The *Berkshire Eagle* has an article today that goes into more detail about translation from Latin than any newspaper article I’ve seen in a long time. It seems that the motto of the town of Pittsfield (as well of Pittsburgh), “Benigno Numine,” has been translated by St. Joseph’s Central High School Latin teacher Kathleen Canning as “Under Protection of the Goddess.” The paper quotes her as saying that

benignus has masculine, feminine and nongender endings, and that “benigno” is the “neuter form” of the noun.

Because both words in the city motto contain endings with no specific gender, they could be used to refer to a “goddess,” Canning said.

I’ll be charitable here and assume the reference to the adjective *benignus* as a noun is the paper’s mistake and not Canning’s, but the idea of translating *numen* as “Goddess” is just silly, I don’t care what they told her in civics class. The story goes on to say that “Mary C. Quirk, who teaches Latin at Miss Hall’s School, found 17 possible translations for the city motto, ranging from ‘propitious divine will’ to ‘with kind-hearted favor or approval (of the gods),’ to ‘with benign power,’ to ‘by beneficent authority'”; any of them would be a great deal better. (Thanks for the tip, Leslie!)

The whole phrase is “benigno numine Iuppiter,” in Lang & Shorey, eds., Horace,

Carmina, Book 4, Poem 4, line 73. Heh heh. I love the Internet.Well done, O b.v.i.! The stanza is:

nil Claudiae non perficient manus,

quas et benigno numine Iuppiter

defendit et curae sagaces

expediunt per acuta belli.

‘There is nothing that shall not be accomplished by Claudian hands, which Jove defends with his friendly

numen(majesty or what have you) and wise counsels guide through the crises of war.’ Or words to that effect; my Latinity is rusty. At any rate, I think we can rule out the Goddess. Nunc est bibendum!Sounds good to me. Yeah, there’s no reason that

numencouldn’t mean Goddess if it came floating in out of nowhere (still not the first thing that comes to mind)… but town mottoes rarely do come floating in out of nowhere. Hell, evenE Pluribus Unumcomes from somewhere, unlikely though that source may be.It’s amazing how often this sort of thing comes up- some organization forgets what its motto means, then contacts a random person who knows Latin, who is forced to fall back on dictionaries, rather than on the original intent of the people who picked the motto. This is unfortunate as Latin mottos are deliberately over-pithy and over-succinct, to the point that you often really

needcontext to get the point.Justin: When you say “from somewhere,” do you mean from this somewhere (Pseudo-Vergil, Moretum, 102)? Or were you thinking of the

Gentleman’s Magazine?Ack, yes, I did mean that somewhere, and in fact I had meant to include that very URL as the link! Dunno how I screwed that up.

heh. i find this *incredibly* ironic since i grew up in pittsfield, ma and received my public school education there. in 8th grade, we were given the option of taking either spanish, french or latin. though, we could only take latin if we had placed high enough in math to qualify to take algebra (instead of pre-algebra). i was an A student in everything but math, so i was not allowed to take latin because i was placed in pre-algebra. go figure.

After a bit of websearching I was able to find this rather silly example which had been floating in the back of my head from when I originally read it in the Isthmus.

I particularly like this quote: “

Some could find no example in classical Latin of the two words standing together to make a phrase, and concluded that it was untranslatable.” I guess that means Latin isn’t so much a language as a corpus.Justin! You did it again! Fortunately, you provided a Googlable quote, so I was able to reconstruct the link. But it’s back to HTML school for you!

This time I definitely put the URL in. Either my browser is allergic to your site, or vice versa. Still, and odd thing to happen even by error.

Perhaps Justin omitted “” in the <a> tag. Let’s see if this works . . . .

As I suspected,

BENIGNO NUMINEwas the armorial motto of William Pitt, first earl of Chatham. (I looked in Pine’sMottoesand the Rietstap Armorial.)A personal male God with an impersonal or semi-personal force of some sort associated with him that is personified as female is not unknown in IE-land: consider

Shiva/shaktiandJupiter/Juno.Junois personal in the late materials we have, but there are two plausible etymologies for it, and one is from*dyúh₃onh₂-(or its zr grd equivalent) ‘heavenly authority’, which sounds quite impersonal. If true,Zeus/Dionewould be another such pair. In this context, then, Jupiter’snumenwould be his (female-identified) power: the noun is neuter, but that doesn’t signify.Then there is God the Father and the Holy Spirit….

