/sci/ - Science & Math

[Anonymous No.12331955]

2 comes after 1 in the natural numbers. 2 comes after 1 in the set of integers. What number comes after 1 in the set of reals? Its not 2, because 1.5 exists. Its not 1.1 either. Nor is it 1.000000001. The number that comes immediately after 1 in the real numbers can then only be 1.000...01, the number which is infinitesimally larger than 1. This infinitesimal value is 0.000...1, since 0.000...1 + 1 = 1.000...1

Consider the number that comes before 1 now. This number should be ininitesimally smaller than 1. 1-0.000...01 = 0.999...9. Since this number comes before 1, it must not be the same as 1. Thus 1=/=0.999... QED

[Anonymous No.12332017]

>0.000...1
=0

[Anonymous No.12332032]

>didn't ask about rationals.
YIKES

[Anonymous No.12332035]

>>12331955
Have you heard about our Lord and Saviour, Georg Cantor?

[Anonymous No.12332339]

>>12331955
There are no adjacent numbers in the set of real numbers. Can we stop this already? Those retard 0.999... =/= 1 threads happen literally every day

[Anonymous No.12332343]

>>12332035
No, tell me his entire life story

[Anonymous No.12332345]

>>12331955
Holy based

dogmatics mathlets are heading your way

[Anonymous No.12332394]

>>12331955
There is no real number which can be said to come immediately after 1.
1.000...1 is terminating, and can always be undermined by a number with an additional zero before the terminating 1.

In contrast, 0.9999... is non-terminating, and by definition there is no number which can be greater than 0.9999... and less than 1.0. You cannot compare 1.000...1 and 0.9999... because one is in the realm of countable infinity, and the other is in the realm of uncountable infinity.

[Anonymous No.12332397]

>>12331955
.00...1 is impossible unless the 1 is an ordinal,not a cardinal.

[Anonymous No.12332424]

>>12332339
>>12332394
Let [math]\leq[/math] be a well-ordering on [math]\mathbb{R}[/math]. By the well-ordering theorem, this is guaranteed to exist. Let [math]R^* = \{x : 1 \leq x\}[/math]. Now consider the least element [math]S(1)[/math] in the set [math]R^*\setminus \{1\}[/math]. It is clear from the definition of a well-order that [math]1 \leq S(1), 1\neq S(1),[/math] and there does not exist an element [math]x \in R^*[/math] not equal to 1 or [math]S(1)[/math] such that [math] 1 \leq x \leq S(1)[/math].

[Anonymous No.12332478]

>>12331955
The reals form an Archimidean field, so nothing comes after 1.

[Anonymous No.12332485]

>>12332017
no. 0.000...1 is meaningless nonsense.

[Anonymous No.12332488]

>>12331955
ACTUALLY IT'S .000...1*10 BECAUSE THERE HAS TO BE EXACTLY ∞-1 0S NOT ∞ 0S BECAUSE THERE NEEDS TO BE ROOM FOR A 1 AT THE END
>>12331955
>1=/=0.999...
CORRECT
>>12331955
>QED
I BET YOUR MOM IS GOOD AT QUANTUM ELECTRODYNAMICS, is she?

[Anonymous No.12332845]

>>12332488
cringe

[Anonymous No.12332954]

>>12331955
the real numbers are a densely ordered set, which means there is no number "after" 1. The integers are not densely ordered.

[Anonymous No.12332958]

>>12332424
false, the reals are densely ordered so there always exists an x /in R* such that 1<x<S(1). Hence S(1) doesn't exist.

[Anonymous No.12333181]

>>12331955
there is no number that comes immediately after 1
suppose x > 1 comes right after one. then 1 < (1+x)/2 < x, contradiction.

[Anonymous No.12333241]

>>12332394
>You cannot compare 1.000...1 and 0.9999... because one is in the realm of countable infinity, and the other is in the realm of uncountable infinity.
oh my

[Anonymous No.12333243]

>>12331955
Iiiiits MACHINE EPSILON

[Anonymous No.12333301]

>>12331955
>an absurd premise leads to an absurd conclusion
bravo

[Anonymous No.12333321]

>>12333241
1.000....1 isn't a well defined number period

[Anonymous No.12333324]

>>12333301
At least OP is consistent

[Anonymous No.12333325]

>>12333324
and the definition of the reals isn't?

