Symbol for strict subset

[Benoit]

The symbol "⊂" (latex \subset) is used by many authors to mean "strict subset" (also called "proper subset") and by many (probably even more) authors to mean (non-necessarily strict) "subset".  Therefore, it is ambiguous.  To remove this ambiguity in set.mm, I propose to use the symbol "⊊" (latex \subsetneq, unicode ⊊).  I can carry out the change in the althtmldef and latexdef if you agree.

To be clear: my point is not to decide whether it's better to give the symbol "⊂" the meaning of strict or non-strict; my point is that, as a matter of fact, this symbol is ambiguous, and that there is an easy way to alleviate this ambiguity.

Benoît

[David A. Wheeler]

On Wed, 10 Oct 2018 08:10:30 -0700 (PDT), Benoit <benoit...@gmail.com> wrote:
> The symbol "⊂" (latex \subset) is used by many authors to mean "strict
> subset" (also called "proper subset") and by many (probably even more)
> authors to mean (non-necessarily strict) "subset". Therefore, it is
> ambiguous. To remove this ambiguity in set.mm, I propose to use the symbol

> "⊊" (latex \subsetneq, unicode ⊊). I can carry out the change in

> the althtmldef and latexdef if you agree.
>
> To be clear: my point is not to decide whether it's better to give the
> symbol "⊂" the meaning of strict or non-strict; my point is that, as a
> matter of fact, this symbol is ambiguous, and that there is an easy way to
> alleviate this ambiguity.

Historically, up through about 1910 or so, I think \subset always meant
"non-strict subset". But that seems to have changed, and I think \subset
practically universally means "strict subset" in the last ~50 years.
I think the correlation with "<" is pretty strong.

So I don't think this *needs* to be done. But if others agree to it, I'm fine with it.

--- David A. Wheeler

[Giovanni Mascellani]

Hi,

Il 10/10/18 17:49, David A. Wheeler ha scritto:

> So I don't think this *needs* to be done. But if others agree to it,
> I'm fine with it.

I personally never use \subset in my mathematical writing because of
this ambiguity, so I would welcome this change. However, that is a
matter of taste (to my personal taste, the usage of \mathbb{N} to denote
natural numbers without the zero is much worse...).

Giovanni.
--
Giovanni Mascellani <g.masc...@gmail.com>
Postdoc researcher - Université Libre de Bruxelles

[Norman Megill]

I don't have a strong preference, but if we do this should we also change "less than" for consistency?
http://us.metamath.org/symbols/lneq.gif seems like the style corresponding to your proper subset suggestion.  I think I have seen an older work on lattice theory where "<" was used for non-strict partial order (less than or equal to).

I will say that I have doubts that anyone has been seriously confused or misled by this.  In a book, the author usually states the symbol convention up front, and we have a link to the proper subset symbol at below essentially every proof using it in case there is even the slightest uncertainty.

Also, I don't recall a given book using both \subseteq and \subsetneq.  It's either \subseteq and \subset, or its \subseq and \subsetneq.  We would not be completely consistent with either convention.

Norm

[Benoit]

The symbols < vs \leqslant are standard and should remain as they are.  Indeed, this is not fully consistent with \subsetneq vs \subseteq, but it is more important to ensure non-ambiguity.  If most books use either \subset vs \subseteq or \subsetneq vs \subset, then it is reason enough to use \subsetneq vs \subseteq to avoid ambiguity (this was my original point).

What I can tell you for a fact is that in a majority of cases, when a mathematician sees "⊂" in an article, a book, or on a blackboard, he assumes "\subseteq".  The proportion is probably closer to 90% of cases (though maybe among set theorists, the situation is different; see however https://en.wikipedia.org/wiki/Wikipedia_talk:WikiProject_Mathematics/Conventions#Literature_survey).  This is mainly because the notion of strict subset is much less important and fundamental than that of subset, similarly as the notion of "non-square rectangle" is much less important than the notion of rectangle; use the simpler symbol for the more important notion.

As for me, I use \subseteq (and so far I haven't needed a symbol for strict subset) when I'm on my own, but my coauthor forces me to use \subset (he has more seniority, so I don't get to choose...), and it obviously means "(non-strict) subset" for him.

As for the risk of confusion, indeed, no one would be seriously confused, only sligthly confused for a little while, like I was for a few seconds when wondering why the proof of nthruc was so long (by the way: nthruc is actually four different theorems, but that's another story).  But no confusion is better than little confusion.

Finally, I agree with Giovanni for natural numbers being denoted by \mathbb{N} (and the (strictly) positive ones by \mathbb{N}_{>0}): one should use the simpler symbol for the more important object.  The situation may be less clear cut than for \subset, but when no indication is given, it is probably majoritarily assumed that \mathbb{N} contains zero.  But maybe here, it is more a matter of taste (and it would be overkill to avoid \mathbb{N} altogether).

Benoît

[David A. Wheeler]

On Thu, 11 Oct 2018 02:08:51 -0700 (PDT), Benoit <benoit...@gmail.com> wrote:
> The symbols < vs \leqslant are standard and should remain as they are.

I agree. There's no ambuity with < vs. <=, so let's use standard symbols.

> What I can tell you for a fact is that in a majority of cases, when a
> mathematician sees "⊂" in an article, a book, or on a blackboard, he
> assumes "\subseteq".

The evidence doesn't support that assertion. Indeed, I think the
evidence clearly shows that there is NOT any kind of universal agreement.

First, the ISO standard for mathematical notation (80000-2)
and the US NIST standard for mathematical notation
use ⊆ for "proper subset or equal to" and ⊂ for proper subset.
That's the same convention we currently use.
I think it's pretty defensible to use the mathematical
notation standardized by standards bodies :-)... and
so clearly NOT everyone thinks that ⊂ means "proper subset or equal".

I greatly appreciate the link to this:
https://en.wikipedia.org/wiki/Wikipedia_talk:WikiProject_Mathematics/Conventions#Literature_survey
I added some more information to that, esp NIST, ISO, and publication dates.