(TIL that

mulier‘wife’ may be a variant ofmelior‘better’, meaning first ‘the better woman’ and then ‘the best woman (of the household)’. Other explanations aremollior‘more tender’ or a connection withmulgere‘milk’.)The Creator in the traditional Kusaasi scheme is called

Win, which like the cognates in the surrounding languages doesn’t belong to the distinctively human noun class; although there are human-reference nouns in the class itdoesbelong to, they are mostly explicable as having been transferred from the distinctively-human class for phonological reasons, which doesn’t apply toWin. The self-same word is used for the spiritual essence ofanything, including but not limited to people.You greet an person sitting quietly alone with

(Bareka) nɛ sɔnsiga!“Blessing on your conversation!”, the implication being that they are conversing with their ownwin, which seems to imply that such awinis a “person” (in our terms, anyway.) But the kind ofwinathat are the focus of everyday religious activity are not regarded anthropomorphically at all: they’re much morenuminathandei. They are associated with places and spiritually powerful objects like trees. Just to confuse the issue even further, though, treesareregarded as “personal” in the traditional worldview, and referred to with animate pronouns …I suppose the moral is that “person” is

itselfa surprisingly culture-bound concept.You’re reminding me of how thrilled I was by my first anthropology class in college — people were so much more diverse than I had ever imagined! (It turned out to be my only anthropology class, because the teacher said such idiotic things about language.)

I wished when I lived in that area that I’d had more (or any, really) anthropological knowledge. I faked it as far as I could by listening a lot and keeping as open a mind as I could.

The best account of the the local regional culture in many respects that I’ve seen to date remains Ernst Haaf’s

Die Kusasefrom the early 1960’s. He was a doctor in the same hospital as me, and evidently had an extraordinary ability to get people to explain things to him about their culture. His accuracy is such that I can piece together nearly all of his many phrases in Kusaal, which he recorded with remarkable fidelity given that he evidently didn’t know the language and there wasn’t even an accepted orthography at that time. I’m sure it’s no coincidence that he was still remembered with affection thirty years after he had left.I suppose the moral is that “person” is itself a surprisingly culture-bound concept.Gosh, next you’ll be saying that reality is socially construed. Which, of course, is only another construction that can be put on what you say about “person”.

You merely have to take care that the construction doesn’t suddenly tumble about your ears while you’re dining. A traditional way of doing that is to invite only like-minded friends to dinner.

Reality is socially construed.

Well, that was fast. Too fast, in fact. I wonder whether to construe it as a tease. I suppose I should let it pass on probation for now. Can’t always be rocking the pillars of cognition.

@de

Watch yourself, Samuel Johnson may come back from the dead to kick a stone…

Tempted though I am to relapse into Sphinx-like silence in the vain hope of being taken for profound, I will instead relent and say that the key word there in my view is

construed.I don’t at all think that reality is

determinedby our understanding of it (though this does evoke the uncomfortable question: “How could you tell?”)Johnson’s argument, though fun, is of course bollocks (if I may be forgiven a traditional Scholastic technical term.)

I don’t at all think that reality is determined by our understanding of it (though this does evoke the uncomfortable question: “How could you tell?”)It’s Pyrrhonian Scepticism, all good fun. Not for the faint of heart, natch. I’m convinced it makes one a nicer person. Unfortunately the final results are not yet in as concerns

meine Wenigkeit.Speaking of key words – I decided that “constructivism” misleads the etymologically unwashed masses into thinking “determined”, so I avoid it now in favor of “construe” as being more infinitive.

“How could you tell?” In a free moment you might review Kant’s twistings and turnings on these matters. He ended up with the

Ding an sich. You have to admit it’s a good punchline – just mysterious enough to leave you in great expectations of a sequel, Son of Kritik.It reminds me of poor old Gödel; though he demonstrated that most formal problems in any system broad enough to at least include arithmetic cannot be resolved as true or false within this system, he did believe that the solutions

existed: it’s just that we have no way togetat them.You have to ask whether such a belief can actually be said to interact at all with the world, and if not, what you

meanby claiming to possess such a belief (it was in fact reflections along these lines that led to me first deciding, in my teens, that I wasmistakenin supposing I believed in God. I still think my logic was good.)A variant of the “but what

differencedoes it make” argument underlies that clever Bishop Berkeley’s position which Johnson deliberately failed to understand. It was regarded as uncontroversial at that time that (say) colour was not in fact inherent in objects, but generated by the interaction between object and observer: Berkeley just pointed out that if you can do that with colour, there’s no reason not to do it with extension, and ultimately the “object” as such becomes redundant.Good point about Gödel. From my reading over the years, I get the impression there are many mathematicians who think that way. It doesn’t really matter, in my view. They’re all grist to the mill of … hmm, not “progress” – lets call it “time”.

