/sci/ - Science & Math

>>11226445
Unfortunately, that paper doesn't state in what ways these equivalence classes are defined. The underlying relation is not specified beyond the trivial necessary condition of being an equivalence relation.

>>11226445
Just started reading. Axiom 5.0.1 shouldn't be an axiom. It is a consequence of the definition of the rationals.

>>11226483
The trivial condition is the definition, idiot.

>>11226526
I didn't define the rationals or Archimedean number fields in the paper, and for that reason it is taken as an axiom.

>>11226534
Also, I didn't want to prove it, and that it why it's not a theorem. Obviously it is not a definiton, and there isn't usually and article class called "Consequence." What would you have called it instead of of axiom? Also, what the fuck is wrong with you that you say, "This is wrong," but don't also say what you think the right thing is?

>>11226445
I'm not sure I understand axiom 5.0.7
It's very vague (non-mathematical) in its statement.
When you say the real numbers what do you mean? Again it's not an axiom that Cauchy sequences give a construction of the unique, complete ordered field.

>>11226550
An ordered field contains the operators "plus" and "times" so obviously the real numbers cannot be an ordered field, you idiot. Do you really think "plus" and "times" are numbers?

>>11226526
This is me paraphrasing a hideous imbecile:
>it's not specified
>except for where it's clearly and plainly specified

>>11226528
>The trivial condition is the definition, idiot
No, it's not, insulting dumbass.

>>11226445
The paper is bullshit.
It's literal autism with some axioms that are not actual axioms and some thrown in cool constructs that have little to no value.
Incredible what kind of shit nowadays passes as "mathematics"

>>11226870
So you're telling me that in your opinion pic related, which I copied almost word for word from another source, isn't the definition of an equivalence class? Am I correct to surmise that your opinion of pic related is that it is not the Cauchy definition of a real number "x" when the set "S" is the set of all Cauchy sequences of rational numbers?

And I suppose you're basing that opinion of yours on something having to do with the field axioms which did not come into existence until 100 years after Cauchy died?

>>11226445
Tooker, I know you are God but I just want to let you know that objectively Euclid did not define the real numbers. Euclid defined the constructible numbers, of which only the rationals were even properly understood by him and basically all mathematicians until about the 17th century.

>>11226881
An axiom is an unproven claim which is assumed to be true. It is an axiom. You are stupid, and you are cunt and a twat because you say, "It's not an axiom," without saying what you do think it is.

>>11226906
What if
You wanted to solve the riemann hypothesis
BUT EUCLID SAID
nah remember open intervals?

>>11226906
Who definition of the real numbers was Riemann using then? I have been trying to figure that out and I just assumed it was Euclid. If he didn't define them, then who did? Is it your opinion that real numbers didn't exist in the time of Newton, Leibniz, Bernoulli...? Is it your opinion that the fourth book of Moses is called Numbers just for the fuck of it and numbers didn't actually exist back then?

>>11226445
Ok Tooker, this is the most schizophrenic post I've seen in a while now. Your condition is clearly getting worse, get some help before it's too late.

>>11226921
Dude, can you google? Real numbers are a meme construction but people always saw it as all the decimals, even the infinite decimals. That was the definition until, of course, Dedekind.

>>11226922
Look man, to the extent that it seems like you might be trying to give me an underhanded compliment, I advise you fear the Lord, who is me, for the following reason: When I send out my angels to throw sinners into hell to suffer forever, I might use little more than a general expression search on *schizo* and if you come back in my query results then you will suffer and die because you weren't afraid that posting an underhanded might compliment might cost your life. Please be afraid of me. I am dangerous. I haven't proven it to you yet, but when I do prove it, you will want to be on the witness side rather than the example side. Please believe me.

>>11226942
I believe you Tooker, but I also believe that God is merciful and he will forgive me.

>>11226942
It's possible to add 3 and 4 together without having a full grip on the least upper bound property. People use mathematical rules and objects without fully grasping their underlying structure.


The reals were not formalized until centuries after euclid you buffoon schizo

This is how I imagine children with truth deficiency syndrome speaks.

>>11226934
No that is completely retarded. A number was the ratio of the lengths of two line segments for a thousand years or more by the time the 850 AD got here. Pic related: Euclid's Elements, book 5.