For completeness, here's the current list:
Uses ⊆ for "proper subset or equal to" and ⊂ for proper subset.
Devlin, The Joy of Sets, 2000.
ISO 80000-2, Quantities and units - Part 2: Mathematical signs and symbols to be used in the natural sciences and technology, 2009-12-01, section 5 definition 2-5.7 and 2-5.8.
Suppes, Axiomatic Set theory (June 1, 1972).
Stoll, Set Theory and Logic (October 1, 1979).
(US) National Institute of Science and Technology (NIST), Digital Library of Mathematical Functions (Introduction, subsection Common Notations and Definitions), 2018-09-15
Just and Weese, Disovering Modern Set Theory (December 5, 1995).
Takeuti and Zaring, Introduction to Axiomatic Set Theory (1982).
MathWorld
Uses ⊂ for "proper subset or equal to". No special notation for proper subset.
Folland, Real Analysis (May 1, 2007).
Kunen, Set Theory (November 2, 2011).
Jech, Set Theory (Apr 28, 2006).
Halmos, Naive Set Theory (August 17, 2011).
Halmos, Measure Theory (February 28, 1978).
Rotman, An Introduction to the Theory of Groups (1995).
Lang, Algebra (2002).
Hungerford, Algebra (February 14, 2003).
Quigley, Manual of Axiomatic Set Theory (1970).
Willard, General Topology (February 27, 2004).
Lipschutz, Seymour, Outline of Set Theory and Related Topics (July 22, 1998), ISBN 978-0070381599.
Uses ⊂ for "proper subset or equal to" and ⊊ for proper subset.
Munkres, Topology (January 7, 2000).
Uses ⊆ for "proper subset or equal to". No special notation for proper subset.
Kechris, Classical Descriptive Set Theory (1995).
Riesz and Sz-Nagy, Functional Analysis (June 1, 1990).
Uses ⊆ for "proper subset or equal to" and ⊊ for "proper subset".
Moschovakis, Descriptive Set Theory (June 30, 2009).
Wikipedia article on "subset" on 2018-10-11

The Wikipedia article on "subset" at https://en.wikipedia.org/wiki/Subset
currently uses ⊆ for "proper subset or equal to" and ⊊ for "proper subset", saying:
"Some authors use the symbols ⊂ and ⊃ to indicate subset and superset respectively; that is, with the same meaning and instead of the symbols, ⊆ and ⊇.[2] So for example, for these authors, it is true of every set A that A ⊂ A.
Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and proper superset instead of ⊊ and ⊋.[3] This usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤ y then x may or may not equal y, but if x < y, then x definitely does not equal y, and is less than y. Similarly, using the convention that ⊂ is proper subset, if A ⊆ B, then A may or may not equal B, but if A ⊂ B, then A definitely does not equal B."

In short: I think our current notation for subset is entirely defensible.
There are a number of set theoretical books that use it, and it's
standardized by ISO and NIST.

HOWEVER, if the argument is that "the notation should be as unambiguous as possible", I think that's a decent argument for displaying ⊆ for "proper subset or equal to" and ⊊ for "proper subset". It appears that's what the Wikipedia article did, for example, because different sides couldn't otherwise agree on a notation.

--- David A. Wheeler

[Norman Megill]

If Wikipedia does it, that's good enough for me. :)  And it seems no one here has any actual objection.

Therefore, Benoit, you may issue a pull request with the htmldef and althtmldef updated.  I don't think we should change the ASCII token "C.".  For the htmldef, use the symbol subsetneq.gif
http://us.metamath.org/symbols/subsetneq.gif  (12 x 19)
and I will modify install.sh so it will get copied to the mpegif directory during site build.  (If you prefer another gif, let me know.)

Leave the old htmldef and althtmldef commented out, and add your initials to a dated comment above the new one.

Norm

On Thursday, October 11, 2018 at 10:59:02 AM UTC-4, David A. Wheeler wrote:

[fl]

 

 (to my personal taste, the usage of \mathbb{N} to denote
natural numbers without the zero is much worse...).

I agree with that. Personally I'm always forgetting the exact

definition of NN. I propose that we use NN0 and NN1 instead

in set.mm or much better ZZ>. NN is a historical and useless legacy 

that should disappear.

--

FL

[David A. Wheeler]

(This is a new thread for a different topic that started in "Symbol for strict subset".)

Giovanni Mascellani:

> I personally never use \subset in my mathematical writing because of
> this ambiguity, so I would welcome this change. However, that is a

> matter of taste (to my personal taste, the usage of \mathbb{N} to denote

> natural numbers without the zero is much worse...).

Benoît:

> Finally, I agree with Giovanni for natural numbers being denoted by
> \mathbb{N} (and the (strictly) positive ones by \mathbb{N}_{>0}): one
> should use the simpler symbol for the more important object. The situation
> may be less clear cut than for \subset, but when no indication is given, it
> is probably majoritarily assumed that \mathbb{N} contains zero. But maybe

> here, it is more a matter of taste (and it would be overkill to avoid
> \mathbb{N} altogether).

"'fl' via Metamath" wrote:
> I agree with that. Personally I'm always forgetting the exact
> definition of NN. I propose that we use NN0 and NN1 instead
> in set.mm or much better ZZ>. NN is a historical and useless legacy
> that should disappear.

I'm a big fan of symbols with "obvious" semantics.
Since there are two different meanings for "natural", clearly
"natural" isn't very natural :-).

I'll try to find some information about the prevalence of different
symbols and semantics for "natural numbers".
If anyone knows of anything, please reply to this thread!

--- David A. Wheeler

[Norman Megill]

NN was originally chosen to start at 1 because that's what most analysis books did, and I figured complex numbers were the start of analysis. This contrasts of course with early set theory where om starts at zero, and computer science where I think nat usually starts at 0.

I'll change it to NN1 (ASCII token) and \mathbb{N}_1 (displayed symbol), and there will be no NN, so we can put this to rest.

Norm

[David A. Wheeler]

I tried to find out about symbols & semantics for "natural numbers".

Basically, there's no agreement; when someone sees "N" or says
"natural number", nobody can be sure what it means and you have to
look it up. In the text below, where I write "sub" it
means "subscript", and "sup" means "superscript".

We currently use Fraktur N to mean { 1, 2, 3, ....} and Fraktur N sub 0 to mean { 0, 1, 2, ...}.

ISO 80000-2:2009 defines "N" (set of natural numbers) as:
N = {0, 1, 2, 3, ...}. Note that this is NOT the same as our current notation.
N* = { 1, 2, 3, ...}.
N sub >5 = { 6, 7, 8, ... }
(see section 6, item 2-6.1).
In general they add a "*" superscript to indicate "omits 0", so R* is "the set of reals excluding 0", C* is "the set of complex numbers excluding 0", and so on.

The NIST "Digital Library of Mathematical Functions"
https://dlmf.nist.gov/front/introduction
defines Fraktur "N" as the "set of all positive integers" (as we currently do).
I don't see any simple expression for {0, 1, 2, ...} in the NIST document.

There was nothing here:
https://en.wikipedia.org/wiki/Wikipedia_talk:WikiProject_Mathematics/Conventions

The Wikipedia discussion of Natural number notes the controversy:
https://en.wikipedia.org/wiki/Natural_number
"Some definitions, including the standard ISO 80000-2,[1] begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, ….". It also discusses ways to eliminate the ambiguity:
To be unambiguous about whether 0 is included or not, sometimes an index (or superscript) "0" is added in the former case, and a superscript "*" or subscript ">0" is added in the latter case:[1]
ℕ sup 0 = ℕ sub 0 = {0, 1, 2, …}
ℕ sup * = ℕ sup + = ℕ sub 1 = ℕ sub >0 = {1, 2, …}.

As already noted, we currently use Fraktur N sub 0 to mean { 0, 1, 2, ...}.

