Research diaries #10: The real line, gaps in the rational numbers and the axiom of completeness by mathowl

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hive-196387 · @mathowl · Dec 5 '22

$13.11

Research diaries #10: The real line, gaps in the rational numbers and the axiom of completeness


Historically mathematics did not start at the true fundamentals. The fundamentals were only completed around the 19th century. All of the work done before that was done with some intuitive understanding of the basics, in particular the nature of the real line which we will denote by **R** in this post. 

<center>
![a0da6c02a6004d522069673cd50ba87.jpg](https://files.peakd.com/file/peakd-hive/mathowl/AKAqmiadhGNqtzxuVHMatAu6yoEZhWnkgUfw4AHBStjVECE9LG1jXrNSzNpXRxH.jpg)
*Not sure what images to paste. So here is my cat*
</center>


Mathematicians like Cantor showed that our intuitive understanding of **R** is flawed. Hence, it requires a rigorous formulation so that we can proceed in a proof theoretical manner to obtain properties of **R**. Properties of  **R** are of specific interest in relation to the rational numbers which we will denote by **Q**. Rational numbers are all numbers in **R** that can be expressed as a fraction of two whole numbers. There exist methods to construct **R** from **Q** but that is not the approach we will take as it is rather technical. For the interested you can have a look over [here](https://en.wikipedia.org/wiki/Construction_of_the_real_numbers#Construction_from_Cauchy_sequences)



What kind of fundamental properties can we assume on **R**? We assume that that we have the natural addition and multiplication so that we can perform all the natural algebraic operations that we did in primary school (more rigorously we are assume that **R** is a [*field*](https://en.wikipedia.org/wiki/Field_(mathematics))). We also assume that there exist an ordering on **R** this means that the elements are ordered but that also the basic operations, addition and multiplication, apply to this ordering. So if a < b then also a+c < b+c (this means that it is an [ordered field](https://en.wikipedia.org/wiki/Ordered_field))

<center>
![08c659cbe6ad0594dee6c7942336f99.jpg](https://files.peakd.com/file/peakd-hive/mathowl/EpXJa5qaoQRyQTDjCdg4R7bSpkzHjXQ6R8pDyp9kSEzHg53sWYyjT39Wr2T55CaQ1bg.jpg)
*More of my cat*
</center>

Although we have not proven the existence of numbers which are not in **Q**. We expect that there are numbers that are in **Q** but not in **R**. These numbers will create gaps in **Q** in relation to **R**. So how do we express that **R** is gapless. We phrase this as an assumption:

**Axiom of completeness** *Every nonempty set of real numbers that is bounded above has a least/smallest upper bound*

Here bounded above, also called upper bounded, means that there exists a largest element in **R** which is greater than all the elements in the set. There can exist many such bounds but the axiom guarantees that there exists a single smallest one. Finally, it important to note that the least upper bound does not necessarily need to be contained in the set. For example if we consider the set generated by the numbers -1/n where n=1,2,3, ... Then 0 is the least upper bound but 0 is not contained in the set. 

To get a better understanding of the axiom let's look at it in the context of an example. Consider the set
<center>
A = { r ∊ **R** : r<sup>2</sup> < 2 }
</center>
Then the axiom tells us that is has a least upper bound. However, if we consider
<center>
B = { r ∊ **Q** : r<sup>2</sup> < 2 }
</center>
and imagine that we are only allowed to work on **Q** then there is no least upper bound in **Q**. Or more specifically, square root is approximately 1.4142... and consequently can be approximated from above by finer and finer rational approximation meaning 2, 15/10, 142/100 ... but there is no smallest one in **Q**. It goes on indefinitely. Hence, at least heuristically we observe how the axiom of completeness makes the real line *complete*


<center>
**More cat tax**
![904200987181f2ee1f16e08d921b76c.jpg](https://files.peakd.com/file/peakd-hive/mathowl/EoyQ3UNfDEEB11VEvwdkeQSVZqVWUzM5n55aWgw5QXynv6z9nL3nr9UFpQpLwnLnPyP.jpg)
</center>

