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12044835 No.12044835 [Reply] [Original] [archived.moe]

0.999... is NOT 1

No number other than 1 is equal to 1.

9 is not 10 doesnt matter if there are .999999999999... numbers. This whole problem just proves that infinity isnt real and just mental wankery

>> No.12045479

>he doesn't understand what 1 means from a mathematical standpoint

>> No.12045496

>>12045479
You're not wrong, but the true interesting observation is that people posting these threads reliably never seem to understand what 0.999... means from a mathematical standpoint, and yet for some reason still feel knowledgeable enough to judge what it does and doesn't equal.

>> No.12045505

>>12044835
>just mental wankery
all of math and science is that.
there's nothing real about math or logic.
rationalizing experimental data is just schizoprenia.

>> No.12045512

>I'll smugly and confidently assert something, that will ensure those anonymous people on the internet know I'm right!

>> No.12045525

This is true in the hyperreal numbers, but I strongly doubt that the anons who start [math]0.999...[/math] threads even understand what the hyperreals are.

>> No.12045528

>>12044835
8/9+1/9= 9/9
0.8...+0.1...=?

>> No.12045533

>>12045525
>This is true in the hyperreal numbers,
no it isnt
1 - .999... isnt 1/w
its just 0
limits are a geometric concept, and infinitesimals are an algebraic concept
they never mix

>> No.12045546

>>12045533
>no it isnt

Yes it is

>> No.12045560

1/inf=0

>> No.12045563

1/3 = .3333....
2/3 = .6666....
3/3 = 1
1/3 + 2/3 = 1
.333.... + .666.... = .999.... = 1
OP dumb!
also what does
lim x->1 x equal?
1

>> No.12045565

>>12044835
0.999... is by definition lim n->oo sum i=1 to n of 9/10^n, which is lim n->oo 1-1/10^n=1. It is a simple calculation from the very definition of what 0.999... is. It isn't hard. If they aren't equal, what number is in between them, lol?

>> No.12045591

>>12045546
Why do you retards do this shit? Do you know what the transfer principle is? .9...=1 in the reals so it is true in the hyperreals.

>> No.12045602

>>12045591

Bruh you can explicitly construct infinitesimals in an ultraproduct of the reals.

>> No.12045608

>>12044835
0.999.. is not a number in the formal sense. It is a shorthand hand way of expressing an infinite series.

>> No.12045611

>>12045602
Infinitesimals don't split equal numbers you mongoloid.

>> No.12045612

>>12045602
Okay and...?

>> No.12045618

>>12044835
based

>> No.12045621

>>12045611
>>12045612

You can construct [math]\varepsilon[/math] such that [math]0 < \varepsilon < 1 - 0.999...[/math].

>> No.12045624

I swore to stop visiting /sci/ until mods started deleting these threads. I just checked into see if something had changed. Back to redit.

>> No.12045636
File: 3.16 MB, 424x498, laugh.gif [View same] [iqdb] [saucenao] [google] [report]
12045636

>>12045621
He thinks 0<0 in the hyperreals.

>> No.12045648

>>12045621
Says it can be constructed
Doesn't even try to construct it

>> No.12045667

>>12045621
No, you can't. You can find a hyperreal h such that 0<h<r for every non-zero real number r. By the definition of 0.999..., it follows it is real and is 1, so 1-0.999... is 0, so there is no such h.

>> No.12045672

>>12045667
>o, you can't. You can find a hyperreal h such that 0<h<r for every non-zero real number r. By the definition of 0.999..., it follows it is real and is 1, so 1-0.999... is 0, so there is no such h

Actually, I'll add the question of what in the hell you mean by '0.999...' for this to work...

>> No.12045675

>>12045648
I'm phoneposting, so I won't type out all the latex, I'll just bullet point it:

-take direct product of countably many copies of [math]\mathbb{R}[/math] indexed by the natural numbers

-form ultrafilter of all cofinite sets of [math]\mathbb{N}[/math]

-form ultraproduct using this ultrafilter (note that by identifying any real number with its constant tuple you can see this contains [math]\mathbb{R}[/math])

-set [math]\varepsilon = (\frac{1}{n})_n[/math]

-[math]1[/math] is no longer the limit of [math]0.9,[/math] [math]0.99,[/math] [math]0.999[/math] etc. because the difference always exceeds [math]\varepsilon[/math] (obviously [math]<[/math] is defined as per the ultrafilter.

>>12045667 read the last point bud

>> No.12045692

>>12045563
Limit is not a number. It will never reach 1.
0.9... =/= 1

>> No.12045694

>>12044835
1 > .9 + .09 + .009 + .0009 +...