[Anonymous No.12333331]

I was having a pleasant chat the other day, with a young lad who thought he could arrange the two strings -1/n and 1/n as a single string, such that all the numbers were ordered from least to greatest and such that every number in the string had both a successor and a predecessor except for the first and last.

If you're still around, and would like to continue your education, please feel free to continue elaborating your idea.

[Anonymous No.12333333]

>>12333333

[Anonymous No.12333336]

>>12333333
I am Neo.

[Anonymous No.12333340]

>>12333333

[Anonymous No.12333348]

>>12332958
There's no such thing as a dense well-order, you fucking faggot. Get off my board.

[Anonymous No.12333352]

>>12333333
oh shit

[Anonymous No.12333358]

>>12333348
Yeah, pretty sure reals can't be well ordered because, for example, a subset of reals (0, 1) does not have a least element.

[Anonymous No.12333361]

>>12331955
what is 1.000..01 * 10
what is 1.000..01 * 0.1

[Anonymous No.12333363]

>>12333358
Every set can be well-ordered. This is the well-ordering theorem.

[Anonymous No.12333364]

>>12333333
This post checks all the posts that doesn't check it.

[Anonymous No.12333369]

>>12333358
can't be well ordered is not the same as the standard ordering being well ordering

[Anonymous No.12333371]

>>12333369
Ah, I see. I get it, thanks.

[Anonymous No.12333422]

>>12333363
>well-ordering theorem
Fake and gay. It pretends you can shoehorn a non-existent least element into a set, by wrapping subset brackets around a bunch of DIFFERENT least elements that do exist.
Garbage logic.

[Anonymous No.12333427]

>>12331955
1/∞ + 1> 1 > 1/∞ > 0

[Anonymous No.12333430]

>>12331955
>>12333333
Depends what formalism mr. quints wants

[Anonymous No.12333437]

>>12332424
>Now consider the least element S(1) in the set R∗∖{1}
Prove you can pick up such an element.

[Anonymous No.12333441]

>>12331955
based and guaranteedbumplimitpilled

[Anonymous No.12333443]

>>12333437
it's bounded from below

[Anonymous No.12333450]

>>12333443
By what?

[Anonymous No.12333451]

>>12333450
by 1

[Anonymous No.12333454]

>>12333358
Well-ordering Principle iff Axiom of Choice iff Zorn's Lemma

[Anonymous No.12333464]

>>12333451
So what? How does that prove there's a smallest element?

[Anonymous No.12333475]

>>12333333
>recursion
2spooky4me

[Anonymous No.12333476]

>>12333363
It's not a theorem, it's an axiom.

[Anonymous No.12333482]

>>12333464
do you know what a well ordering is?

[Anonymous No.12333487]

>>12332424
>no one has pointed out that [math]\geq[/math] is not the usual order on [math]\mathbb{R}[/math] and [math]S(1)[/math] could be 42 for all we know
The absolute state of /sci/

[Anonymous No.12333492]

>>12333487
why would anyone need to point that out? it's obvious

[Anonymous No.12333625]

>I will suppose an axiom that will allow me to take an element from a set that is not an element from this set!
>wtf i proved a paradox, how could this possibly happen to me?

[Anonymous No.12333650]

>>12333625
What axiom are you talking about?

[Anonymous No.12333676]

>>12333650
The axiom which states that well order and usual order on the reals are the same.

[Anonymous No.12333680]

>>12333676
There's no such axiom and nobody claims there is

[Anonymous No.12333686]

>>12333680
>There's no such axiom
Yes there is. Paradoxes can be axioms.
>nobody claims there is
Nobody in this thread, yes.

[Anonymous No.12333731]

>>12333333
BOOBAS

[Anonymous No.12336306]

>>12333333
Nice

[Anonymous No.12336315]

>>12333333
and despite this, it does not equal 12333334

[Anonymous No.12336316]

>>12333333
†

[Anonymous No.12336528]

>>12331955
Thank you OP. Praise Jesus!