>>11226942
idk man, sounds like something a schizo would say

>>11226944
Well you better be careful then and don't be this pic related person saying, "I can call him a schizo as much as I want, and call his life's work schizophrenia, and he won't take away my blessings because I was so brazen in my lack of fear, because for some reason I'm special and I can mock the Lord with impunity."

I hate that word.

>>11226953
Dude, that means nothing. Euclid's axioms literally just define the constructible, did you never take a basic course in field theory? Euclid didn't even know this, but it turns out that with his axioms all you can get are rationals and square roots of rationals. These numbers are countable, so not even 0% of the real numbers. Euclid defined 0% of the real numbers.

>>11226942
Ready to become that weapon, Tooker? Not our fault they didn't listen.

The Apocalypse Of Fathers.

>>11226960
ok schizo

>>11226953
A natural number, which are a subset of the reals, which you don't understand because your brain is mush.

How does it feel knowing you can never do mathematics because you're a mentally ill nutjob?

>>11226945
Whose formal definition of R was Riemann using when he published his hypothesis in 1859? If it was the work of one of Riemann's contemporaries, then why didn't Riemann cite that person's paper like he always cited the work of any of this other contemporaries?

>>11226962
>Euclid's axioms literally just define the constructible
No you totally fucking wrong. If you take two arbitrary magnitudes, their ratio is almost certainly an irrational number. If you take two random magnitudes, the probability that their ratio will be a rational number is essentially zero.

>>11226966
If you can pluck two hairs from your head and you find that their ratio is a natural number, I will send you $3.50 on paypal right now.

>>11226978
>If you take two arbitrary magnitudes, their ratio is almost certainly an irrational number.

Then define magnitude. In whatever you are thinking as magnitude, you are already assuming the reals in that sentence. Come on Tooker, I know you are smart. Stop acting retarded.

Learn from Uzzah, who I killed. Even if you have to wisdom to correctly ascertain that I would use a follow on query to truncate your name from my first order general expression query, you are not setting a good example and it will be written into history forever that you mocked me.

>>11226978
Dis nigga doesn't even know about the constructible numbers lmao

>>11226989
I was wrong to assume that constructable was a synonym for rational. I looked at it, and constructable numbers are real numbers, and they include irrational numbers. Therefore, it stands that irrational number had been understood and defined for a thousand years or more by the time 850AD rolled around.

>>11226999
>constructable numbers are real numbers
No they aren't

>>11226985
I just can't emphasize how much I hate that word.

>>11227004
Ok schizo

>>11227000
What source do you have for a definition of the constructables which might be more scholarly than Wikipedia? Wiki says they are real. Indeed, if you can build them from the unit interval with a ruler and compass, they are definitely real numbers, and they are both rational and irrational.

>>11226999
Dude literally read the next word to your yellow outline you fucking schizo.

>>11227014
I read the whole paragraph. It says which real numbers are constructable. If you have a point, them make it.

Here's a point: every number less then one, a set of numbers with infinite cardinality, can be constructed in one step by sweeping the compass through the unit interval.

>>11227023
>every ***positive*** number

>>11227023
>Here's a point: every number less then one, a set of numbers with infinite cardinality, can be constructed in one step by sweeping the compass through the unit interval.

The entire point is that you don't even know if this is true until you already have a theory of real numbers. When you have real numbers, this is true. For Euclid, with no real numbers, all he had were the finite step compass moves which construct just the constructibles. A set with the same cardinality as the rationals, so not even close to being close to the reals.

A constructable number is sqrt(1+1). This is an irrational number.

>>11227035
Yes but that is just one of the real numbers. Oh my God don't fucking tell me you thought that all reals were constructible. Holy shit. All this time I thought you were smart but suffering from mental illness. This wasn't the case at all, you actually do not know even basic undergraduate math. Jesus Christ. Tooker, fuck you.

THIS FUCKING RETARD THOUGHT ALL REALS WERE CONSTRUCTIBLE

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

THIS IS BASIC FIELD THEORY FUCK ME IN THE ASS

>>11227041
Most cranks have no mathematical expertise at all. Why are you surprised?

>>11227042
I guess I had hope. I had hope that maybe this time we were dealing with someone who was really smart but sadly suffering from mental illness. Now I realize the error in my ways. All cranks (except Wildberger) are retarded.