>> No.12045707 [DELETED] 

>>12045692
>Limit is not a number
with infinite series, it it
>lrn2math

>> No.12045709

>>12045692
>Limit is not a number
with infinite series, it is
>lrn2math

>> No.12045724

>>12045692
Wait, so what do you think the limit of
1+ 1/2 + 1/4 + 1/8 ..... is?

>> No.12045733

>>12045724
2^1/2, still the sum of numbers in that sequence will never be equal to the limit, it's a limit.

>> No.12045734

>>12045675
>-set ε=(1n)n
thats 0
its not an infinitesimal
you dont form infinitesimals through a limiting process
quoting my first post
>limits are a geometric concept, and infinitesimals are an algebraic concept
>they never mix

>> No.12045739

>>12045733
>2^1/2
how?

>> No.12045741

>>12045675
Well, we've done the ultraproduct construction so far so fine, but then the issue is near the end where there are some implicit assumptions.

Okay, great-we've created the standard model of 'the' hyperreals. So far, it appears not that useful if we don't clearly define what is '0.999...' and furthermore what is 'limit' in hyperreals. 0.999... is generally notation for real numbers, representing the limit of the sequence sum 9/10^n, which exists. How are we extending this notion of 'limit'-after all, the derivative 'realifies' the hyperreal quotient, does the limit do so as well? How about 0.999..., do we intend this instead to be based on a notion of decimal expansion in hyperreals, in which case it will mean something different then above.

0.999... as itself being a real number, being found to be 1 in the reals, is, by extension, also 1 in the hyperreals. If you don't start with 0.999... as being a prior defined number but instead defined by some process you are extending into hyperreals, then it will of course be possibly different from 1, but you have to define what this process is. 0.999... is notation I've only seen for real numbers, in turn, it should be associated with this prior real number, and then embedded in the hyperreal.

>> No.12045742

>>12045733
>2^1/2
kek

>> No.12045757

>>12045742
kek. Now I understand how treacherous it was for the Wolfshekl Committee to sieve through the vast number of alleged proofs of Fermat's Last Theorem by Amateur Mathematicians.

>> No.12045764

>>12045602
You don’t know what the fuck you’re talking about, bud. Stop using words you don’t understand.

>> No.12045771

>>12045525
I've heard about surreal numbers, but not hyperreal nummbers, is there a practical difference?

>> No.12045773

>>12045734
>thats 0
How is 0 if not a single coordinate of the tuple is 0?

>>12045741
Come on man, I know it was just a bullet point explanation, but surely you could fill in the gaps? Limit is defined exactly the way it is in the reals. [math]0.999...[/math] is defined at the limit of [math]0.9,[/math] [math]0.99,/math] [math]0.999[/math], ...

>> No.12045774

>>12045621
>>12045675
In the hyperreals, the limit extends into the hypernatural.

>> No.12045777

>>12045773
Yes, and the ... notation extends beyond the naturals in the hyperreal context. See the transfer principle.

>> No.12045778

>>12045773
Wow I should've previewed the latex in that post. It's always the simple posts where you make errors...

>> No.12045781

>>12045773
who cares if you have some sequence in the hyperreals
we're talking about .999... in the reals, which are embedable into the hyperreals
this entire tangent is unrelated to the original topic

>> No.12045785

>>12045771
In truth, there is no '1' hyperreal field. A hyperreal field is a field [formalization of 'number'] that otherwise satisfy the axioms [or rather, first-order implementations] of the real numbers [called real closed fields] and contain elements that can be considered infinitesimal and infinities. When you hear about 'the' hyperreals, they are generally referring to what is known as the ultraproduct construction. Surreals are when one considers how large one can go with hyperreals, which uh, contains all ordered fields and all transfinite ordinals and yeah [I'm no expert though].

>> No.12045787

>>12045778
0.999... still equals 1 in the hyperreals because ... is meant to extend over the *entire* sequence which is now hypernatural in length. The 9s don’t stop at the naturals.

>> No.12045793

>>12045781
You are correct, but it was to refute >>12045533

>>12045787
This post is utter handwavy garbage

>> No.12045800

>>12045773
Oh I am not the one making the bullet point. He is referring to some more advanced mathematical constructs, which are in truth quite necessary for a concrete construction of hyperreals. I am responding that while it looks all technical and rigorous, implicit reference to 0.999... or 'limit' without a clear meaning of what either of these mean in the hyperreals is the error. 0.999... is notation quite unique to the real numbers and so you would be correct in that being the 'real' limit. Defined as a real number, it is then contained in the hyperreal as a real. However, the person I am responding to seems to be indicating 0.999... as not being a real number defined by real means by as some type of notation extendable into the hyperreals, which will give an entirely different number. I am challenging of what in the world this means, since as you mentioned, 0.999... is purely real number notation.