One of my favorite books is The Pilgrim’s Progress. I see things as follows: some fall by the wayside in their teens, others almost reach the Pearly Gates but are waylaid by cultish thieves in the night. A few get in, only to remember suddenly that they left the oven on at home.

I suppose the moral is that “person” is itself a surprisingly culture-bound concept.I may have told the story of Gale arriving at her therapy appointment looking very shaken. The therapist wanted to know what had happened, and Gale described (probably in tears) how she had seen a cat struck by a car and hurled high into the air, then crashing down in ruin.

The therapist said, “You’re reacting as if you’d seen a person killed.” When I heard the story that evening, I knew that therapeutic relationship was doomed. We cat people know that cats

arepersons; not human, of course, but persons. (Gale eventually left, but only after some months of deterioration; starting over with a new therapist can be a very daunting prospect.)bollocks“Complete bollocks”, one of my favorite Wikipedia meta-articles. But I actually think Johnson’s

argumentis pretty good: not logically, of course, but rhetorically.Pyrrhonian ScepticismI’m pretty sure, reading what little we know about Pyrrhon, that his skepticism was of the ordinary scientific sort: don’t claim you

know, because all you have is a theory more or less well confirmed. “The first principle is that you must not fool yourself and you are the easiest person to fool.” He extended this beyond the physical sciences: his key termataraxiais not ‘calmness’, still less ‘impassiveness’ or (Ghu forbid) ‘apathy’; it’s ‘non-attachment’.Gödel […] did believe that the solutions existed: it’s just that we have no way to get at themMost mathematicians do. But the alternative possibility (h/t Hofstadter) is that there actually is a finite proof of ~G. In that case, G is unequivocally false as well as unprovable, and there is no bifurcation of number theory at all, since neither G nor ~G can be added as an independent axiom. Which means supernatural numbers are inherently part of that undivided theory, and so are their reciprocals, which are infinitesimals. And in that case, Newton and Leibniz were right after all, and Johnson was quite right to refute Berkeley!

@jc

For muller I have not seen *mal+yos “very/rather small” proposed; it would be like “babe” or “chick”, but maybe it is impossible to schwa an a in a Latin first syllable. Etienne would know. I also found Germanic *magaths, Celtic *maguessa for maid, these would go back to a PIE root for “young”, I suppose, but the Latin reflex would not be l for g or g-t.

@John Cowan: The truth of falsehood of the Gödel sentence G is in some ways less complicated than people make it, and in some other ways, it is more complicated. For a countable first-order logical system S, G is a sentence that is

equivalentto a statement that “G is unprovable in S.” This statement is, by any reasonable reckoning, true. The system S contains only a countable number of possible proofs, so if there is a proof of G in there, we could eventually find it. Since we cannot prove or disprove G (in first order logic), no such proof exists; therefore, G is indeed unprovable under the rules of S. However, this last argument cannot be framed in the first-order language of S itself.That is the sense in which things are simpler: G is unquestionable true in S. Where things get more complicated is when we can append either G or ~G as a new axiom and still have a consistent system (assuming S was consistent itself—not always a trivial assumption!). The relative consistency of ~G as an axiom seems to contradict the statement that G is actually true, but it does not actually. Adding ~G to S creates a new first-order system S’, and G is

notequivalent to a statement that “G is unprovable in S’,”—unprovable in the new system. So there is no contradiction, although it still may seem weird that a statement G that is true in S can be false in an extension of S. However, that should not really be too surprising; since G is a negative statement, adding to the theory can make it false. If I say, “There are no women in this room,” that is a true statement, but if I extend the system under study—either in space, to “There are no women in this building,” or in time “There have been no women in this room in 2020,”—the statement becomes false. So a statement G, which asserts that there is no proof of something (G itself, in this case) can likewise be made false if we pass to a larger system S’, which contains more possible proofs, because one of the new proofs may be a proof of G.Separately, something that this discussion thread reminded me of, was that I had recently come across a couple of individuals claiming that falsificationism, as described by Popper, had been declared philosophically dead or unsound. When I looked into this, I found that the apparent basis of the claim was that Popper’s solution to the demarcation problem (of what is scientifically meaningful and what is not) of falsifiability was unsound, because the falsifiability criterion was itself not falsifiable.