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goblinknackers	31,347,848,448	7%
vixmemon	1,650,187,974	4.2%
honeycup-waters	592,005,233	2.8%
yaelg	1,155,165,990	1.68%
kylealex	4,562,497,921	10%
gasaeightyfive	946,451,312	100%
cleanplanet	15,614,039,512	1.68%
otp-one	2,740,279,581	35%
fran.frey	4,083,832,766	50%
perpetuum-lynx	2,143,177,160	68.6%
thelittlebank	17,813,567,770	2.8%
foodfightfriday	23,149,361,553	25%
chops.support	8,870,184,305	30%
pboulet	97,218,934,890	56%
javyeslava.photo	716,085,755	2.24%
cercle	1,097,515,313	33%
josesalazar200	954,718,543	5.6%
stem-espanol	25,963,622,857	100%
voter003	19,255,736,423	9.6%
voter000	253,029,072	0.7%
babysavage	927,779,877	6.22%
ravensavage	771,085,037	6.22%
cribbio	1,320,575,960	100%
palasatenea	2,428,822,399	2.8%
the.success.club	3,073,577,659	2.8%
admiralbot	6,838,373,873	100%
javier.dejuan	533,612,682	70%
meanroosterfarm	1,398,283,106	35%
agrovision	12,884,872,362	35%
tommyl33	1,467,948,234	5.6%
kuku-splatts	25,060,542,245	50%
itharagaian.net	800,237,636	33%
brianoflondon	31,952,813,360	0.84%
giulyfarci52	1,668,657,161	50%
kristall97	23,343,125,976	100%
inpursuit	2,007,263,972	19.72%
steemcryptosicko	12,412,392,088	1.12%
certain	750,744,345	0.66%
alvin0617	491,052,078	2.8%
multifacetas	771,671,219	2.8%
cakemonster	4,932,842,475	5.6%
cowpatty	1,456,554,858	35%
olivier.jacobs	830,580,628	33%
stem.witness	4,149,070,082	70%
incinboost	2,939,212,706	4.7%
witkowskipawel	927,042,152	2.8%
devann	8,956,151,353	3%
autobodhi	800,475,439	5.6%
mechanicalowl	1,367,091,723	100%
regularowl	2,401,626,447	100%
robmojo	1,649,999,458	7%
double-negative	521,412,271	20%
bluerobo	148,391,995,512	100%
wilmer14molina	825,234,730	50%
jacuzzi	1,135,497,391	0.7%
vaultec	2,615,769,855	12%
steemstorage	6,695,229,869	5.6%
aqua.nano	1,740,446,172	100%
steemegg	1,000,103,376	2.8%
ragnarhewins90	492,068,000	10%
jtm.support	5,423,551,567	70%
edithbdraw	679,589,714	2.8%
apokruphos	3,562,057,045	1%
hairgistix	3,104,732,334	2.8%
limka	260,071,277	81.31%
peter-bot	693,723,728	100%
bluemaskman	601,118,513	2.8%
pursuant	1,982,755,269	19.46%
steemean	9,998,189,832	5%
proxy-pal	1,076,315,622	5.6%
deveney	492,384,012	100%
thelogicaldude	933,847,465	1.12%
yoyo-jp	1,112,259,560	100%
filosof103	797,223,857	2.8%
elements5	1,884,277,485	2.8%
photographercr	2,501,330,271	1.12%
memehub	93,524,878,009	100%
epicdice	3,054,343,738	1.68%
iamsaray	1,291,083,196	2.8%
monsterjamgold	1,775,325,504	10%
robibasa	25,672,727,339	10%
beerlover	2,045,351,953	1.68%
borjan	352,207,246,640	22.7%
qwerrie	3,190,044,914	0.42%
tinyhousecryptos	491,087,832	5%
aninsidejob	6,806,121,895	100%
tggr	1,243,350,562	2.8%
aicu	34,660,413,773	5.6%
zeruxanime	11,716,803,211	35%
afarina46	1,943,572,645	35%
davidtron	795,788,186	4.48%
imbartley	825,678,713	26.25%
mind.force	1,276,981,819	1.4%
stemgeeks	12,124,838,029	60%
freddio.sport	517,887,787	5.6%
stemcuration	1,378,002,826	60%
babytarazkp	4,111,316,621	40%
aicoding	571,269,932	2.8%
abh12345.stem	529,315,463	30%
knightsunited	980,859,111	5.6%
sbi-tokens	2,863,142,720	6.22%
bastionpm	779,752,582	33%
nazer	2,618,543,658	35%
elianaicgomes	3,773,819,587	5%
idig	5,173,927,272	12.5%
wanderingmoon	3,355,275,651	50%
urtrailer	388,095,834,433	50%
precarious	493,414,089	50%
steemstem-trig	5,730,560,556	70%
yggdrasil.laguna	0	30%
mrhaldar	712,315,112	2.8%
ibt-survival	34,041,111,425	10%
riazud	1,316,970,531	2.8%
apineda	14,610,470,969	100%
splatts	107,308,103,043	25%
chicoduro	581,141,521	25%
sacrosanct	702,375,453	1.68%
delilhavores	1,311,570,679	7%
hjmarseille	2,267,484,666	49%