>> No.12045806
File: 5 KB, 454x88, Oh no no no.png [View same] [iqdb] [saucenao] [google] [report]
12045806

>>12045793

>> No.12045814

>>12045800
Clarification: by [math]0.999...[/math] in the reals we mean the limit of the sequence [math]0.9[/math], [math]0.99[/math], [math]0.999[/math] and so on. Now map these real numbers to their constant tuples in the hyperreals. Then [math]0.999...[/math] in the hyperreals is the limit of this new sequence. This limit is NOT [math]1[/math]

>>12045806
So you kicked off the whole tangent then

>> No.12045818

>>12045814
>So you kicked off the whole tangent then
no you did
you were the first person to mention hyperreals
hence the fact that i fucking quoted you talking random shit about the hyperreals

>> No.12045826

>>12045793
*Your* post is utter hand wavy garbage, retard.
https://en.wikipedia.org/wiki/Transfer_principle

>> No.12045827

>>12045818
I mentioned hyperreals as an interesting aside (because OP's statement is true in the hyperreals) and that would have been that. Instead you tried to insist that it is not the case, so I had to post a bunch of stuff explaining it to you

>> No.12045837 [DELETED] 

>>12045826
>>12045787
>over the *entire* sequence which is now hypernatural in length

This is what's garbage. The sequence is *exactly* the same length as it is in [math]\mathbb{R}[/math] because it's the same sequence with constant tuples instead of real numbers

>> No.12045845
File: 1.82 MB, 4032x3024, A86B6CF9-35C5-4C6A-9A6A-2BA9478EEC75.jpg [View same] [iqdb] [saucenao] [google] [report]
12045845

Where does the logic in pic related fail in the hyperreals? Math newbie here.

>> No.12045846

>>12045814
Okay, I figured that much out (though 0.999... could have referred to some hyperreal decimal expansion with more 9's). Now how is limit being defined.[I don't even think the limit in a naive sense would exist here, which I mean typical epsilon but using hyperreals. Let {s_n} be the sequence {0.9,0.99,...} Let us try epsilon limit definition. Then, L=1+h for some infinitesimal hyperreal h and 1 both satisfy the epsilon condition [when the epsilon is considered real], so the limit isn't unique. If we try hyperreal epsilon, then we need to find N such that n>N implies |L-s_n|<e. Since L can't differ in real part, it is at best 1+h for some hyperreal h, but then we just have 1/10^n+h<e given n>N which is impossible since it has to hold when e is less then every other real. So, by naively extending the notation of ... into hyperreals like this, have made it unusable.

>> No.12045850

>>12045826
I think we're talking about different numbers. See my clarification here >>12045814. I'm talking about the [math]0.999...[/math] of the reals when embedded in the hyperreals. Seems like we've been at each others throats over a miscommunication. Apologies

>> No.12045859
File: 57 KB, 500x369, 1337850440812.jpg [View same] [iqdb] [saucenao] [google] [report]
12045859

>>12045505
I mean, there is some fundamental reality to math and logic, because we can use mathematical formulas to predict real-world physical interactions down to the atoms.

Physics is just reality behaving according to mathematics that we've discovered.

>> No.12045866

>>12045859
That's like saying the Vatican and Italy are one and the same.
Math is infinitely larger than physics.

>> No.12045875

Anon, you haven't answered why the sum of
1 + 1/2 + 1/4 + 1/8 .... = 2^1/2

>> No.12045876

>>12045866
And all of it is reflected in reality, even if we can't observe some of it.

>> No.12045878

>>12045827
>because OP's statement is true in the hyperreals
no its not
.999... doesnt stop meaning the real number once we go the hyperreals
call f the embedding of R into the hyperreals
then 1=f(1)=f(.99...)=.999...
.99... is not a hyperreal sequence, its a real sequence
you cant just redefine agreed upon notions and then say "ackshually in the hyperreals this isnt true"
youre no longer talking about .999..., but youre still calling it .999...
you mean to say that (.9...9)_n as a sequence in the hyperreals is not (1)_n in the hyperreals, which is different from (.999...)_n

there is a distinction here you do not understand
real numbers are classes of cauchy sequences of rationals
.999... is the sequence of .9 .99 .999 etc
elements of the hyperreals are sequences of reals
taking the sequence .9 .99 .999 etc as a sequence of reals to define a hyperreal is not the same as taking the class of .9 .99 .999 etc as a sequence of rationals
they live in totally different spaces

>> No.12045883

>>12045876
>all of it
lolno

>> No.12045891

>>12045878
Not him, but do you have a PhD in mathematics?

>> No.12045893

>>12045883
What branches of math have absolutely no basis in the physical reality in which we live?