I am sure that whatever jackass philosopher came up with this argument thought that he was being incredibly, deviously clever, but it honestly made me pretty angry. That argument is really only slightly more advanced than, “If you love falsificationism so much, why don’t you marry it?” I am no fan of Popper (as I have noted here more than once previously), but he really did codify a useful criterion for deciding if some line of inquiry was scientific or merely metaphysical. He was not especially good at applying that criterion in practice—at least in the interesting edge cases where having a precise demarcation criterion would actually be useful—and he spilled far too much ink on this one idea, but the basic idea was a very good one and fundamentally sound. The reason that a rule of falsifiability does not need to be itself falsifiable is that it not a statement about the nature of the world, but belongs to the realm of epistemology. Like statements about uncountable sets, it is not an assertion about the nature of the universe, but a purely logical rule, which can nonetheless be used to distinguish potentially meaningful empirical statements from useless ones.

I am sure that whatever jackass philosopher came up with this argument thought that he was being incredibly, deviously clever, but it honestly made me pretty angry.Yes, that’s right up there with “If you linguists think standard language is arbitrary, how come you write in it, huh?”

Science theory is part of philosophy, not of science. That’s why it’s called “philosophy of science” and why it doesn’t need to be falsifiable. 😐

(Also, there’s a lot of parsimony hidden in falsification.)

Here’s what Hofstadter actually says: better to critique that than my rendering of it. This is after he explains that either G or ~G can be added as an axiom, which is what you are addressing above:

I’ll add for the benefit of those who need it what ω-inconsistency is. Suppose that for some formula F, you can prove that F(0) is true, F(1) is true, and so on forever, but you

cannotprove “For all natural numbers n, F(n) is true.” So the truth of this last statement (call it G) is open, and we can add either “G is true” or “G is false” to the theory. If we add “G is false” we get ω-inconsistency: F is true for 0, for 1, for … and yet there are numbers for which it isnottrue: these numbers (which are not negative or fractional or imaginary or any other such well-known notion) are called supernatural.Fair enough. If supernatural numbers exist, then my argument for the truth of G does not hold. A great many other theorems also fall to pieces as well.

Oh, that kind of supernaturals. WP has an article on another kind.

John, how do you even take the inverse of such a number? Naturals that are not ordinals… are they necessarily greater than all ‘mundane’ naturals? Obviously that would make the notional inverse non-zero but less than any real number, thus infinitesimal.

@Lars Mathiesen: The supernaturals are indeed all greater than the canonical natural numbers. The supernaturals are defined by their prime factorizations, which look like the factorization of ordinary naturals, except that they have an infinite number of factors. Moreover, you can append to this theory additional supernatural primes, in additional to the canonical primes, which lead to even more supernaturals.

However, you cannot really define consistent theories of division (or even addition!) of supernaturals. Multiplication is easy to define, just adding the powers of the prime factors, but that’s about the only operator available with them.

I’m reasonably sure that the supernaturals you linked to and the ones we’ve been talking about are the same.

Here’s Hofstadter again on addition and multiplication:

It is not obvious to me that the supernaturals of an ω-inconsistent theory are necessarily the ones that Steinitz invented / defined, and the WP article for those does not talk about number theory or indexing schemes with triples.

I haven’t read the Hofstadter book (I think — which one is it?), but if he starts off with Steinitz’ definition then of course ω-inconsistency is possible. It’s the other direction I’m wondering about.

https://en.m.wikipedia.org/wiki/Hyperinteger

https://en.m.wikipedia.org/wiki/Talk:Supernatural_number

GEB chapter 14. The first quotation is at pp. 452-53, the second one at p. 449 of the first edition.

From that talk page (thanks, PP): “Hofstadter’s discussion of ‘supernatural numbers’ comes on pp. 451–455 of the 20th Anniversary Edition of GEB. My very uneducated guess is that the entities he’s talking about are what Wikipedia calls hypernatural numbers, i.e. hyperintegers.”

The hyperintegers make more sense, at a glance. There is a link from that WP article claiming that the hypernaturals (non-negative hyperintegers) constitute a Skolem-type non-standard model of Peano arithmetic and that article states that Gödel’s incompleteness theorem can be used to prove the existence of such models. Still nothing about indexing schemes. (However the hyperreals are a field so division is well defined —

contraHofstadter’s remark — and any hyperinteger that is not an integer will indeed have an infinitesimal as its inverse in that field).I did read GEB 40 years ago, but this part would not have been interesting to me back then and my copy is long gone (and it was not the Anniversary Edition anyway). If the quoted sections are all that Hofstadter wrote about what his supernaturals actually are, I don’t think we’ll get any closer.

Madness!

Ultrapower.

Jerzy Łoś. I just like how his name looks.