roamingsparrow	7,301,303,057	2.8%
steemdiamond	685,808,025	5.6%
abachon	1,523,356,554	5.6%
zirky	2,086,337,459	4.48%
gloriaolar	1,301,024,306	1.68%
caturday.curator	56,909,958,095	44.37%
ignet	1,088,954,137	33%
peterale	1,974,016,249	2.8%
hive-127039	491,080,200	25%
brofund	962,246,376	2.32%
monica-ene	3,494,071,060	2.8%
stuntman.mike	5,639,085,620	100%
blue-witness	1,508,106,707	100%
laruche	331,304,993,184	7%
stemsocial	589,761,972,491	70%
kyleana	528,808,278	35%
peterpanpan	864,307,498	1.12%
the100	1,364,277,511	2.8%
hiveonboard	2,878,537,122	2.8%
authentika	3,588,360,582	33%
hive-143869	446,402,600,037	33%
taniagonzalez	1,535,189,645	10%
catinthewindow	2,059,222,031	25%
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w-splatts	25,877,223,656	50%
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hivecoffee	577,242,965	5.6%
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nbs.gmbh	421,138,686,928	5.6%
dorkpower	912,592,578	30%
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stemcur	907,647,217	100%
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bubblegif	1,957,876,431	2.8%
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emeraldtiger	1,473,041,334	16.5%
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stemline	3,844,585,283	30%
he-index	3,636,038,503	10%
meestemboom	457,853,354	25%
dcrops	40,559,832,643	2.8%
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krishu.stem	820,081,549	100%
ruari	720,441,507	100%
entraide.rewards	1,076,255,481	33%
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cookaiss	2,670,884,226	16.5%
cbridges573	700,017,185	2.5%
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limn	766,324,344	2.8%
brofi	309,424,527,197	2.32%
failingforwards	2,891,509,337	2.8%
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miroslaviv	2,145,310,631	2.8%
trippymane	1,671,733,730	5.6%
thomashnblum	9,084,485,621	49.7%
nfttunz	18,467,285,087	1.16%
merit.ahama	875,961,971	1.68%
holovision.stem	1,313,003,584	50%
mirteg	1,396,076,157	5.6%
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podping	4,093,966,692	0.84%
jessicaossom	676,524,937	2.8%
drhueso	1,238,677,191	2.8%
star.stem	1,780,324,024	50%
solominer.stem	925,253,599	100%
clinton19	4,117,046,923	100%
mooontivated	152,598,268	9%
jaydr	737,648,293	2.8%
germangenius	448,604,154	35%
menzo	1,128,080,940	5.6%
seinkalar	3,590,005,723	5.6%
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rencongland	550,588,985	2.8%
cugel	1,298,181,153	2.8%
chuiiiiiiii	1,489,890,767	65%
rynos-6	683,539,744	100%
yixn	3,621,909,257	2.8%
alt3r	876,778,163	2.8%
ragk88	7,652,005,239	100%
iumac03	625,754,818	2.8%
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vyb.pob	1,247,565,678	24.85%
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saboin.stem	495,928,676	60%
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albuslucimus	501,511,927	2.8%
wrestorgonline	433,795,891,546	5%
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dyptre	565,187,722	2.8%
anhdaden146	154,775,190,759	2.8%
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dondido	23,651,069,275	5.6%
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archangel21	646,868,927	5.6%
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vragolana	554,886,439	2.8%
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sc000	1,311,626,484	5.6%
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rent-an-ape	607,915,248	2.8%
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ambicrypto	11,535,099,133	5.6%
superuser3	16,465,710,874	35%
reverio	819,496,390	5%
profwhitetower	0	100%