>> No.12045896

>>12045878
>taking the sequence .9 .99 .999 etc as a sequence of reals to define a hyperreal

That is NOT what I've been doing. The [math]0.9[/math]s and [math]0.99[/math]s etc. are all shorhand for the hyperreals that those real numbers map to when we embed [math]\mathbb{R}[/math]. Those are sequences *of* hyperreals, NOT sequences defining hyperreals

>> No.12045899

>>12045896
I even said this in my clarification >>12045814:
>Now map these real numbers to their constant tuples in the hyperreals

>> No.12045967

>>12045893
any of it that doesn't use the same axioms as physics

>> No.12045971

>>12045967
Physics generally doesn't have 'axioms'.

>> No.12045973
File: 112 KB, 953x613, 1595913191512.jpg [View same] [iqdb] [saucenao] [google] [report]
12045973

>>12044835

>> No.12045974

>>12045971
lol ok sweetie

>> No.12045980

>>12045973
Thanks anon, do you think you could make that image a bit harder to read though? There's not enough jpeg artifacting, I can still barely make out the text. Also, the colors need to contrast more, I only have a mild headache from trying to read those boxes.

>> No.12045982

>>12045974
Please, show me an instance in physics where we go 'Ax. 1 blah blah, Ax. 2 bluh bluh', and after some derivation, we get blub. No, physicists have a certain intuitive connection between physical world and corresponding mathematics, physicists then may use some less rigorous mathematics [say, working with dirac-delta as function] to get answers that can be interpreted physically. Not axioms of an abstract formal system.

>> No.12045984

>>12045982
>'Ax. 1 blah blah, Ax. 2 bluh bluh', and after some derivation, we get blub.
Impressive, did you quote that from your PhD?

>> No.12045987
File: 3 KB, 125x114, 1589316322831s.jpg [View same] [iqdb] [saucenao] [google] [report]
12045987

>>12045984

>> No.12045989

>>12045984
As you can clearly tell, since I don't think physics actually uses 'axioms', I can't actually insert such a sentence that would make a convincing example (since I don't think they exist) I have to term it in terms of 'blah' 'bluh' and 'blub' as placeholder terms for whatever axioms YOU think physics is based on.

>> No.12045990

>>12045989
bro he ethered you, it's over just close the tab and do something else with your night

>> No.12045994

>>12045990
>ethered
https://youtu.be/PDNZX2nql2Y?t=2

>> No.12045995

>>12045994
Okay now I officially choked rip https://www.youtube.com/watch?v=PDNZX2nql2Y

>> No.12045997

>>12045994
not an argument. kek. You've lost. you're an embarrassment to human kind. Return to the wild and live among the chimps.

>> No.12046003

>>12045997
Too much of a degen to do that. Now STFU and just give the axioms of physics or whatever.

>> No.12046015

>>12045896
>That is NOT what I've been doing.
sick, if thats not what youre doing then .999... is unequivocally 1

>The 0.90.9s and 0.990.99s etc. are all shorhand for the hyperreals that those real numbers map to when we embed R.
call f the embedding
1 = .999... in the reals implies f(1) = f(.999...) in the hyperreals by welldefinedness
we just follow convention and call f(1)=1 and f(.999...)=.999..., so 1=.999... as the embedded hyperreals
.999... is not 1 - 1/w in the hyperreals, it is still always 1

>> No.12046018

>>12045982
>working with the dirac delta as a function
It's a distribution and if you still don't understand that then you're rarted. Colloquially calling it a function does not change how you use it (aka like a distribution).

>> No.12046019

>>12044835
The universe is discrete and finite and so are numbers. Anything else is the work of the Devil.

Fuck this. I am sick of all these arguments. Its time we put our faith to the test and settled this issue with SWORDS.

ON THE HOLY FIELD OF BATTLE!

I have no doubt that we shall prevail, for GOD is on our side. We shall march upon the CURSED SODOMITES who have defiled our world with their heinous lies and infinities and infinitesimals and complex number planes and and and shit! They shall perish at our hands and then THEY WILL discretely BURN IN HELL!...for a finite amour of time.

Brothers and Sister of the ONE TRUE FINITE UNIVERSE, say it with me...

DEUS VULT!
DEUS VULT!
BY GOD!
DEUS VULT!

>> No.12046020

>>12046003
>Axioms play a key role not only in mathematics, but also in other sciences, notably in theoretical physics. In particular, the monumental work of Isaac Newton is essentially based on Euclid's axioms, augmented by a postulate on the non-relation of spacetime and the physics taking place in it at any moment.

>In 1905, Newton's axioms were replaced by those of Albert Einstein's special relativity, and later on by those of general relativity.

