/sci/ - Science & Math

>>10809129
Do you still like you're constantly being anally raped? You might wanna get that checked out Tooker.

Why are you more interested in my anus that the definition of real numbers?

An equivalence class of rational Cauchy sequences or a Dedekind cut on the rationals.

>>10809151
hobo anus is substantially more interesting than your mathematics

>>10809159
Euler died in 1783. Cauchy was born 1789 and Dedekind published in 1872. Do you think real numbers didn't exist during Euler's lifetime?

>>10809163
If you want to discuss my anus and you all's constant rape going back as far as to when I thought it was just my body falling apart with old age causing that discomfort, then please make a dedicated thread for my anus, or more properly your fascination with my anus. Since you and everyone who abided the implants to begin with are all going to end up in the same toilet, maybe you can all commiserate about it 'gether in the same thread now.

This guy slaps your fake number system in the ass, what do you do?

>>10809187
I let some of the prisoners out of Fulton County jail to rape Paul Merritt's wife and children.

>>10809129

Its a moot point. The entire conceptual framework of mathematics needs an overhaul.

As it stands Maths is bean counting. Operators are how we count those beans. Higher math results from dealing with the absurdities that logically result from such a primitive system of thought. At this point its a waste of time as all it does is reveal more curiosities about entirely artificial absurdities. Instead we should be focusing all our mind power on understanding the reality of the Universe and devising an entirely new framework for describing it, the very concept of "numbers" as being abstracts or representations of physical objects may even become obsolete. Yes, we are talking about overhauling at that sort of fundamental level. Exactly what is anyone's guess, but there are some clues, some of which may be trivial, some which may not. For example excluding infinities.

>>10809129
>What is a real number?
A real number is a member of the unique-up-to-isomorphism structure that satisfies the axioms of the complete ordered field.

>>10809346

Good parrot, here, have a cracker.

>>10809362
>how much is 2 + 2?
>and don't give me 4, give me a real answer
The fact that a certain solution to a problem is the popular option that everyone agrees on does not mean it is wrong, you know.

>>10809187
I wish we had Norman posting here. Jon's retardation is just sad.

>>10809346
So then do you think that real numbers didn't exist until complete ordered fields got invented?

Do you think all progress in improving the definition of R has ended, and that although every previous definition has been replaced with a "better" one, the one which is popular now will be the popular one into perpetuity, never changing and never allowing any progress?

>>10809576
>So then do you think that real numbers didn't exist until complete ordered fields got invented?
No, not at all. The structure was something that people had conceived of long before that. The axiomatization of the real ordered field is a more recent innovation to *communicate* what structure we are talking about.

The structure of the real numbers is distinct from any particular axiomatization of the real numbers, just like the structure of the natural numbers is distinct from the Peano axioms. Be careful not to confuse those two concepts.

>Do you think all progress in improving the definition of R has ended, and that although every previous definition has been replaced with a "better" one, the one which is popular now will be the popular one into perpetuity, never changing and never allowing any progress?
No, I think that there may well arise better ways of communicating what structure we are talking about. But the structure itself has not changed. All the changing definitions have been better, clearer ways of describing and explaining what the real numbers are, but they have all been *equivalent*, in that they are different ways of describing the same structure.

I see room for continued progress in how we communicate the properties of the structure we have in mind. But *what structure we are talking about* will not change.

>>10809303
infinity and 0 do not exist

Speaking of ignoring history, Euclid's Elements dismisses magnitudes "in the neighborhood of infinity". That's the history of where real numbers came from.

>>10809838
>*what structure we are talking about*
Yes, I think the structure is the real line with real numbers being cuts in that line. What do you think the structure is?

>axiomatization of the real ordered field
For me when I look at this, the I like to separate the field axioms from the field axiom definition of R which is:
AXIOM: R is a set such if R' = {R,+,x}, then R' satisfies the axioms of an ordered field.

I am greatly irked by the common failure to distinguish R by itself from another set containing R and also the plus and times operators. It's not that it's a "failure" so much as a rampant imprecision in the language used to describe something which is very precise. My point which cannot harp on enough is this: How can R be an ordered field when R is only numbers but a field contains numbers and operators.

>>10809845

Correct. Zero is another clue.

>>10809890
Where does it dismiss them? Here's a nice translation which each little thing numbered so you can precisely identify where Euclid dismisses them.
http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf

I think anything you could cite as "dismissing" real numbers in the neighborhood of infinity could be equally well be used to dismiss numbers like pi, Phi, and root 2. Let's see if you can come up with something quoted from the text which "dismisses" numbers in the nbhd of infinity but doesn't equally dismiss irrational numbers. Maybe, you can! I'll be interested in your specific citation of Euclid's dismissal of such numbers.

>>10809918
>>10809845
>no zero
What is this thing in pic related field axioms then?

>>10809933
Book 5, definition 4. But you knew that, because this isn't the first time we've done this dance. This comes from the long history of dismissing infinite numbers.

>>10809913
>What do you think the structure is?
I told you -- when I am talking about "the real numbers", I am talking about a member of the unique structure satisfying the axioms of the complete ordered field. The same holds for other mathematicians.

>Yes, I think the structure is the real line with real numbers being cuts in that line.
I'm not sure what you mean by that, but sure. I can't tell whether that's the same structure I (and by extension, most other mathematicians) have in mind, though.

>For me when I look at this, the I like to separate the field axioms from the field axiom definition of R which is:
Indeed. Which is why I worded by interpretation of the real numbers as "members of the structure satisfying those axioms".

>I am greatly irked by the common failure to distinguish R by itself from another set containing R and also the plus and times operators.
That's not a set, that's a structure -- a set of elements, combined with relations between those elements.