@Brett:

Do you remember whose claim is this? I’m more familiar with the Neurath/Feyerabend/Lakatos critique: falsificationism, unlike earlier epistemologies,

couldin principle be the basis for a science, but it sure as hell doesn’t account for science as she is actually practisedProofs and Refutationsis one of the half-dozen most interesting books I have ever read.(And far and away the funniest work on the philosophy of mathematics – against stiff competition, of course.)

OK, now I’m intrigued; I’ve found this excerpt (pdf), and will check it out forthwith.

It took me days to get through it when I first read it (despite its brevity) because I had to stop and think so often.

Be reassured, one gradually gets used to thinking. It’s not a matter of practice, though. Habit is inimical to light-bulb moments.

Back in the days when I was plowing the

œuvresof the Feyerabend, Kuhn, Foerster crowd, I think I read only one something by Lakatos on the fly. Somehow the idea (from the name) that he was Greek reduced my interest. I was already getting a little fed up with Feyerabend’s repetitiveness. BTW there are a few video interviews with him that show him as not so wigged-out after all – like those of Luhmann do for Luhmann.Gosh, how I do ramble on.

One gradually gets used to thinkingOK. I’ll just have to take your word for it …

Of Feyerabend, I’ve only read

Against Method, which I thought entertaining enough but not especially thought-provoking. I agree about the repetitiveness. I suppose it’s not easy to write at length on the topic without repetition when your message boils down to not much more than “thereisno method/spoon.”Heh, I was taking for granted the name was Hungarian…

gelernter Österreicher“Austrian by trade/training”gelernter ÖsterreicherThat’s cute. It could also be a typo for

gelehrter Österreicher, which is almost pleonastic when allowances are made for Schwarzeneggers and Würste.I know only one gelehrter Österreicher on this site.

So there is hope for those of us who are not Austro-Hungarian

ab ovo? We, too, might contribute to the philosophy of science one day afterbecomingAustrian?[Obviously it is not possible to become Hungarian.]

Provided you are sufficiently

gelehrig. [Word For Today]Or

anstellig. [Synonym For Tomorrow]Conchita Wurst

I might try to become mildly Austrian; that level of advanced Austrianity is, alas, beyond my waning powers. Indeed, to be brutally honest with myself, I might never really have been in the running, even in my fiery youth.

I

dohave the beard, though.I imagine you bearded the best, at that inflammable age.

Owê, war sint verswunden alliu mîniu jâr?

iemer mêre ouwê !

A bit of poking about shows that

supernatural numbersis indeed an alternative term for Robinson’s hypernaturals (perhaps promulgated by a peever opposed to Latin-Greek hybrids). I have yet to find anyone saying in so many words that the Steinitz supernaturals are or are not the same as the hypernaturals.Obviously it is not possible to become Hungarian.Not at the present moment, perhaps (a million people have gained Hungarian citizenship under the 2011 law, but they were ethnic Hungarians already). But I also find a 2017 article “Signs That You Are Becoming Hungarian” (part 1, part 2). There is also Daniel Viragh’s 2014 dissertation,

Becoming Hungarian: Jewish Culture in Budapest, 1867-1914.I remember Robinson”s terms as “standard” and “nonstandard” numbers. Whatever you call the elements of his field extension of R, it is a field.

The Steinitz “supernaturals” can’t be part of a field, since they can’t be added, but only multiplied.

Robinson’s numbers are not the same as Steinitz’ numbers.

“Signs That You Are Becoming Hungarian”That’s the headline, meant to sucker you in; the piece itself talks about becoming “a little bit Hungarian,” which is a very different matter.

The context for

gelernter Österreicheris that it’s a common comment, especially by journalists and politicians, on political, bureaucratic or similar situations: “as an Austrian by training, you know what comes next” – “they don’t tell you when you come in, it’s not innate knowledge, and it doesn’t make enough logical sense that you could figure it out on your own, but you’re familiar with it because you’ve experienced it many times before”.See also:

österreichische Lösung“weird compromise that has probably been made permanent but isn’t really a solution –Lösung– at all”.Are you sure

warhad a short vowel? This interesting paper argues it’s been long ever since Proto-Germanic (…while solving the ancient problem of the vowel history ofhereand demonstrating the importance of phonetic pendantry in historical linguistics: sound changes happen to sounds, not to phonemes… well, the difference between [ɛ] and [e] will hardly strike anyone familiar with West Africa as pedantic, but…).Heh, I was taking for granted the name was Hungarian…Indeed, there are few if any Greeks named Imre, though there’s a speculation that it’s a Magyarization of

Amalareiks/Emmerich/Emory/Emericus/Americus/Amerigo.Robinson’s numbers are not the same as Steinitz’ numbers.Thank you for being definitive.