>Another paper of Albert Einstein and coworkers (see EPR paradox), almost immediately contradicted by Niels Bohr, concerned the interpretation of quantum mechanics. This was in 1935. According to Bohr, this new theory should be probabilistic, whereas according to Einstein it should be deterministic. Notably, the underlying quantum mechanical theory, i.e. the set of "theorems" derived by it, seemed to be identical. Einstein even assumed that it would be sufficient to add to quantum mechanics "hidden variables" to enforce determinism. However, thirty years later, in 1964, John Bell found a theorem, involving complicated optical correlations (see Bell inequalities), which yielded measurably different results using Einstein's axioms compared to using Bohr's axioms. And it took roughly another twenty years until an experiment of Alain Aspect got results in favour of Bohr's axioms, not Einstein's. (Bohr's axioms are simply: The theory should be probabilistic in the sense of the Copenhagen interpretation.)

>As a consequence, it is not necessary to explicitly cite Einstein's axioms, the more so since they concern subtle points on the "reality" and "locality" of experiments.

>> No.12046025

>>12046019
superpositions are anything but discrete

>> No.12046027

>>12046025
Say that to my Sword, HERETIC!

>> No.12046028

is 0.8888888..... the same as 0.99999999999.....?

>> No.12046029

>>12046028
no.

>> No.12046043 [DELETED] 

>>12046028
8/9

>> No.12046047

>>12046028
8/9 and 9/9

>> No.12046051

>>12046018
I'm aware. I'm looking at the 'sloppy mathematics' seen in physics.

>> No.12046052

>>12045876
>>12045893
Where do you think transfinite set theory, e.g. large cardinals and the continuum hypothesis, are reflected in reality? Genuinely curious.

>> No.12046117

>>12046020
Copy pasta from wiki is fine and all but they are hardly what considered 'axioms' in the mathematical sense [which, is what is being referred to]. Generally speaking, for Newton they are referred to as laws, for Einstein as postulates. Only Bell's theorem, I would say, really truly counts as a theorem of certain 'axioms', perhaps.

Axioms are formal statements used to derive other ones by matter of derivation. When I consider something like Einstein's postulates, namely that the speed of light is invariant and that the laws of physics are the same in all inertial frames, when we derive the Lorentz transformations, we are making implicit assumptions of what in the hell a change of frames is and what counts as 'laws of physics' as 'being the same'. These are based on an intuitive understanding of the connection between math and physics and not from rigorous proof. So, the original person I was responding to was complaining that mathematics concepts not based on physics axioms had no basis in physical reality is 100% correct because your notion of 'axiom' [meaning 'assumed truth', in the loosest sense] is entirely distinct from the mathematical one.

>> No.12046127

>>12044835
Everyone is making fun of OP, but he's right. I can say that 1 = 2 + 2 if I'm working over modulo 3. The point is, without which equivalence relation is being used, the only equality that matters is literal equality, of which these symbols are different and can represent different things.

However, using the equivalence rule that a ≈ b iff |a - b| < 1/n for any positive integer, we find that 0.9999... ≈ 1.

>> No.12046152

>>12046117
>Newton they are referred to as laws
But didn't we build the whole of classical mechanics from just 3 laws (this may be an understatement but the general idea still holds)?

>> No.12046269

>>12045546
why do you think the definition of 0.999... should change when you move from reals to hyperreals ?

>> No.12046308

Why wouldn't 0.000...1
Always in between 0.999... and 1?

>> No.12046319

>>12046308
>let's pretend finite is infinite

>> No.12046321

>>12046308
because 0.000...1 doesn't represent a real number

>> No.12046367
File: 660 KB, 1432x4127, 1585406223787.jpg [View same] [iqdb] [saucenao] [google] [report]
12046367

>>12044835

>> No.12046881

>>12046308
This is a not at all rigorous way to look at it but in essence, you would get, in this view [if it even makes sense and it doesn't really make sense I think] 0.999... 1. This 1 at the end can't pair with any 9 at the end because there is no 9 at the end. Naturally, when it comes to hyperreals, 0.999... 1 doesn't actually exist and neither does 0.000... 1, though the idea between splitting between real decimal and infinitesimal decimal exists.

>> No.12046928

>>12046152
I mean, yes, in a certain sense-though to be clear, what exactly counts as Newtonian mechanics isn't really fixed. Newtonian mechanics could mean Newton's original mechanics which means no galilean relativity-with galilean relativity, we'd also be fine and this is generally the type of mechanics of introductory physics. If we abstract instead into working with real space rather then euclidean geometry, we get a different framework in doing physics [generally the type used for classical mechanics] How about Lagrangian and Hamiltonian and stuff that is considered 'classical mechanics'? Unlike if I were talking about the axioms of real numbers, which tend to define a quite clear system, the laws of Newton don't really do the same. Looking at Newton's book, we see that euclidean geometry is really the vehicle used to prove results, with the physical arguments moving forward by his laws.