If we want to have an unambiguous symbol for { 1, 2, 3, ...},
the obvious symbol is "Fraktur N sub +". A "+" is unambiguously "positives",
and a subscript would be consistent with N sub 0 and our
ZZ sub (condition) notation. Oddly, while "+" is listed in
Wikipedia, subscripting "+" is not - but I suspect that's just an omission.

We could also use superscripts: N sup 0 and N sup +.
Both of those notations *do* appear in the Wikipedia article.
However, that'd change N0 as well, and I like the consistency
of having these restrictions always in the subscript.

Bottom line: I *like* the idea of tweaking the typography of "NN"
to show Fraktur N sub +. The subscripted + wouldn't be distracting,
and it would eliminate a current ambiguity in the symbols.
Yes, you can look it up... but it'd be nicer to use symbols that
didn't require that in the first place.

None of this would change any theorems; this would solely be
a change in the typography used to display final proofs.

--- David A. Wheeler

[David A. Wheeler]

On Thu, 11 Oct 2018 11:02:22 -0700 (PDT), Norman Megill <n...@alum.mit.edu> wrote:
> NN was originally chosen to start at 1 because that's what most analysis
> books did, and I figured complex numbers were the start of analysis. This
> contrasts of course with early set theory where om starts at zero, and
> computer science where I think nat usually starts at 0.
>
> I'll change it to NN1 (ASCII token) and \mathbb{N}_1 (displayed symbol),
> and there will be no NN, so we can put this to rest.

Okay. That works too, and the pair "NN sub 0" and "NN sub 1" is also listed at:
https://en.wikipedia.org/wiki/Natural_number#Notation

I think a challenge that happens in set.mm is a specific symbol
must have one meaning, but different branches and communities sometimes
use the same symbol with different meanings.
I think having a *single* meaning for a given symbol is valuable
(for both computers and humans), so intentionally choosing unambiguous symbols
is a short-term pain and long-term gain.

--- David A. Wheeler

[Benoit]

> What I can tell you for a fact is that in a majority of cases, when a
> mathematician sees "⊂" in an article, a book, or on a blackboard, he
> assumes "\subseteq".

The evidence doesn't support that assertion.

I think that I've read enough books and articles for my research, I've read enough textbooks for my teaching, I've been to enough conferences, seminars, department colloquiums, I've talked to enough colleagues at department teas and around blackboards, to know what I'm talking about. Note that I wrote *mathematicians* and *majority* (which means, more than 50%), which renders your following paragraph moot.

Benoît

[Benoit]

> I'll change it to NN1 (ASCII token) and \mathbb{N}_1 (displayed symbol),
> and there will be no NN, so we can put this to rest.

Please, by all means, don't use \N_1.  The symbol \N_0 is not very good either, and superscripts are to be avoided for obvious reasons.  If I were to choose, I would use \N, the main symbol, for the more fundamental object, which is the set of natural numbers (which contains 0).  For the strictly positive ones, I prefer \N_{>0} (but \N_{\neq 0} is also possible).  If you want to explicitly say that 0 is included, you can use \N_{\leq 0} (but again, I think it is not necessary).

I don't remember exactly what is in set.mm, but the same notation should be used for e.g. \Z_{\neq 0} and \R_{\neq 0} and \R_{> 0} and \R_{\leq 0} and so on (since \R_+ is ambiguous).

Benoît

[David A. Wheeler]

On Thu, 11 Oct 2018 11:26:44 -0700 (PDT), Benoit <benoit...@gmail.com> wrote:
> If I were

> to choose, I would use \N, the main symbol, for the more fundamental object...

But that doesn't solve the problem of ambiguity.
Some communities are certain that N contains 0, and others are certain it does not.

Some sort of additional marking, such as a superscript or subscript,
is necessary if we want people to understand the symbol without
having to look it up. I think that's a valuable property in a symbol.

--- David A. Wheeler

[Norman Megill]

What is wrong with NN_1?  It is perfectly unambiguous, consistent with NN_0, and one of the recommendations on the wikipedia page David mentioned.  I dislike both \N_{>0} and \N_{\neq 0} which seem excessively complex and distracting symbols for such a simple notion.  The cases of ZZ and RR are very different because they don't "start" anywhere.

Norm

[Mario Carneiro]

I don't have any skin in this game, but I would caution against disambiguating symbols too much. It is hard enough to have a consistent and globally unambiguous notation for all the things in set.mm, without also trying to be unambiguous with all ways people use all symbols. Mathematics is far too context dependent for this to work. So I am opposed to adding disambiguating markings on symbols for which *in set.mm* there is nothing they could be confused with. I think David's attempts to survey the literature to find the most commonly used symbols and just use those is the right approach.

P.S.: My personal favorites are N = NN0, N+ = NN1, \subseteq = C_, \subset = C.

Mario

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[Benoit]

There is a pattern \N_{>0}, \Q_{>0}, \R_{>0} (for the index {\neq 0}, you can add to the list \Z, \C, \H etc.).  There is no such thing for \N_0 or \N_1. The sign + is ambiguous (strictly or nonstrictly positive?), superscripts can be mistaken for exponents or a symbol for the dual.

But now I'm sorry I started this picrocholine war, and I'm fine with whatever (current state, Mario's favorite, etc.).  I will simply recommend this talk by Serre for your enjoyment (it is worth the hour) https://www.youtube.com/watch?v=ECQyFzzBHlo

Benoît

[David A. Wheeler]

On Thu, 11 Oct 2018 14:41:24 -0400, Mario Carneiro <di....@gmail.com> wrote:
> I don't have any skin in this game, but I would caution against
> disambiguating symbols too much. It is hard enough to have a consistent and
> globally unambiguous notation for all the things in set.mm, without also
> trying to be unambiguous with all ways people use all symbols.

I would certainly agree that trying to pick symbols that are
unambiguous in all circumstances worldwide is far too much to ask.

However, I think you can go far by just focusing & fixing "worst case" situations.

The only symbols I know of in set.mm that are *notoriously* ambiguous
are "natural number" (NN) and "subset of". The Wikipedia pages for both
of them *expressly* mention that the symbols are ambiguous, and these
are used *all* *over* set.mm (not just in local areas).

I suggest that we identify the few worst cases,
try to fix them, and leave it at that. I only know of those 2 cases.

--- David A. Wheeler

[David A. Wheeler]

On Thu, 11 Oct 2018 11:26:44 -0700 (PDT), Benoit <benoit...@gmail.com> wrote:
> For the strictly positive ones, I prefer \N_{>0} (but \N_{\neq 0} is also
> possible). If you want to explicitly say that 0 is included, you can use
> \N_{\leq 0} (but again, I think it is not necessary).

I presume you mean \geq, not \leq :-).

The symbol ZZ>= is defined by definition df-uz to use >= as the comparison,
and no matter what you want to make it obvious how these sets differ.
So if you want a single symbol that typographically embeds a comparison,
a reasonable one would be:
NN0: \N_{\geq 0}
NN1: \N_{\geq 1}

That said, I think representations using N sub 0 and N sub 1 are okay;
Wikipedia seems to suggest they're common, and they're simple.