That’s the headline, meant to sucker you inI cheerfully confess to not reading the article.

Oh yeah, I had forgotten he’s an Imre. And yes, that was traditionally equated with Emmerich back when anyone bore

thatname.It is certainly well known that the best way for a Jew from Russia to become Russian is to move to the U.S., where the transformation is instantaneous.

Are you surewarhad a short vowel?Ah! well caught. I was thinking in Gothic, as usual. But if the paper is correct, I may have to modify my pronunciation of

ƕar. These historical prescriptivists, always telling us how we ought to have spoken!Like that of Trevor Noah, who moved in from South Africa and suddenly found himself black.

@Alon Lischinsky: I have seen that supposed refutation of Popper alluded to a few times, but I have never had the time or the inclination to track the argument back to it’s source.

Popper did have one good idea about the philosophy of science, which is one more than most philosophers. However, he, along with almost all the contemporary philosophers who debated his work, had so little knowledge of statistics and so little idea how scientists actually worked, that any specific applications of falsifiability that they tried to work out were generally useless.

Obviously everybody will have seen this already, but it would be

wrongnot to link to it:https://xkcd.com/2078/

Well, I at least had not seen it. Or so I think.

(*dusts off and dons mathematician hat*)

The supernaturals are defined by their prime factorizations, which look like the factorization of ordinary naturals, except that they have an infinite number of factorsPerhaps we should say that they have a finite-but-supernatural number of factors 😉

It’s really quite hard to formalize that the naturals are supposed to be “just 0 and numbers that follow it”. For another approach, of the sort where addition is easy but multiplication is hard: it’s entirely possible to affix to the usual naturals additional countable series of the shape {… ö-3, ö-2, ö-1, ö, ö+1, ö+2, ö+3, …}. Ruling these out by various naive approaches doesn’t work; for example, if we claim that a natural number must be expressible by at most N iterations of the successor function applied to 0, where N is a smaller natural number — then still nothing stops us from treating e.g. ö as S^(ö-1)(0), that is, S applied to 0, just (ö-1) times. Peano’s axiom of induction doesn’t help either since perhaps induction, too, naturally spills over into this “second helping” of natural numbers. Nonstandard models of arithmetic can be explicitly constructed where this is indeed the case. To keep arithmetic working, they end up requiring indeed at least a countable number of such additional series, so that for one such supernatural ö and any rational p/q, there will have to be a separate series “centered on” ö·p/q. E.g. if we can apply S(x) ö times, which is to say always calculate the sum n+ö, we can do this again and acquire ö·2, which cannot be in the same series as ö.

And although supernaturals could seem “infinite”, this is incorrect: “infinite” is not a primitive notion! Infinite numbers by definition have to be larger than every natural number. Yet the supernatural series just keep going. ö is not larger than ö+1 or ö·2, which would be themselves also natural numbers. Or, if you want a natural number not defined in terms of ö: we can rename the indexing to start at ö-1 = ä, and then establish that ä·2 > ö.

The nonexistence of supernaturals can only be stated by bringing in heavier guns: second-order axioms such as “there does not exist a nonempty proper subset of the natural numbers closed with respect to the successor function” or “the natural numbers ordered by the successor function must have order type ω”.

But if Peano induction potentially includes the supernaturals, won’t order type ω do so as well? Ordinals being constructed by induction and all.

If i remember rightly (this was 1979) my undergraduate course notes in the construction of the number system just made the existence of (N, 0, succ) an axiom where succ was injective, and surjective on N \ 0. I’m pretty sure that doesn’t rule out supernaturals but they were not mentioned, much less was their existence attempted to be disproved.

(Maybe they saved the bacon by specifying that N was the unique such object (up to bijection, obvs). I don’t know if it helps, though).

@Lars Mathiesen: Well, that’s the thing about second-order logic. Since you can quantify over sets, you need to have at least some minimal notion of the Universe of sets already, before you start to state propositions.

Sure, and I can see where the thing about no closed proper subsets is second order and excludes supernaturals. Clearly I know too little about order types to see how the other thing works.

In set theory, “induction” is defined purely as closure-under-successor. This however ends up being

transfiniteinduction, which applies not just to the naturals but to all limit ordinals (e.g. the successor of every countable ordinal is also countable, and thus ω₁ is inductive). The naturals are then defined as the smallest inductive set. So there will be no separate notion of “Peano induction”.Peano’s second-order formulation of the axiom of induction restates the smallest-inductive-set condition as indeed the demand that there are no proper subsets closed under successor and including 0. This will exclude the possibility of supernaturals (the “mundane naturals” would be such a proper subset). Trouble only begins if we try to abandon the second-order formulation.