[Anonymous No.12336529]

>>12332485
>no. 0.000...1 is meaningless nonsense.
But infinity is perfectly fine.

[Anonymous No.12336623]

>>12336529
In the context of repeating digits, one has a perfectly consistent definition and the other doesn't

[Anonymous No.12336636]

>>12336623
Have numbers, which have foundation in reality
Have infinity, an entirely imaginary concept
Expect both to seamlessly integrate.
>But its okay, we have a definition of infinity! Sure, its totally made up, but its consistent!

[Anonymous No.12336735]

>>12336636
>Sure, its totally made up, but its consistent!
this is exactly how math works, yes

[Anonymous No.12336743]

>>12336636
All of mathematics is basically made up. No need to get upset about that.

[Anonymous No.12336762]

>>12336735
Numbers are things which can have a physical representation. Infinity cant.

[Anonymous No.12336768]

>>12336762
numbers are abstractions, they dont have real physical representation

[Anonymous No.12336779]

>>12336762
Doesn't matter. Real numbers have certain properties and the consequence of those properties is that 0.999... represents the same number 1 does.

[Anonymous No.12336812]

>>12336768
One apple. Two apples.
>>12336779
Look at two parallel railway lines and they appear to converge.All the way to infinity and they will meet. Except its impossible to reach infinity, and hence any derivative of that process is meaningless nonsense.

[Anonymous No.12336841]

>>12336812
We've been over this. There's no number between 0.999... and 1 and that makes them the same number under reals.

[Anonymous No.12336855]

>>12336812
>Except its impossible to reach infinity,
reach by doing what?

[Anonymous No.12336880]

>>12331955

[Anonymous No.12336889]

>>12336812
>Except its impossible to reach infinity, and hence any derivative of that process is meaningless nonsense.
Well then it doesn't make any sense to talk about 0.999... in the first place, because you can't define a number with infinite digits if infinity doesn't exist.

[Anonymous No.12336916]

why do these stupid threads stay up? This is not science, just bait

[Anonymous No.12336924]

>>12331955
There is no real number that comes after 1.
Infinitesimals aren't real numbers.

[Anonymous No.12336925]

>>12336916
I can believe some people just can't get it. Kids in school sometimes remain unconvinced as well.

[Anonymous No.12336927]

>>12336812
>One apple. Two apples.
infinity apples

[Anonymous No.12336938]

>>12331955
> The number that comes immediately after 1 in the real numbers can then only be 1.000...01
1.000...001 is closer

[Anonymous No.12336967]

>>12336812
>parallel railways lines don't intersect
>therefore we shouldn't use infinity
right

[Anonymous No.12336980]

>>12336925
>Kids in school sometimes remain unconvinced
Only because of activist teachers who like infinitesimals or dislike infinity, and use their students as a platform to push their own alternate math ideas. It really is child abuse.

[Anonymous No.12336982]

>>12336812
>>12336967
Parallel railways don't even converge so

[Anonymous No.12336986]

>>12336980
Even with normal teachers, some students are like "0.000...1". I don't think I have met an infinitesimal loving teacher yet but I can believe they exist.

[Anonymous No.12336997]

>>12336986
Yes, if 0.999... is explained as an ongoing process—i.e., arbitrarily long finite—then the arbitrarily long finite decimal elision followed by a 1 makes perfect sense.
That's not what's happening though. The nines are simply there and stay the same forever. The only snags in intuition arise from poor or misleading education, not from the children themselves.

[Anonymous No.12337031]

>>12336997
Yeah, I agree. Just saying that there are some "smart"-ass students who make those arguments to appear smart.

[Anonymous No.12337122]

>>12337031
Students who make that argument should be praised as smart, and explained that what they've discovered is the idea of arbitrarily large finite.

[Anonymous No.12338198]

>>12333333
did you just check yourself?

[Anonymous No.12338255]

>>12333333
Witnessed

[Anonymous No.12339353]

>>12333333
Rare

[Anonymous No.12340575]

>>12333333
prove it

[Anonymous No.12340579]

>>12336636
>Have infinity, an entirely imaginary concept
Wrong

[Anonymous No.12340627]

>>12340579
no u