--- David A. Wheeler

[Benoit]

When you speak, you say "the positive natural numbers" or "the nonzero natural numbers"; you don't say "the natural numbers starting at 1" or 'the natural numbers at least 1".  Hence my proposals \N_{>0} and \N_{\neq 0} but not \N_1 or \N_{\geq 1}.

Benoît

[David A. Wheeler]

Perhaps, but if one symbol includes a "1" or "+", and the other has a "0",
they are visually more distinct & more obviously indicate the difference.

I increasingly avoid using the term "natural number" because nobody agrees on what it means.
I instead say "positive integers" and "nonnegative integers", which are unambiguous.

--- David A. Wheeler

[fl]

When you speak, you say "the positive natural numbers" or "the nonzero natural numbers"; you don't say "the natural numbers starting at 1" or 'the natural numbers at least 1".  Hence my proposals \N_{>0} and \N_{\neq 0} but not \N_1 or \N_{\geq 1}.

It is a way to remember what's the first element of the set. Not a clue as to how to pronounce the symbol.

The shorter the better.

--

FL 

[Norman Megill]

Although I said I'll change \mathbb{N} to  \mathbb{N}_1, I'm going to postpone that because there seems to be no consensus.

As David and FL indicate, we usually say "nonnegative/positive integers" not "nonegative/positive natural numbers".  If we want to write it as we pronounce it, we would logically use \mathbb{Z}_{\nless 0} and \mathbb{Z}_{> 0} and dispense with \mathbb{N} altogether.

One purpose of a separate symbol \mathbb{N} is to simplify notation, and having 2 characters for the subscript (like we would need with \mathbb{Z}) is not simplification.  \mathbb{N} also provides immediate visual recognition of the important concept that it is an inductive set (unlike \mathbb{Z}), where starting at 0 and starting at 1 are by far the most common cases.  Because it's an inductive set, the subscript has an implicit "starting at", which is the most important information for an inductive set.

While marginally acceptable, I don't really like \mathbb{N}^+ because it is inconsistent with the "starting at" notion.  If I saw \mathbb{N}^+ I'd probably look it up to be sure, since it also suggests the possibility of "\mathbb{N} plus something" such as +oo.

For subscripts other than 0 or 1, we could simplify the notation for our inductive set function \mathbb{Z}_\ge (ZZ>=) by replacing it with the newly-avalaible bare  \mathbb{N}, with the numeric argument again meaning "starting at".  I didn't chose this initially because having a negative starting point for an \mathbb{N} variant seemed to me a little odd, but might it be worth considering?  Then ~ nnuz would be stated as NN1 = ( NN ` 1 ).

I still think that most analysis texts start at 1 for natural numbers.  It would be nice to have a survey like the one David found for \subset   I'll note that Apostol's Calculus (p. 22) starts at 1, as does Schechter's Foundations of Analysis (p. 180).  Interestingly, rather than introduce a different symbol for nonnegative integers, Schechter uses \mathbb{N} \cup \{0\} because it is needed so infrequently.  By the way, it was only in Dec. 2002 when Raph Levien (mostly a computer science person) added df-n0.

Norm

[Benoit]

I agree that we shouldn't change anything hastily.

It would be nice to have some survey.  The one I found randomly on wikipedia (for subset) is limited.  We should select a representative corpus.  I have access to the whole GTM series (https://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics) which I think can be considered as representative.  I could look at (a sample of) the notation indexes to get an idea.

Benoît

[Mario Carneiro]

As a point in defense of NN+, it is consistent with the already existing RR+ where + means >0 (not >= 0). It also encourages reading it as "natural numbers" and "positive natural numbers", which is fine provided you are consistent about the convention "0 is a natural number". If you want to be "formally ambiguous" about this convention, then NN0 and NN1 seems like the easiest way to unobtrusively say which is which without reference to the other one (and I think this is all that we should need to do - it does not need to match what we say or anything like that, being vaguely suggestive is good enough).

I will remark that my personal preferences are just that. If the world could agree on a convention this is the one I would want, but we have to work with the way things are. I remember seeing the \subset symbol used for strict subset in metamath and thinking "yes, take a stand!" because the dumb convention isn't going to go away if people remain formally ambiguous, but that's just my personal take on it. As for NN, I admit to sometimes taking advantage of the ambiguity by making it exclude or include 0 according to whichever is more convenient in the moment, but all things considered I have come to view NN0 as the more basic one. But again, I support evidence based decision making here - feel free to take my preferences with a grain of salt.

Mario

--

[Alexander van der Vekens]

According to the German Wikipedia (article "Natürliche Zahlen"), Richard Dedekind was the first one defining natural numbers by axioms. He used the symbol N, denoting the natural numbers starting with 1. This plain N was stylized to  \mathbb{N} later. I verified this by looking at the original article Was sind und was sollen die Zahlen? Braunschweig 1888.

Later, in 1894, Giuseppe Peano presented his axiomization of the natural numbers, starting at 0, denoting them with N_0 (N with subscript 0) - this can also be checked in the original article "Opere Scelte", Volume III). This also evolved to \mathbb{N}_0 later.

So the symbols currently used in set.mm are perfectly in line with history!  Therefore, they should not be changed.

Alexander

[David A. Wheeler]

On October 12, 2018 4:07:24 PM EDT, 'Alexander van der Vekens' via Metamath <meta...@googlegroups.com> wrote:
>So the symbols currently used in set.mm are perfectly in line with
>history! Therefore, they should not be changed.

I don't think that's a strong argument. We are not arguing what the symbols used to mean, you're arguing whether or not the symbols used are easily understood in the same way by all.

I did not have time to do a deep literature search, but the fact that the iso standard and nist have different definitions for the natural number symbol makes it clear that not everyone understands that the same way.

>
>Alexander


--- David A.Wheeler

[David A. Wheeler]

> you're arguing whether or not the symbols used
>are easily understood in the same way by all.

s/you're/we're/


--- David A.Wheeler

[Giovanni Mascellani]

Hi,

I did not imagine to start such a discussion with my comment that was
mostly meant to be tongue-in-cheek. While the current notation is not my
favourite one, it is definitely not a big problem for me.

Il 12/10/18 20:01, Norman Megill ha scritto:

> One purpose of a separate symbol \mathbb{N} is to simplify notation, and
> having 2 characters for the subscript (like we would need with
> \mathbb{Z}) is not simplification.  \mathbb{N} also provides immediate
> visual recognition of the important concept that it is an inductive set
> (unlike \mathbb{Z}), where starting at 0 and starting at 1 are by far
> the most common cases.  Because it's an inductive set, the subscript has
> an implicit "starting at", which is the most important information for
> an inductive set.

I agree here: if there is a subscript, the most sensible choice is the
minimum element. However, I still would retain the symbol \mathbb{N} for
the situations in which a reference inductive set must be considered and
its "starting point" is not relevant.