I’m now confused on a higher level. WP was helpful for once in explaining how to get from the existence of some inductive set to the naturals — basically take the union of all inductive sets to get the smallest one which has to be the ‘mundane’ naturals. (Or we assume it corresponds to that notion).

I think what confused me is that induction only works for the naturals (and/or the ordinals), not (necessarily) for any inductive set — programming language semantics likes what it calls ‘inductively-defined’ sets like the sets of lists of numbers over which you can indeed prove stuff by induction.

And now we’re back at lists, Stu, just with a fancier word.

Oh, I think I get it now. A proof by induction ‘really’ only shows that a proposition is true on

someinductive set (but since the naturals are (by construction) a subset of all inductive sets, Bob’s your uncle). That nugget is so obvious that Google can’t find it, but it explains the ‘inductive set’ naming.Us country folk think of “induction” as a straightforward consequence of the axiom of regularity (“foundation”):

# Given the other axioms of Zermelo–Fraenkel set theory, the axiom of regularity is equivalent to the axiom of induction. #

However, Aczel’s AFC (anti-foundation axiom) is cool, and even comes with an intuitive model: “accessible” (every node is reachable from the root) rooted (“pointed”) directed graphs. The following sentence at the link says something I did not know, but I suppose makes the AFC easier for Trump supporters to accept:

# Virtually all results in the branches of mathematics based on set theory hold even in the absence of regularity; see chapter 3 of Kunen (1980). #

“Virtually all” – for some value of “virtual” that excludes generalized induction, I guess. The natural numbers were always good enough for me. Sometimes “list” is a useful way to think about or implement something, sometimes not.

Yes, but, to prove that there’s a minimal inductive set to do induction over you have to prove that there is one, i.e., have the ‘axiom’ of induction as a lemma. (It’s not that straightforward to me, as you may have been able to tell). And regularity does not help me understand why inductive sets are called that.

(I don’t know how to go the other way and get regularity from induction either, but anyway I don’t think it’s germane to the potential existence of supernaturals).

I’m sure that if the lecturer had gone all foundational on us and derived the naturals from the axiom of foundation he would have had fewer exams to grade. But they chose to do the winnowing by introducing metric spaces in the first lecture of Math 101. (It was very effective).

I have not yet found myself convinced that I need to know what “inductive sets” are – if they are anything more than succ-closed sets. I probably haven’t followed the discussion closely enough. As for (non-zero) supernaturals, they are dead to me because they make everything into a zero-divisor and can’t be added anyway. And as for metric spaces – that’s like sleeping with pyjamas.

There are big holes in my math knowledge. Last year I bought Topics in Algebra by i.n. herstein. Yesterday I was skimming it again, as I do, and discovered that normal subgroups R of a group S are simply the kernels of homomorphisms of S. I had never understood the motivation behind that aRa(-1) = R

definitionalcrap that you find in older books.Well, succ-closed sets that contain zero (an element that is not succ of anything). In set theory (as opposed to Peano arithmetic) succ(

x) =x∪ {x} and zero is ∅ (which is the equivalent in the von Neumann construction, of course) and I can vaguely see how the existence of an inductive set might be used to prove all sorts of things, but I wouldn’t be up to doing it myself.EDIT, short of time: It seems that the axiom of induction is about epsilon-induction, which I can’t tell if what relation it has to natural numbers.

An empty set and a set with only the empty element walk into the bar. “What are you drinking guys?” — “There is only one of us.”

More like

“There are zero of us”

von Neumann numbers get complicated fast, but not that fast.

0 = ∅

1 = {0}

2 = {0, 1} ( = {∅,{∅}} as per D.O.)

3 = {0, 1, 2}

and so on. Every number is a set with that number of elements.