Perhaps I was being a bit too gung-ho in my initial statement, but I really like to characterize the difference in approaches between physics and mathematics. If I open up a math textbook, it shouldn't take me more then like 10 seconds to find a direct declared 'axiom'. I don't think I can do the same in physics. There are postulates and all that, sure, but I prefer to not imagine physics as being so rigorously based on these as absolutely foundation of physical systems but rather as principles that connect the physics to math, so that whatever math we are using as our vehicle physics, is well, physics-it has physical meaning. For instance, one can prove all kinds of results about vector calculus and one can consider E and M as arbitrary vector fields but this isn't physical-restrain these by Maxwell's equations and we haven't necessarily made the mathematics, the vehicle we carry out proofs, any different-we've just added physical conditions which can be used to actually solve problems [say, the wave equation]

>> No.12046960

>>12046928
>verbal diarrhea
cope

>> No.12046982

I'm waiting for fags to prove [math]0.999... \neq 1[/math] in the real numbers. All anybody has done is reject proofs of [math]0.999... = 1[/math], but not prove the negation.
After showing ellipses notation to be contradictory and vague, then we can finally get rid of ellipses notation and use superior formal symbols like >>12045845 that makes clear that it was all limits, anyways.

>> No.12047002

>>12046982
>ellipses notation
it's fine, autist

>> No.12047008

>>12045505
I said infinity isnt real

What number is before 0.999...? Nothing

>> No.12047012

>>12044835
900.900900900... = 1000
Cope harder faggot

>> No.12047014

>>12047008
-13

>> No.12047026

>>12046982
It's literally decimal notation. There is a correspondence between real numbers r and sequence of integers. For a real number r, choose the greatest integer a_0 such that r-a_0>0. Then, pick the largest integer a_1 such that r-a_0-a_1/10>0. Pick next largest integer a_2 such that r-a_0-a_1/10-a_2/100>0, and so forth. This process gives us a well defined sequence {a_0,...,a_n,...} associated to the real number.

In reverse, consider a sequence like this and consider the sum from i=0 to n of a_i/10^i. The limit of this sequence exists and is considered the real number determined by the decimal expansion. The number associated to this decimal expansion {a_0,a_1, a_2...} is sometimes notated as a_0.a_1a_2... where the sequence is understood before hand. So, play this number with the sequence {0,9,9,9,...} We call the real number associated 0.999... as a matter of notation, and by the definition, it evaluates to 1.

>> No.12047028

>>12047012
1000x=900900.900...
x=900.900...
999x=900000
x=900000/999=100000/111

>> No.12047031

>>12046960
something something 4 second attention span

>> No.12047037

>>12047031
something something liquid shit

>> No.12047057

>>12047037
na man this is spaghetti-o shit

>> No.12047170

>>12046928
What about QM? I'm not good at QM, but doesn't it rest on fundamental laws?

>> No.12047305

>>12047170
I mean, there are sometimes 'postulates' of quantum mechanics, but I am emphasizing that these are not so hard and fixed-they serve as physical guidelines.

Generally, the postulates of QM are something like each quantum state is associated with a vector [actually a ray] in hilbert space, each observable is associated with a hermitian operator with the state of a particle of measured property value a as being an eigenvector of that operator with eigenscalar a, the measurement of an observable collapses it onto an eigenstate, the existence of a probability function of sorts on a quantum state and observable, and the time-dependent Schrodinger equation or some variations of equivalence. In fact, see Dirac-von Neumann axioms for probably the closest rigorization of this I've seen [though I haven't actually studied it this far]

But these axioms don't serve to actually elucidate what is physically going on. That connection is entirely unmathematical and based on physical insight.

You can't use this to actually 'show' that momentum operator in position space is iħ∇ for instance [or better said, perhaps, this is an arbitrary example of an operator in the formalization-the physical meaning isn't granted by the axioms] Or, for perhaps an even stronger example, of spin. You could imagine QM without spin, but as it happens spin exists and so we introduce it based on physical insight.

Trust me, as a mathematically oriented person, I like seeing more and more detailed descriptions of physics. I've seen classical mechanics formalized in quite an interesting way by arnold in his book, quantum mechanics in the above mentioned way-there was a book on relativity by Reichenbach making an axiomatization of relativity. I got interested in this as 'einstein synchronization' isn't the only method [see Reichenbach synchronization] and all of this is fine but I really don't think much mind is payed attention to how rigorous our axiomatization of physics is.

>> No.12047333

>>12044835
Which is bigger, 0/1, or 0/2?

>> No.12047338

>>12047305
https://youtu.be/NwcVJMvVWDA?t=40

>> No.12047361

>>12047338
so uhm yeah fun fact physics requires talking about physics

>> No.12047373

>>12047361
[HAND WAVING INTENSIFIES]

>> No.12047374

>>12047373
just another day in the works of a physicist

>> No.12047377

>>12047305
Ok, so there's no true axiomatic understanding of physics, rather pseudo axiomatic could be a better term I guess.