> For subscripts other than 0 or 1, we could simplify the notation for our
> inductive set function \mathbb{Z}_\ge (ZZ>=) by replacing it with the
> newly-avalaible bare  \mathbb{N}, with the numeric argument again
> meaning "starting at".  I didn't chose this initially because having a
> negative starting point for an \mathbb{N} variant seemed to me a little
> odd, but might it be worth considering?  Then ~ nnuz would be stated as
> NN1 = ( NN ` 1 ).

I personally don't think that would be very useful. One-letter symbols
are very scarce, I would not waste them on a definition that is rarely
bound to be useful.

So I believe that \mathbb{N} should either be defined as \mathbb{N}_0 or
\mathbb{N}_1, leaving the explicit subscript for the other one. Between
the two possibilities, I prefer taking \mathbb{N} to be \mathbb{N}_0,
because it seems to me that zero is "naturally" a member of the natural
numbers. For example, natural numbers with the zero are naturally in
bijection with finite cardinalities (if someone asks me "How many apples
do you have?", zero is a totally natural answer; or even "How many
iterate derivatives do I have to take?", or "How many indices does this
tensor have?"). Natural numbers with the zero are also in a natural
bijection with "shifts" of natural numbers. It seems to me that having
an additional algebraic property for free (the existence of an neutral
element for the addition) is good thing.

Of course none of these properties is needed to have a set of natural
numbers, as the current set.mm shows. However, they contribute to give,
in my view, a certain form of aesthetics to the set of natural numbers,
which the set \mathbb{N}_1 does not have.

> I still think that most analysis texts start at 1 for natural numbers. 
> It would be nice to have a survey like the one David found for \subset  
> I'll note that Apostol's Calculus (p. 22) starts at 1, as does
> Schechter's Foundations of Analysis (p. 180).  Interestingly, rather
> than introduce a different symbol for nonnegative integers, Schechter
> uses \mathbb{N} \cup \{0\} because it is needed so infrequently.  By the
> way, it was only in Dec. 2002 when Raph Levien (mostly a computer
> science person) added df-n0.

Funny, I have a totally different feeling here. Thinking at my
mathematical studies, I am pretty sure that most of the times \mathbb{N}
was defined with the zero, and when my classmates and I would find a
book with natural numbers starting from 1, we would consider that odd
(and would probably check if it is an old book with old conventions).
Maybe there are different customs in different places (for the record, I
did undergraduate and PhD studies in Pisa, Italy).

Maybe it is an US vs Europe thing?

Have a good day, Giovanni.
--
Giovanni Mascellani <g.masc...@gmail.com>
Postdoc researcher - Université Libre de Bruxelles

[Alexander van der Vekens]

Hi again,

I scanned through a lot of math books I have in my bookshelf, and without exception I found that the natural numbers start at 0. So this seems to be the de facto standard.

Therefore, I would agree to change NN0 into NN, with symbol \mathbb{N}. However, I wonder about the value of the current definition NN for proves/theorems. Contentually it can completely be replaced by ( ZZ>= ` 1 ). Of course NN (or the future NN1) is shorter by 9 characters (including blanks).

In my recent work, I used NN0, NN and ZZ>= a lot of times, always being annoyed to convert between these notations. So reducing them from 3 to 2 would have helped (and would allow for removing a lot of auxiliary theorems and shortening proofs). On the other hand, this would blow up the theorems itself.

As a conclusion, I would vote for one of the follwing alternatives (in arbitrary order):

1. leave everything as it is
2. change NN0 into NN, with symbol \mathbb{N}, remove current NN completely and replace it by ( ZZ>= ` 1 )
3. change NN0 into NN, with symbol \mathbb{N}, and change current NN into NN1, with symbol \mathbb{N}_1

Alexander

[fl]

Hi again,

Hi Alexancer,

I advocate that we use NN0 and NN1. Just to have a clue about the first element of the set.

Simply a matter of psychological comfort. Not a problem of hypothetical de facto standard.

--

FL

[Mario Carneiro]

On Sat, Oct 13, 2018 at 8:56 AM 'Alexander van der Vekens' via Metamath <meta...@googlegroups.com> wrote:

Hi again,

I scanned through a lot of math books I have in my bookshelf, and without exception I found that the natural numbers start at 0. So this seems to be the de facto standard.

Therefore, I would agree to change NN0 into NN, with symbol \mathbb{N}. However, I wonder about the value of the current definition NN for proves/theorems. Contentually it can completely be replaced by ( ZZ>= ` 1 ). Of course NN (or the future NN1) is shorter by 9 characters (including blanks).

I believe there are reasons for that besides just notation. NN is defined in a way so that it is not obviously a set, and it makes sense even if you are doing finite mathematics, while ( ZZ>= ` 1 ) requires the axiom of infinity to talk about meaningfully, because ZZ>= itself is only a function if NN is already a set. I realize this is a minor nitpick and we stop caring about it not long after introducing ZZ>=, but I would prefer that the definition of NN stays as is (or possibly NN0 is defined first with a similar definition as current NN and NN1 is defined in terms of NN0).

 

In my recent work, I used NN0, NN and ZZ>= a lot of times, always being annoyed to convert between these notations. So reducing them from 3 to 2 would have helped (and would allow for removing a lot of auxiliary theorems and shortening proofs). On the other hand, this would blow up the theorems itself.

You will notice that a lot of the theorems about sequences have a Z = ( ZZ>= ` M ) assumption. This is so that you can substitute nnuz, nn0uz or eqid as desired depending on the context, so that you can uniformly deal with all these cases.

Mario

[fl]

 

In my recent work, I used NN0, NN and ZZ>= a lot of times, always being annoyed to convert between these notations. So reducing them from 3 to 2 would have helped (and would allow for removing a lot of auxiliary theorems and shortening proofs).

You can even reduce them to one.

 

You will notice that a lot of the theorems about sequences have a Z = ( ZZ>= ` M ) assumption. This is so that you can substitute nnuz, nn0uz or eqid as desired depending on the context, so that you can uniformly deal with all these cases.

You can define NN using Peano's axioms renaming it as NN1. Add a (New usage is discouraged). Them show its equivalence in presence of the axiom of infinity with (ZZ>= ` 1). Then use (ZZ>= ` 1) and (ZZ>= `0) everywhere.

That way everybody will be happy.

Mario: the inductive definition will be there.

Alexander: no more conversion.

I: it will be easy to remember the symbols.

-- 

FL

[Glauco]

Hi Norm,

are you going to also change labels like nnnn0 in nn1nn0 ?

And there are many comments referring to NN as "Natural numbers". For instance

"A natural number is a nonnegative integer." Is it going to be changed? (it seems hard to be done semiautomatically)

Glauco

[David A. Wheeler]

On Sun, 14 Oct 2018 12:49:02 -0700 (PDT), Glauco <glaform...@gmail.com> wrote:
> Hi Norm,
>
> are you going to also change labels like nnnn0 in nn1nn0 ?

I suggest *just* changing the typography of NN when displayed in HTML
(if there's a change at all). There's no need to change labels (names).

