what is to follow, no use will be made of the “axiom of completeness.”
:Modern presentations of Hilbert's axioms include both the Archimedean axiom and some completeness axiom or other. My remarks on this talk page were made from my knowledge of several modern presentations, before I had seen the English translation of Hilbert's original. --[[User:Whayes43|William Hayes]] ([[User talk:Whayes43|talk]]) 01:50, 26 August 2015 (CEST)
::Yes. Archimedean axiom does not forbid "holes" in a line (such as missing irrational points). Rather, it forbids coexistence of two or more "worlds" that are infinitesimal/infinitely large w.r.t. one another. In this situation completeness cannot hold, since it is always possible to increase the number of such "worlds" (beyond any given cardinality). But in presence of Archimedean axiom completeness can hold, and selects one and only one model (up to isomorphism); such model that coordinates of points are real numbers, no less, no more. Which means, no more "holes". [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 12:03, 26 August 2015 (CEST)

== References ==

In the "Notes" we see "Boyer p. XX", but is it Boyer 1939 or Boyer 1968?  Also, "238. Hilbert (1921-22) cited in Venturi pp. 10-11" — which Venturi?  And maybe more problems like that? [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 23:28, 3 September 2015 (CEST)

:Thank you. Yes there were more, some of which I have noticed and fixed, but others may also need attention. Mea culpa. --[[User:Whayes43|William Hayes]] ([[User talk:Whayes43|talk]]) 01:06, 4 September 2015 (CEST)

== Dedekind’s theory of numbers ==

"Axioms 1 and 3, however, are “set-theoretic” in character, not at all elementary, and tend to require second-order logic for their formalization." — Really? About Axiom 3, I understand it. But what is the problem with Axiom 1? Just "φ(n) belongs to N, for every n that belongs to N". [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 22:14, 25 September 2015 (CEST)

:Good question -- which I should have asked myself. Here is the whole of F's comment (''emphasis'' added WH):
:::Peano tended to impose ''conditions on the behaviour of his individuals'', the natural numbers, and the operations on them. These are elementary conditions which most often are amenable to formalization within first-order logic. Dedekind establishes ''structural conditions on subsets of the (structured) sets he is defining, on the behaviour of relevant maps, or both things at a time''. Axiom 1. says that N is closed under the map ϕ, axiom 3. says that N is the minimal closure of the unitary set {e} under ϕ. Such axioms are non-elementary and ''tend to require second-order logic for their formalisation'', as happens with axiom 3 ''(however, 1. is easy to formalize in first order'').
:SO: I will (somehow) revise the text. I, myself was uneasy over F's use of the phrase "tend to require," wondering how he determined that there was such a tendency and, if there was, what its significance was. --[[User:Whayes43|William Hayes]] ([[User talk:Whayes43|talk]]) 16:54, 28 September 2015 (CEST)

::I am glad to see this "however, 1. is easy to formalize in first order". Maybe F wants to say that Dedekind did not bother at all about first order or higher order? Or maybe, that, having one axiom (3) that needs second order, we have little motivation to reformulate other axioms in first order? After all, the highest order matters. [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 22:53, 28 September 2015 (CEST)

== The theory of transfinite numbers ==

"a new process of taking limits of increasing sequences, which yields, after any given sequence of numbers without a last element, a number b" — Really? Sequences work only before the first uncountable ordinal. [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 17:49, 6 October 2015 (CEST)

:Yes, and yet Tait writes this:
:*  To obtain ''all'' of the second number class, much less the higher number classes, a new idea was needed, namely the idea of introducing numbers as limits of ''arbitrary'' increasing sequences of numbers (''ω''-sequences in the case of the second number class). As far as I know, at no time before his letter to Dedekind did Cantor explicitly introduce this idea. (p. 7, note 7)
:* That Cantor speaks of ''sequences'' of numbers rather than sets is inconsequential, since the sequences in question are in their natural order and so are determined by the set of their members. (p. 19, note 14)
:I will investigate!--[[User:Whayes43|William Hayes]] ([[User talk:Whayes43|talk]]) 01:18, 7 October 2015 (CEST)
:::It may happen that Cantor was mistaken here and there, but I think, in such case, when quoting his erroneous claim, it is needed to comment that it is erroneous. It looks too strange to a mathematician, to see a wrong statement in a math encyclopedia (even in historical context). [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 21:00, 6 October 2015 (CEST)
::::Yes and Thank you: I understand your point. Perhaps (philosopher) Tait's description of Cantor's usage of "sequences" rather than "sets" as "inconsequential" is his way of saying that it is not an error, but an extension, i.e. part of the "transfinite paradise" that his work has revealed. I don't know enough to say that this make any sense, but I will investigate further. --[[User:Whayes43|William Hayes]] ([[User talk:Whayes43|talk]]) 01:18, 7 October 2015 (CEST)

"# the power (cardinality) of $\mathbb{N}$ is smaller than thwithe cardinality of its power set, $2^{2^{ℵ_0}}$" — something is wrong here. [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 17:54, 6 October 2015 (CEST)
::Thank you. I will look more closely at this! My eyes are bleary from dealing with ups (^) and downs (_). --[[User:Whayes43|William Hayes]] ([[User talk:Whayes43|talk]]) 01:18, 7 October 2015 (CEST)
::Yes, something was wrong -- not merely the "thwithe" bit. I had introduced the confusion by inserting and then incompletely removing the (unnecessary) statement that $card(p(\mathbb{N})) > card(\mathbb{N})$. Perhaps my revision has improved things. --[[User:Whayes43|William Hayes]] ([[User talk:Whayes43|talk]]) 01:18, 7 October 2015 (CEST)

By the way: the set theory is impredicative to extreme! I just happened to read [http://www.logicmatters.net/yet-more-logic "Induction and predicativity"] by Peter Smith... (item 20 of his list). [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 23:16, 6 October 2015 (CEST)
:Ah, something for the "non-expert audience"! I will examine.