>>12047305
>I really don't think much mind is payed attention to how rigorous our axiomatization of physics is.
Do you think physics could ever be truly axiomatized? Or is the nature of physics itself such that there may not exist true axioms upon which GR, SR, CM and QM are built?

>> No.12047396

>>12045814
The limit is 1. The set however, doesn't include 1.

>> No.12047408

>>12047396
But the notation is defined to be the limit [see what I wrote about the decimal expansion {a_0, a_1,...}.

>> No.12047416

Go post this on reddit or something, you are not telling/informing shit in here

>> No.12047527

>>12047377
The core issue is that the connections established between abstract math and the physical world isn't really one suitable to formalization. For instance, if I want to say F=ma, I would first need to establish a primitive concept of 'force' and 'mass' associated to a given particle and then a mapping of 'force' to a vector, mass to a real number, as well as space and time to define acceleration. Then, our axioms would be this correspondence in addition to F=ma. Establishing such abstract connections based on very real physical things under hot philosophical and scientific debate, based on the inductive methods of science rather then 'axiom based' is why I view this as being not being done.

That being said, there is an interesting paper I saw that formalized the principle of relativity by way of set theory and 'results' of an experiment, I think this was it: https://www.researchgate.net/publication/51937070_Formal_statement_of_the_special_principle_of_relativity [published in Springer] I'm sure it isn't impossible to do more things like this but if I am working in physics, it's kind of more like 'what's the point'. In physics, all our formal models don't matter if they fail experimentation. We could have made the most robust version of Newtonian mechanics by axioms and it wouldn't account for the most basics of modern physics discovered. Not even electromagnetism.

>> No.12047704

>>12045479
other than that we can make 1=2?

>> No.12047856

>>12046015
OK, but how would you define the limit of the sequence in >>12045814 in the hyperreals*? It definitely isn't 1, by the argument in >>12045675

*i.e. [math]f(0.9),[/math] [math]f(0.99),[/math] [math]f(0.999),[/math] and so forth, where [math]f[/math] is the embedding.

>> No.12047941

>>12045528
1/9>0.111...

>> No.12047984

If you claim that the infinite sequence of numbers makes sense, then you already postulate that infinity is possible. Then you shouldn't complain about infinitesimals.

>> No.12048018

>he doesn't know
kek

>> No.12048228

>>12047984
"real numbers don't include infinitesimals" is not complaining, it's a fact

>> No.12048325

>>12047984
>then you already postulate that infinity is possible
this is unrelated to the stuff inside of infinite sets, the real numbers have a definition that doesnt include infinitesimals

>>12047856
it is one
you still arent realizing the difference between the kinds of sequences here

you are taking a sequence of hyperreals and limiting to another hyperreal
you arent constructing a hyperreal as a sequence of reals

youre taking .9 + 0/w
.99 + 0/w
.999 + 0/w
...
at the end of the limit, the infinitesimal part is still 0
with that being said, you technically cant even define limits in the hyperreals, since they dont form a metric
but the obvious way, taking limits on real parts and infinitesimal parts separately would be one of the better ways

but even easier is just noticing that f(.9) are all real numbers themselves, so we can just use the transfer principle

>> No.12048479

>>12047527
thank you anon

>> No.12048504

>>12048228
Why should we prefer your "real numbers" to numbers containing infinitesimals? Are you claiming that they describe the actual universe better?

>> No.12048527

>>12048504
no, why would you think that

>> No.12048544
File: 58 KB, 704x414, hyperreals.png [View same] [iqdb] [saucenao] [google] [report]
12048544

>>12048325
I think I've found out why we are at loggerheads; see pic related (yeah I know it's r*ddit, but the people on r/math know their stuff). The limit of the sequence I defined is not 1, because it doesn't have a limit at all.

We are each talking about different sequences and thus we are both right about our own points. Neither of us is wrong (about our own sequences), but neither of us is 100% correct.

>> No.12048605

>>12048504
I mean, literally every physics book I have ever seen uses calculus and real numbers so... yes.

>> No.12048617

>>12048544
[Different person giving my though] Personally, I think hyperreals are above most people to even argue about. 0.999... is quite strictly terminology used in the reals, assigned to a real. This real is embedded in hyperreals. Only if one wants to define a different hyperreal by the notation 0.999... [in the context of .9,.99 and so forth being considered in the hyperreals and reals and considering limiting questions that why] can one get a different answer, but this requires a higher level of tech. Personally, all I've actually seen about the hyperreals are in their use as real closed fields in mathematical logic. The terminology is not well defined for hyperreals is the ultimate answer.

>> No.12048658
File: 120 KB, 400x333, 1584418870693.png [View same] [iqdb] [saucenao] [google] [report]
12048658

>>12048617
>The terminology is not well defined for hyperreals is the ultimate answer

Yeah [math]...[/math] is a vague and imprecise way of representing a hyperreal number. A massive chunk of this thread is a debate which arose from 2 different interpretations of what those dots mean.