In my mind, the purpose of a typography change is to make it easy to read the
*final* displayed (HTML) proofs and know exactly what was meant.
All the ASCII labels are abbreviations and you have to know a lot more to create proofs anyway.

--- David A. Wheeler

[Norman Megill]

On Saturday, October 13, 2018 at 8:56:41 AM UTC-4, Alexander van der Vekens wrote:

Hi again,

I scanned through a lot of math books I have in my bookshelf, and without exception I found that the natural numbers start at 0. So this seems to be the de facto standard.

What is the subject matter of these books?  Are any of them on analysis?

I checked Choquet-Bruhat et. al. "Analysis, Manifolds and Physics" (1982), which is a monumental reference work in the field, and they define NN as positive integers.  Since they are French, it doesn't seem to be an American vs. European thing.  Maybe it's a physics vs. math thing?

Norm

[Norman Megill]

On Sunday, October 14, 2018 at 3:49:03 PM UTC-4, Glauco wrote:

Hi Norm,

are you going to also change labels like nnnn0 in nn1nn0 ?

Well, I've postponed any decision to change anything yet.  If we change NN to NN1, then they should probably be changed, but it isn't absolutely necessary, and label changes are somewhat low priority.  I'd likely post a list of proposed changes for commentary before doing so.
 

And there are many comments referring to NN as "Natural numbers". For instance

"A natural number is a nonnegative integer." Is it going to be changed? (it seems hard to be done semiautomatically)

Given this thread, I think changing "natural number" to "positive integer" everywhere in the comments (when it refers to NN) is a good thing to do anyway, regardless of any symbol or label change.  That can be done at any time.  If someone wants to volunteer, great, otherwise I'll add it to my to-do list.  df-nn would be reworded to state something like, "Define the set of positive integers.  Some authors, analysts in particular, call these the natural numbers, whereas other authors choose to include 0 in their definition of natural numbers." (Is "analyst" the right word for a mathematician specializing in analysis?)
 
Norm

[David A. Wheeler]

On Sun, 14 Oct 2018 19:13:36 -0700 (PDT), Norman Megill <n...@alum.mit.edu> wrote:
> Given this thread, I think changing "natural number" to "positive integer"
> everywhere in the comments (when it refers to NN) is a good thing to do
> anyway, regardless of any symbol or label change. That can be done at any
> time. If someone wants to volunteer, great, otherwise I'll add it to my
> to-do list. df-nn would be reworded to state something like, "Define the
> set of positive integers. Some authors, analysts in particular, call these
> the natural numbers, whereas other authors choose to include 0 in their
> definition of natural numbers."

I like this idea; it's clear and concise.

> (Is "analyst" the right word for a
> mathematician specializing in analysis?)

I think "authors" is the better term (as written). I think "analyst" is too unclear.

--- David A. Wheeler

[fl]

I checked Choquet-Bruhat et. al. "Analysis, Manifolds and Physics" (1982), which is a monumental reference work in the field, and they define NN as positive integers.  Since they are French, it doesn't seem to be an American vs. European thing. 

The book was directly written in English. She has just published her memoirs. I think I will buy them.

-- 

FL

[fl]

 

I checked Choquet-Bruhat et. al. "Analysis, Manifolds and Physics" (1982), which is a monumental reference work in the field, and they define NN as positive integers.  Since they are French, it doesn't seem to be an American vs. European thing.  Maybe it's a physics vs. math thing?

She wrote her book on manifolds in English and her memoirs in French :) I suppose it has a meaning . I'm not it's a heartening one.

-- 

FL

[Alexander van der Vekens]

On Monday, October 15, 2018 at 4:01:02 AM UTC+2, Norman Megill wrote:

On Saturday, October 13, 2018 at 8:56:41 AM UTC-4, Alexander van der Vekens wrote:

Hi again,

I scanned through a lot of math books I have in my bookshelf, and without exception I found that the natural numbers start at 0. So this seems to be the de facto standard.

What is the subject matter of these books?  Are any of them on analysis?

 

Most of the books (all in German) are on analysis, some are "general" mathematics, some (linear) algebra ...

[fl]

Most of the books (all in German) are on analysis, some are "general" mathematics, some (linear) algebra ...

Hi Alexander,

if you have Hausdorff's Mengenlehre could you check for us that Hausdorff invented the axioms for topology (1).

It is clear he invented the axioms of neighborhood but I can't find nowhere confirmation that he also invented

the concept of topology. The Mengenlehre has never been translated as far as I know.

(1) http://us.metamath.org/mpeuni/df-top.html

-- 

FL

[Alexander van der Vekens]

Hi Frederic,
according to Wikipedia (English or German), Johann Benedict Listing intoduced the term "topology"/"Topologie" 1847...
In Felix Hausdorff's "Mengenlehre" (page 213) I find the following definition:

" Unter einem topologischen Raum verstehen wir eine Menge E, worin den Elementen (Punkten) x gewisse Teilmengen U_x zugeordnet sind,
die wir Umgebungen von x nennen, und zwar nach Maßgabe der folgenden Umgebungsaxiome:
(A) Jedem Punkt x entspricht mindestens eine Umgebung U_x; jede Umgebung U_x enthält den Punkt x.
(B) Sind U_x, V_x zwei Umgebungen desselben Punktes x, so gibt es eine Umgebung W_x, die Teilmenge von beiden ist (...)
(C) Liegt der Punkt y in U_x, so gibt es eine Umgebung U_y , die Teilmenge von U_x ist (...).
(D) Für zwei verschiedene Punkte x, y gibt es zwei Umgebungen U_x, U_y ohne gemeinsamen Punkt (...)."

This "topologische Raum" (Topological space) of Hausdorff corresponds to a today's Hausdorff space because of the separation axiom (D): "For two different points x,y,
there are two neighborhoods of x and y without common point" or, as described in Wikipedia article "Hausdorff space", "for any two distinct points
there exists a neighbourhood of each which is disjoint from the neighbourhood of the other".

In Wikipedia, article "Topological space" gives the following definition of a topological space

1.    If N is a neighbourhood of x (i.e., N ∈ N(x)), then x ∈ N. In other words, each point belongs to every one of its neighbourhoods.
2.    If N is a subset of X and includes a neighbourhood of x, then N is a neighbourhood of x.
      I.e., every superset of a neighbourhood of a point x in X is again a neighbourhood of x.
3.    The intersection of two neighbourhoods of x is a neighbourhood of x.
4.    Any neighbourhood N of x includes a neighbourhood M of x such that N is a neighbourhood of each point of M.

1. corresponds to (A), second part: A. x e. X A. N e. N(x) x e. N; the first part of (A) can be formalized to A. x e. X N(x) =/= (/)
3. ( A. y e. x A. z e. x ( y i^i z ) e. x - corresponding to th second part of df-top)almost corresponds to (B), which can be formalized
to A. x e. X A. y e. N(x) A. z e. N(x) E. N e. N(x) n C_ ( y i^i z )
(C) could be formalized to A. x e. X A. N e. N(x) A. y e. N E. M e. N(y) M C_ N

So the system of neightborhoods is the topology of the topological space in both definitions. However, I do not see immediately if both definitions (Hausdorff's
axioms (A)-(C) and Wikipedias axioms 1.-4.) are equivalent.