@agmoore · Dec 8 '22

I don't pay much attention to the math... but that cat could have a post of its own :)

properties (22)

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@agrovision · Dec 6 '22

$0.02

Intuitive understanding is the mother of knowledge, just as important is the understanding of nature and its care.

👍 mathowl, dustbunny, stemgeeks, stemcuration, saboin.stem, yggdrasil.laguna

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properties (23)vote details (6)

voter	rshares	pct
mathowl	38,252,587,214	80%
dustbunny	15,111,415,241	4.55%
stemgeeks	1,171,833,787	6%
stemcuration	93,309,479	6%
yggdrasil.laguna	0	3%
saboin.stem	4,787,146	6%

@mathowl · Dec 6 '22

$0.02

Mothers, like intuitive understanding, can be wrong too :P

👍 dustbunny, stemgeeks, stemcuration, saboin.stem, yggdrasil.laguna

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properties (23)vote details (5)

voter	rshares	pct
dustbunny	53,323,247,843	16.02%
stemgeeks	1,168,929,165	6%
stemcuration	92,969,811	6%
yggdrasil.laguna	0	3%
saboin.stem	4,657,679	6%

@agrovision · Dec 6 '22

That is also true, observation and testing end up clearing up any doubt.

👍 stemgeeks, stemcuration, saboin.stem, yggdrasil.laguna

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properties (23)vote details (4)

voter	rshares	pct
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@carolkean · Jul 25 '23

LOL
"Not sure what images to paste. So here is my cat"
It worked!
Of all your posts, the first I clicked on was the CAT post!
Thanks to @agmoore for leading me to the math owl.
I'm angry and bitter sometimes about having an INTEREST in math but not the APTITUDE
So I will stop right there, and go back and try to read your post.
I did read (and review!) a physics book recently.
[The Magick of Physics by Felix Flicker is well-written, endearing, engaging, and maddening](https://hive.blog/ccc/@carolkean/the-magick-of-physics-by-felix-flicker-is-well-written-endearing-engaging-and-maddening)

But it's a long review, despite my leaving out sooooo many aspects the book addresses.
I'm still mad about the chapter on the three towers (statistical probability)...
Most of all, I'm upset that I had to read read RE-READ and read again and still I cannot summarize or explain much of anything, except possibly a wee bit about monopoles, but that's only because I watched Flicker's you-tube video (many times) and kinda/sorta believe ....

But he says "anyone" can learn physics, and if he weren't so cute, earnest, and endearing, I'd be mad at him for reinforcing that myth, that ANYONE, with motive and persistence and practice, can learn anything.

Kurt Vonnegut had a more useful (to me) view on that - that it is not about mastery, or even getting really good at something (e.g. learning a musical instrument) - just, the PROCESS is good. Yeah, yeah, we all know that...