>> No.12048831

>>12048605
Will the physics in it break if there will be numbers between 0.(9) and 1?

>> No.12048873

>>12048831
Well, math itself would break, so let's say yes.

>> No.12049009

>>12048325
You don't need a metric in order to define a limit.

>> No.12049025

>>12049009
this comment was made by topology gang

>> No.12049485

>>12045973

How is bottom-left a proof by induction?

>> No.12049555

>>12044835
op is retarded

>> No.12049694

>>12044835
2 + 2.999... = 5

>> No.12049705

>>12045973
3*a^n+3*b^n=3*c^n has no natural solutions for natural n>2.
Therefore a^n+b^n=c^n has no natural solutions for natural n>2.
Here, I proved Fermat's theorem in two lines! What was all the fuss about?

>> No.12049987
File: 31 KB, 665x624, 10400000.png [View same] [iqdb] [saucenao] [google] [report]
12049987

Cirno already proved this

>> No.12049997

>>12049705
this is the 4th time youve posted this, and its still dumb as shit
if youve proven the top line, then yes, the second line follows, theres no issue

>> No.12050010

>>12049705
That's why there is the box on the right for the truly skeptical bastardfucker that demands a rigorous proof. The rest are more like persuasive arguments.

>> No.12050967
File: 18 KB, 330x165, swayello.png [View same] [iqdb] [saucenao] [google] [report]
12050967

>>12049987

Cool. I don't think I've seen the proof of the formula for the sum to [math]\infty[/math] of a geometric series before.

>> No.12050973

>>12049997
>if youve proven the top line, then yes, the second line follows, theres no issue
Bingo. Now you can see why your 0.333... proof is as silly.

>> No.12050990

>>12050010
Where do your infinite series and convergence definitions introduce infinitesimals?

>> No.12051142

>>12047941
0.3 written in binary is 0.0100110011001...

Are you going to say that 0.3 != 3/10?

3/10, 0.3, and [0.010011001...]base2 are just different ways of representing the same number.

Same is true for 1.
9/9, 1, 0.999... Are all the same thing written in different ways.

>> No.12051146

>>12051142
0.3000... or 0.2999...?

>> No.12051152

>>12051146
[0.3000..]base10 = [0.2999...]base10 = [0.010011001...]base2

>> No.12051169

>>12051152
[0.3000..]base10 = [0.2999...]base10 + [0.0000...1]base10

>> No.12051224

>>12051169
>[0.000...1]base10

What .00...1??? There is no 1 at the end of infinite series of 0s. If there was, then it wouldn't be infinite. This isn't 0 + epsilon. 0 + epsilon is a number, but it is distinctly different from 0. Similarly, 1 - epsilon is a real number, but it is distinctly different from 1.

One of the properties of the real numbers is that if you have two number, x and y, then you can always find another unique number between them. Put simply,

x < (x+y)/2 < y for all real numbers where x, y are unique elements of the real numbers.

This holds for (1 - epsilon).

(1 - epsilon) < ((1 - epsilon) + 1)/2 < 1

This doesn't hold for 1 and 0.999...

If it did, what number would be in between them?

Therefore, 1 and 0.999... must be the same number.

>> No.12051374

>>12044835
>No number other than 1 is equal to 1.
.999... is not a number is a mathematical expression.

Are so saying cos(0) = 1 isn't valid?

>> No.12051479

>>12051142
0.9...=0!!!

>> No.12052014

>>12050990
They don't.

>> No.12053301

>>12051479
Well, yeah. 0! = 1 = 0.999...

>> No.12053375

>>12050973
you actually arguing against 1/3 = .333... ? you are retard, correct?

>> No.12055655

>>12044835
You're right.
0.99999... doesn't exist.
0.33333... doesn't exist.
It's a flaw in the decimal system.
We roll with it.

>> No.12055714
File: 301 KB, 172x172, 117015-full.gif [View same] [iqdb] [saucenao] [google] [report]
12055714

>>12055655
>It's a flaw in the decimal system

There's nothing special about base 10 (ten) in this regard. The same phenomenon occurs in other bases.

>> No.12055732

>>12055714
>The same phenomenon occurs in other bases.
Not all of them, people usually argue that 12 is objectively the best base to use in but it never stuck because of some bullshit like "we have 10 fingers doe".

>> No.12055751

>>12055732
>1x2x3x4
>What about x5?
You just found out why we have 60 minute hours.

>> No.12055755

>>12055732

In base 12 you have [math]1=0.BBBBB...[/math]

>> No.12055763

>>12053375
Is your worldview shattered now?

>> No.12056035

>>12055763
no, i already knew morons existed before i met you

>>
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