The second definition in Wikipedia, article "Topological space", seems to be most close to df-top:

1.   The empty set and X itself belong to τ.
2.   Any arbitrary (finite or infinite) union of members of τ still belongs to τ.
3.   The intersection of any finite number of members of τ still belongs to τ.

I do not know who was the first one giving this second definition: The English Wikipedia cites Armstrong 1983, the German Wikipedia Fuehrer 1977, but these do not seem to be the originators.

As a consequence, I cannot find any evidence in Hausdorff's "Mengenlehre" for the axioms as used in df-top. Also the term "Topologie"/"topology" does not occur in "Mengenlehre".

I hope this helps a little bit. If this topic shall be discussed further, I would suggest to open a new discussion thread for this.

Best regards,

Alexander

[Alexander van der Vekens]

BTW: In Hausdorff's Mengenlehre (page 47), the natural numbers start at 1: "Menge der natürlichen Zahlen 1, 2, 3, ..."

[Norman Megill]

On Sunday, October 14, 2018 at 10:13:37 PM UTC-4, Norman Megill wrote:

On Sunday, October 14, 2018 at 3:49:03 PM UTC-4, Glauco wrote:

...

And there are many comments referring to NN as "Natural numbers". For instance

"A natural number is a nonnegative integer." Is it going to be changed? (it seems hard to be done semiautomatically)

Given this thread, I think changing "natural number" to "positive integer" everywhere in the comments (when it refers to NN) is a good thing to do anyway, regardless of any symbol or label change.  That can be done at any time.  If someone wants to volunteer, great, otherwise I'll add it to my to-do list.  df-nn would be reworded to state something like, "Define the set of positive integers.  Some authors, analysts in particular, call these the natural numbers, whereas other authors choose to include 0 in their definition of natural numbers." (Is "analyst" the right word for a mathematician specializing in analysis?)

I did this in commits 0b4891d and 98e1d43 on github.  (Will be on us2 server in ~4 hours.)

BTW the technique I used was "set width 9999" then "write source set.mm/rewrap", so that "natural number" would not be split between lines in my editor.  This made matching and replacing simpler (while doing them one at a time by hand to make sure context was NN and not ordinals).

Norm

[fl]

 

As a consequence, I cannot find any evidence in Hausdorff's "Mengenlehre" for the axioms as used in df-top. Also the term "Topologie"/"topology" does not occur in "Mengenlehre".

I hope this helps a little bit.

Yes it helps a lot. It is not Hausdorff who is the inventor of the axioms in df-top. The next step

is who invented this definition?

Thank you Alexander.

--

FL

[Benoit]

Yes it helps a lot. It is not Hausdorff who is the inventor of the axioms in df-top. The next step

is who invented this definition?

The definition in terms of open sets is due to Bourbaki.

Benoît

[fl]

 

Yes it helps a lot. It is not Hausdorff who is the inventor of the axioms in df-top. The next step

is who invented this definition?

The definition in terms of open sets is due to Bourbaki.

Source?

--

FL

[Alexander van der Vekens]

N. Bourbaki, Éléments de Mathématique, Topologie Générale, Chapitre 1 Structures topologiques, § 1. Ensembles ouverts; voisinages; ensembles fermés, 1. Ensembles ouverts, Définition 1. - On appelle structure topologique (ou plus brièvement topologie) ...

[fl]

 

N. Bourbaki, Éléments de Mathématique, Topologie Générale, Chapitre 1 Structures topologiques, § 1. Ensembles ouverts; voisinages; ensembles fermés, 1. Ensembles ouverts, Définition 1. - On appelle structure topologique (ou plus brièvement topologie) ...

Well OK. But it doesn't mean they have invented this notion. They may have simply used a previous notion.

Some paragraphs later in the treatise the concept of neighborhood is exposed and we have just seen it

was invented by Hausdorff in 1914.

They used the book of Alexandroff and Hopf (Topologie 1) a lot. It was published in 1935 in Berlin. If you

have it, can you have a look on the definition used by Alexandroof and Hopf for the concept of topology?

--

FL

[Benoit]

Indeed, the definition is in TG, I, §1.1, Def 1 (!).  See in particular Section 15 of

Gregory H. Moore, The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica 35 (2008) 220–241

(available at https://www.sciencedirect.com/science/article/pii/S0315086008000050?via%3Dihub)

See also https://math.stackexchange.com/questions/70445/origins-of-the-modern-definition-of-topology?noredirect=1&lq=1

Benoît

[fl]

See also https://math.stackexchange.com/questions/70445/origins-of-the-modern-definition-of-topology?noredirect=1&lq=1

The hesitations in this discussion about the  real inventor of the axioms show that the origin of the definition is not commonly 

known and suggest one must add the link to the article to df-top.

-- 

FL

[fl]

Hi Benoît,

> Gregory H. Moore, The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica 35 (2008) 220–241

very good article. So from the beginning of section 14, I conclude the axioms are due to Alexandroff, published

in 1925 and used by Bourbaki at Chevalley's insistance in the first chapters of Bourbaki's topology book published

in 1940. Due to the influence of this book everybody later used the definition given by Bourbaki.

(But one must read the article to understand the whole story).

A link to this article should be added to df-top. And the axioms credited to Alexandroff (1925) and 

(less importantly) Bourbaki (1940).

-- 

FL

[Benoit]

I'll add the references to the comment of df-top.  Indeed, the final formulation (without superfluous separation axioms) is due to Bourbaki (Chevalley), but most of the credit for the shift from the neighgorhoods approach of Hausdorff to the open sets approach is due to Aleksandrov (with intermediate steps by Kuratowski, Tietze and Sierpinski).  I'll be very brief in the comment of df-top since history of math is not the object of set.mm.

Benoît

[fl]

 

Indeed, the final formulation (without superfluous separation axioms) is due to Bourbaki (Chevalley),

I suggest that you write: "The axioms are due to Alexandroff and polularized by Bourbaki".

 

No need to refer to a particular person in the Bourbaki group. It is far from being clear

who did what. The only thing we know thanks to Beaulieu is that Chevalley insisted

energically to take the open sets as the primitive concept. But it might be Weyl who first brought

them up as a derived notion.

but most of the credit for the shift from the neighgorhoods approach of Hausdorff to the open sets approach is due to Aleksandrov

It is not simply a question of shift. It is a question of invention. Aleksandroff did invent the axioms.

(with intermediate steps by Kuratowski, Tietze and Sierpinski). 

You can't say that. The approach of Kuratowski or Sierpinski are different.