@Stu Clayton: I think for a long time abstract algebra textbooks were written with the assumptions that students studying the subject would have already taken linear algebra. In linear algebra, you need conjugation to change the basis, so the potential usefulness of something like

gAg⁻¹ could have been already apparent to somebody with a linear algebra background. Nowadays, many students in pure mathematics do not take a separate linear algebra; instead, they get a very abbreviated discussion of vector spaces and linear transformations as part of a year-long abstract algebra curriculum. I had never seen more than the most rudimentary linear algebra of eigenvalues and eigenvectors before I took Algebra I–II, so when conjugation was initially introduced as an abstract group action, and normal subgroups introduced as subgroups that were invariant under conjugation, it seemed totally arbitrary until we talked about group homomorphisms.Once I understood why normal subgroups were important though, I really appreciated them. I think my favorite homework problem in our abstract algebra textbook was one that really combined almost everything we had learned up to that point about group theory. We were given the character table for the representations of a certain finite group and told to give two proofs, based just on the information in the table, that the group was simple (with no proper normal subgroups). One way to do it was see that there was no union of conjugacy classes whose size divided to the size of the group. The other way was by noting that all the group representations, except the trivial representation, were faithful, so there were no nontrivial homomorphisms from the group with a nontrivial kernel.

Separately, on the topic of encoding things like the integers as sets:

When I took real analysis, although I did not enjoy the course much, I did appreciate one thing that the lecturer told us. We studied a lot of foundational material before turning to the main topic of measure theory, which meant constructing the natural numbers, ordinals, real numbers, etc., starting just from the axioms of set theory. However, Prof. Richard Dudley (who apparently died just a couple months ago) told us that it was wrong—or at least unwise—to thing of the real number as if they were “really” the objects we had constructed. The real numbers were what we had always thought of them as—a continuum of numbers lying in between the separated integers. Likewise, the integers were the set of positive and negative whole numbers we were used to, not some exotic set-theoretic construction. The constructions that we studied were important, because they showed that things like the integers and the reals could be encoded in the language of set theory, meaning that the objects themselves were not pathological. But once that was shown, we could dispense with the explicit set theory model and understand the real numbers as basic objects, in the way we always had before studying analysis.

The reals that can be encoded in set theory are not the real reals.

I always liked adjoint groups. If multiplication is not commutative, one would like to know how much non-commutative it is.

Walter Rudin, Principles of Mathematical Analysis: “Experience has convinced me that it is pedagogically unsound (though logically correct) to start off with the construction of the real numbers from the rational ones.”

@D.O.: I don’t remember that quote from Baby Rudin. However, I do remember that he put Dedekind’s construction of the real numbers in a special supplementary section, and that we skipped over it completely in my first undergraduate analysis course.

I think I was taught to think of the reals as the topological completion of the rationals, and Dedekind was covered later as ‘but you can do this as well’. We didn’t have a set order of courses beyond year one, so for instance measure theory could not presuppose any specific model of the reals.

We didn’t have a set order of courses beyond year oneBut it sounds like you took proper classes, otherwise you wouldn’t know what you do. How did the staff manage a set of them ?

Year two and three were like ‘pass two of four offered courses in each semester’, so there was an order (and probably some recommended prerequisites), just not the same for everybody. I think I took construction of the number system and measure theory in parallel. “This is a university, not our job to tell you what you need to learn, innit? But you’re welcome to ask for advice.” This was the seventies when people didn’t expect to have their hands held. Now my nephew calls it ‘school’.

(Strictly speaking this was all a bit circular, metric spaces were defined using real-valued metrics and the resulting topology on the rationals used to define the reals. But real analysis was a parallel course to the one with metric spaces and they didn’t want everybody to drop out — so we just took the continuous real line as given and partied like it was 1684.

1675. (We also covered linear algebra in that course, however).

You missed my silly pun unfortunately [“proper class”]. Or perhaps you did not deign to give it the time of day.

Sorry, I was in a literal mindset (don’t look for anything, I’m not trying) I guess. But yeah, the classes constituted a partially ordered set so they were clearly not proper.

@Lars Mathiesen: In Analysis I, one reason Prof. Helgason said he skipped the Dedekind cuts definition of the reals was that it was more natural to use the general construction of the completion of a metric space to produce the reals from the rationals. We actually studied two different constructions of the completion—the usual one involving equivalence classes of Cauchy sequences and another lesser known formulation, which Rudin also gives in one of the exercises. However, to truly do the completion of the rationals properly, you need to start with metric spaces in which the metric is only rational-valued, which naturally complete to a more general real-valued metric.

metric is only rational-valued— yes, and that was one of the things that was glossed over. It’s not like it’s hard to do, it’s just easier to start with the naive notion of a real-valued metric.Completion of the rationals as a metric space is exactly the Cauchy sequence equivalence thing, since a metric space is complete when any Cauchy sequence has a limit and testing whether one Cauchy sequence converges to the limit point of another gives the equivalence relation. I think we covered completion for metric spaces in the abstract and applied it to the rationals in a later course.

OT Lars: Dana Scott remembering.

Boy, Dana Scott is a grumpy-looking guy.

He tells great anecdotes and deadpan jokes.