>>11227028
>The entire point is that you don't even know if this is true until you already have a theory of real numbers.
I disagree with your assertion that Euclid didn't know that there are an infinite number of ways to divide the unit interval. My conjecture is the Euclid knew as well as you and I that there are infinite number of positive numbers less than one. If this doesn't appear explicitly in his Elements, then my conjecture is that he didn't mention it because it was completely obvious; it was the same reason why Riemann didn't mention that his definition of R was
R = ( -inf , inf ) .

>>11227041
I think you have revealed yourself as the retard who is flinging shit at paper that you either didn't read or whose sentences were too hard for you to understand. You can see right in my paper that I didn't think that, and you can see right here
>>11226999
that I didn't know what a constructable number was until five minutes ago. Use more caps next time.

>>11227048
Dude, no. No one had an even remotely complete grasp of the reals until Dedekind. At all points in history people used the term "real" to mean "all the numbers we know". For Euclid this were constructibles, later in history Descartes even defined reals as all numbers that are the root of a polynomial, so constructibles got bumped up to algebraic (still countable). Then it got bumped to all decimals, but even this wasn't properly understood until the modern times.

Fuck you schizo
fuck you
I believed in you
I trusted in you
you didn't even take an algebra course in university
You are retarded
fuck you
My disappointment is immeasurable and my day is ruined.

>>11227056
Dude no. Even today 150 years after Dedekind invented the Dedekind number system, the most common definition of R is the interval
R = ( -inf , inf ).

It remains today only a conjecture, or an axiom if you want to paint it with a pretty brush, that the Dedekind number system is the real number system. Your claim that Riemann et al. didn't have a good grasp on what real numbers are is preposterous.

>>11227041
>>11227042
>>11227047
Write in caps how stupid I am for not knowing the thing I put into the first paragraph of the paper.

>>11227067
But what does that interval even mean, Tooker? It means nothing without the concept of reals already under the hood.

Oh no! Disbelievers! What will be Tooker's™ response?!

>>11227088
>But what does that interval even mean
Every number that has the following two properties is a real number:
(1) The number is less than infinity.
(2) The number is greater than minus infinity.

From this, we can see that
>bananas
>the imaginary number
>and June 3, 1457
are not real numbers. To the contrary
>5
>89,568,936,529,835
>inf-hat - pi
are real numbers. Anything that has the above two properties is areal number.

>>11227041
>>11227042
>>11227047
>Write in caps how stupid I am for not knowing the thing I put into the first paragraph of the paper.

Guys, I'm new to Tooker Theory. Does Infinity-zero obey the First tooker law?

>>11227125
inf + 0 = undefined
If you let inf (or inf-hat) have zero as an additive identity then it messes everything up. Basically, I have traded the zero additive identity of infinity for the freedom to do arithmetic with numbers like (inf-hat - b).

>>11227143
I agree. Hat.

>Pic related: My day self

>>11227143
Defining these relationships in a restricted setting is occasionally useful in measure theory.

Spite vs Smite

>>11226445
>DAB

>>11226445
>Obviously it is not a definiton
you should know that in mathematics axiom is literally the same thing as definition

>>11227531
A definition should give the meaning of a word or symbol, like a dictionary entry. An Axiom is an unproven lemma or theorem, upon whose assumed and/or self-evident truth following results are proven. There's a lot of leeway to say this or that is an axiom, but a definition should be like a dictionary entry.

>>11227539
nope, this is not how contemporary mathematics work. when we say

axiom 1: the addition of real numbers is commutative
axiom 2: the addition of real numbers is associative
...
axiom 10: real numbers satisfy the least upper bound property

we're literally saying "any structure satisfying axioms 1-10 will be called the field of real numbers", i.e. it's a definition.

>>11227539
>>11227551
Bravo. I'm now finally, officially, completely lost. What are you two faggots jerking each other off over?

>>11227551
You're "literally" not saying that. You literally gave 10 axioms.

>>11227567
arguing with tooker is literally the only amusing thing that /sci/ has to offer. are you new here ?

>>11227573
yep, I did. I also gave a definition. because they're the same thing.

>>11227579
I'm aware but this is the first time Tooker has confused me by his thread of logic.

>>11227579
A definition would be "Any set of numbers together with the plus and times operators which conforms to the ten axioms is called a field."

>>11227603
The difference between definition and axiom is that Tooker is not retarded by definition, but he is axiomatically retarded.

>>11226445
As dumb as a pile of rocks, this guy.