-- 

FL

[fl]

I'll add the references to the comment of df-top.  Indeed, the final formulation (without superfluous separation axioms) is due to Bourbaki (Chevalley), 

I hadn't understood why you mentionned separation axioms.  Now I see. But no need to mention them. Without much development your allusion

will not be understable. Same thing for your allusion to Sierpinski. A link to the full article suffices. And a sentence like:

"The axioms were invented by Alexandroff (1925) and popularized by Bourbaki (1940)."

-- 

FL

[fl]

I'll add the references to the comment of df-top.  Indeed, the final formulation (without superfluous separation axioms) is due to Bourbaki (Chevalley), but most of the credit for the shift from the neighgorhoods approach of Hausdorff to the open sets approach is due to Aleksandrov (with intermediate steps by Kuratowski, Tietze and Sierpinski).  I'll be very brief in the comment of df-top since history of math is not the object of set.mm.

And there was indeed a shift between Haudorff and Alexandroff due to Tietze. But Union was missing. So we can consider Alexandroff is  the true inventor.

-- 

FL

[fl]

Also the term "Topologie"/"topology" does not occur in "Mengenlehre".

If you read carefully the  article referenced by Benoît, you will understand why Hausdorff does not use the word topology.  For a mathematician of 1914, *topology* was *algebraic topology*. By contrast, neighborhood issues etc.

it's simply  Analysis. It is not surprising either if we find topology in a book of set theory because Cantor has written

about the problems of limit points of a set.

--

FL

[Benoit]

I'll add the references to the comment of df-top.

Done (see http://us2.metamath.org/mpeuni/df-top.html).

Benoît 

[fl]

 

Done (see http://us2.metamath.org/mpeuni/df-top.html).

Thank you.

I've  just discovered Google translate. It is nearly perfect to translate Hausdorff. "Umgebungis" is mistranslated however. 

I'm afraid to say that human translators will be first humans to disappear because of the AI "revolution". 

The great replacement of the others by AI agents will follow soon.

Congratulations to the mathematicians and computer scientists responsible for this progress.

Is it a disaster? I'm not sure. After all human beings were a hack.

-- 

FL

[fl]

"My" English fixed by a neural network:

"I have just discovered Google translate. It is almost perfect for Hausdorff, but "Umgebung" is poorly translated.
I am afraid I have to say that human translators will be the first humans to disappear because of the AI "revolution".
The great replacement of the others by AI agents will soon follow.
Congratulations to the mathematicians and computer scientists responsible for these advances.

Is it a disaster? I'm not sure about that. After all, human beings were a hack."

--
FL

[Norman Megill]

Here's another data point.  I'm tutoring a 7th grade boy, and his text is Smith, Charles, Dossey, and Bittinger, _Algebra 1_, 2006.  I noticed they define "natural numbers" as starting at 1, "whole numbers" as starting a 0, and "integers" as integers.  I wonder if there was any disagreement among the 4 authors and why they decided on this convention.

Norm

On Thursday, October 11, 2018 at 2:08:41 PM UTC-4, David A. Wheeler wrote:

I tried to find out about symbols & semantics for "natural numbers".

Basically, there's no agreement; when someone sees "N" or says
"natural number", nobody can be sure what it means and you have to
look it up.  In the text below, where I write "sub" it
means "subscript", and "sup" means "superscript".

We currently use Fraktur N to mean { 1, 2, 3, ....} and Fraktur N sub 0 to mean { 0, 1, 2, ...}.

ISO 80000-2:2009 defines "N" (set of natural numbers) as:
N = {0, 1, 2, 3, ...}.  Note that this is NOT the same as our current notation.
N* = { 1, 2, 3, ...}.
N sub >5 = { 6, 7, 8, ... }
(see section 6, item 2-6.1).
In general they add a "*" superscript to indicate "omits 0", so R* is "the set of reals excluding 0", C* is "the set of complex numbers excluding 0", and so on.

The NIST "Digital Library of Mathematical Functions"
https://dlmf.nist.gov/front/introduction
defines Fraktur "N" as the "set of all positive integers" (as we currently do).
I don't see any simple expression for {0, 1, 2, ...} in the NIST document.

There was nothing here:
https://en.wikipedia.org/wiki/Wikipedia_talk:WikiProject_Mathematics/Conventions

The Wikipedia discussion of Natural number notes the controversy:
https://en.wikipedia.org/wiki/Natural_number
"Some definitions, including the standard ISO 80000-2,[1] begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, ….".  It also discusses ways to eliminate the ambiguity:
To be unambiguous about whether 0 is included or not, sometimes an index (or superscript) "0" is added in the former case, and a superscript "*" or subscript ">0" is added in the latter case:[1]
    ℕ sup 0 = ℕ sub 0 = {0, 1, 2, …}
    ℕ sup * = ℕ sup + = ℕ sub 1 = ℕ sub >0 = {1, 2, …}.

As already noted, we currently use Fraktur N sub 0 to mean { 0, 1, 2, ...}.

If we want to have an unambiguous symbol for { 1, 2, 3, ...},
the obvious symbol is "Fraktur N sub +".  A "+" is unambiguously "positives",
and a subscript would be consistent with N sub 0 and our
ZZ sub (condition) notation.  Oddly, while "+" is listed in
Wikipedia, subscripting "+" is not - but I suspect that's just an omission.

We could also use superscripts: N sup 0 and N sup +.
Both of those notations *do* appear in the Wikipedia article.
However, that'd change N0 as well, and I like the consistency
of having these restrictions always in the subscript.

Bottom line: I *like* the idea of tweaking the typography of "NN"
to show Fraktur N sub +.  The subscripted + wouldn't be distracting,
and it would eliminate a current ambiguity in the symbols.
Yes, you can look it up... but it'd be nicer to use symbols that
didn't require that in the first place.

None of this would change any theorems; this would solely be
a change in the typography used to display final proofs.

--- David A. Wheeler

[Jim Kingdon]

That'll be what you'll find in US K–12 education, at least if we can go by the definition of "whole number" at http://www.corestandards.org/Math/Content/mathematics-glossary/

So I think you are seeing something bigger than just one author's preference.

[fl]

 

Here's another data point.  I'm tutoring a 7th grade boy, and his text is Smith, Charles, Dossey, and Bittinger, _Algebra 1_, 2006.  I noticed they define "natural numbers" as starting at 1, "whole numbers" as starting a 0, and "integers" as integers.  I wonder if there was any disagreement among the 4 authors and why they decided on this convention.

 This discussion began on October 11. A dozen of posts, hundreds of lines have been written on the subject. 

This is all a reiteration of a discussion that has already taken place. The "revolution" concerned by this avalanche 

consists in adding a 0 or a 1 to the symbol NN to disambiguate it and make reading more effective.

-- 

FL

[fl]

The new arguments concern the practices in the textbooks of the American secondary school. Sounds like an illusion to me.

 -- 

FL

[fl]

 

The new arguments concern the practices in the textbooks of the American secondary school. Sounds like an illusion to me.

And then what does this incursion into textbooks teach us: that ambiguity is also a problem for the American teachers and that we 

must effectively resolve the ambiguity between NN0 and NN1 orally and in writing.

 

-- 

FL