Still. I want to achieve more than mediocrity, and I want to KNOW MATH, if only at the most rudimentary levels.

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@carolkean · Jul 25 '23

"Axiom of completeness," he says.
SHOW ME MORE OF YOUR CAT, I was thinking.
You are good, @mathowl. You are more than good!

![image.png](https://images.hive.blog/DQmWeYstuUDCXgqFENDw4gbtEvcxLdXD2NNrhuM5Na7PS29/image.png)

properties (22)

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@clinton19 · Dec 7 '22 (edited)

$0.02

Interesting post 😁 but i have a series questions that may follow from the one i'm about to ask, it would depend on your response.

>2, 15/10, 142/100 ...

Are these set of values for r² belonging to Q or r belonging to Q ?

👍 mathowl, dustbunny

`author`	clinton19
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dustbunny	0 B		7,880,271,856	2.38%

@mathowl · Dec 7 '22

These are values in Q. They are not contained in B. But they do bound B from above ^^

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@clinton19 · Dec 7 '22 (edited)

$0.02

>These are values in Q. They are not contained in B.

This looks convincing. In fact, that response alone cancels the other questions i would have asked.

>But they do bound B from above ^^

But then this seems to be complicating your response.

The values you presented don't follow the rules of B ( for example, r = √1.5 or √(15/10) isn't a rational number).  So my question, bound B in what context ?

👍 mathowl, dustbunny

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6 replies

@hivebuzz · Dec 13 '22

Congratulations @mathowl! You received a personal badge!

<table><tr><td>https://images.hive.blog/70x70/http://hivebuzz.me/badges/birthday-5.png</td><td>Happy Hive Birthday! You are on the Hive blockchain for 5 years!</td></tr></table>

<sub>_You can view your badges on [your board](https://hivebuzz.me/@mathowl) and compare yourself to others in the [Ranking](https://hivebuzz.me/ranking)_</sub>


**Check out the last post from @hivebuzz:**
<table><tr><td><a href="/hive-102201/@hivebuzz/wc2022-recap-qf-day2"><img src="https://images.hive.blog/64x128/https://i.imgur.com/uNPMgnd.png"></a></td><td><a href="/hive-102201/@hivebuzz/wc2022-recap-qf-day2">HiveBuzz World Cup Contest - Quarterfinals - Recap of Day 2</a></td></tr><tr><td><a href="/hive-102201/@hivebuzz/wc2022-recap-qf-day1"><img src="https://images.hive.blog/64x128/https://i.imgur.com/uNPMgnd.png"></a></td><td><a href="/hive-102201/@hivebuzz/wc2022-recap-qf-day1">HiveBuzz World Cup Contest - Quarterfinals - Recap of Day 1</a></td></tr></table>

###### Support the HiveBuzz project. [Vote](https://hivesigner.com/sign/update_proposal_votes?proposal_ids=%5B%22248%22%5D&approve=true) for [our proposal](https://peakd.com/me/proposals/248)!

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@mooontivated · Dec 7 '22

$0.02

Aww that was too adorable

👍 mathowl, dustbunny

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@mathowl · Dec 7 '22

=^..^=

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@stemsocial · Dec 7 '22

re-mathowl-research-diaries-10-the-real-line-gaps-in-the-rational-numbers-and-the-axiom-of-completeness-20221207t003138902z

<div class='text-justify'> <div class='pull-left'>
 <img src='https://stem.openhive.network/images/stemsocialsupport7.png'> </div>

Thanks for your contribution to the <a href='/trending/hive-196387'>STEMsocial community</a>. Feel free to join us on <a href='https://discord.gg/9c7pKVD'>discord</a> to get to know the rest of us!

Please consider delegating to the @stemsocial account (85% of the curation rewards are returned).

You may also include @stemsocial as a beneficiary of the rewards of this post to get a stronger support.&nbsp;<br />&nbsp;<br />
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properties (22)