>It's not that it's a "failure" so much as a rampant imprecision in the language used to describe something which is very precise.
Indeed, which is why I am being very careful in my wording.

>How can R be an ordered field when R is only numbers but a field contains numbers and operators.
Well... how are the natural numbers distinct from the integers or the rational numbers? As bare sets, they are all isomorphic. If you want to distinguish between those three structures, you HAVE to consider more than just the bare sets.

>>10809943
Ah, the Archimedes property. Numbers in the neighborhood of infinity absolutely satisfy this property.

Let me say, however, I am considering changing my axioms about the operations such that they will no longer satisfy this property, and then I will have to resort to claiming that Euclid was only describing reals in the nbhd of the origin.

Right now I have both of these operations defined:
(inf - b) + (inf - a)
(inf - b) - (inf - a)

I am thinking about getting rid of the sum definition and making it be undefined. To do this, I will also strip infinity of its multiplicative absorptive property when it is hatted. Then the sum will be undefined via some mechanism I create for calling something "larger than infinity." Anyways, if I take away the multiplicative absorptive property too, which I really ought to because it is slightly funny to only get rid of the additive absorption, then, pic related I would have
zx = (1/6)inf - (5/6) not greater than y = inf - 1

due to some notion that (1/6)inf < inf

>>10810012
"Multiply" in this context means adding to itself repeatedly, i.e. integer multiples in modern language. Once again, we've been over this before. For x=1 and y=inf-hat - 1, what integer can you multiply x by so that it'll end up bigger than y?

>>10810021
>integer multiples in modern language
In the modern language, "multiplication" isn't restricted to integer multiples.

Also, where do you see context that suggests Euclid was only implying multiplication by naturals? I don't doubt it's there, but I haven't been able to find it.

>>10810012
>Numbers in the neighborhood of infinity
oh boy here we go again

>>10810031
>In the modern language, "multiplication" isn't restricted to integer multiples.
The Archimedean property, which is what we're talking about, is still defined by multiplication of positive integers. And in general if you want to change a proof, you'd better not change the meaning of terms without checking the consequences.

>Also, where do you see context that suggests Euclid was only implying multiplication by naturals? I don't doubt it's there, but I haven't been able to find it.
Book 5, definition 2 defines what it means for one magnitude to be the multiple of another. Or, you can read the footnotes in the same translation you linked where it gives the definitions in modern notation.

>>10810054
>The Archimedean property, which is what we're talking about, is still defined by multiplication of positive integers.
Where is it defined that way? It's not defined that way in Euclid's book.

>Book 5, definition 2 defines what it means for one magnitude to be the multiple of another.
Pic related, this does not in any way imply restriction to naturals. Indeed, we can use Book 5 Def 1 to set up the case where a greater magnitude is 2.5 as great as lesser magnitude. Please explain you thinking.


>you can read the footnotes
I didn't see anything about naturals there, please explain your thinking.

Consider the footnote to Book 5 Def 1, also pic related. By my reading, this means that "m" is absolutely ~NOT~ restricted to naturals. In so far as Euclid was concerned with what can be measured with a ruler, one magnitude of 10cm is obviously greater than one of 9cm, and therefore the lesser is a part of the greater with
m = ( 10/9 )

Please explain your thinking!

Why I am the only in these threads who at least attempts to make complete and meaningful statements and everyone else, apparently, is tuned to maximum brevity, proof-by-non-existent-evidence quips?

I mean seriously. You can look right at the foot note for Def 1 and immediately see that "m" is not restricted to the naturals.

>What is a real number? What is the thing in itself without reference to the "plus" and "times" operators which are certainly not real numbers?

>>10810190
Yeah... that's Paul Merritt, a/k/a Michael Wolff of the "Gorilla Channel that's a good gorilla," fame. Paul listen to me: if you don't believe me and you let yourself and your family fall into my hands while still proudly proclaiming your disbelief, then you're going to get what unbelievers get. If you kill your family and then yourself in a murder suicide, however, then I will know that you realized what you did was wrong and that you believed me when I told you what you had coming. As a former Red Dog officer, I am sure there are plenty of people in the county jail who will zestily force themselves upon your loved ones, and I will help them do it if you're still breathing by the time I can help them do exactly that... for starters of your slow descent into hell.

(Yes, APD officer Paul Merritt who loves to name drop "Italy" and is some federal cuntbag of one variety or another he is the real person behind the frequent CNN guest Michael Wolff. Indeed, as much as Paul looks like Wildberger, he looks even more like Jake Tapper, a/k/a Jake Hitter, a/k/a Jake Murderer, a/k/a Jake Dirty-Cop, who showed up on CNN right about the same time Paul showed up at Alliance. I will have mercy on some of those people from Alliance because I like them so much, and Paul might have made it into that group because I did like Paul quite a bit, but the "good gorilla" thing, and the destroying my home thing, and the throwing me in the slave hole thing, and the trying to file a mental health detention affidavit against me thing are going to weigh heavier on the side of the scale that sends you to hell, Paul.

t. Yours Truly)

>>10810134
>Where is it defined that way? It's not defined that way in Euclid's book.
Yeah, it is...

>Pic related, this does not in any way imply restriction to naturals. Indeed, we can use Book 5 Def 1 to set up the case where a greater magnitude is 2.5 as great as lesser magnitude. Please explain you thinking.
You don't understand what Euclid means by "measure". Look at a version with diagrams.

>I didn't see anything about naturals there, please explain your thinking.
How about you try looking at the footnote on the cover page of book 5?

>Consider the footnote to Book 5 Def 1, also pic related. By my reading, this means that "m" is absolutely ~NOT~ restricted to naturals. In so far as Euclid was concerned with what can be measured with a ruler, one magnitude of 10cm is obviously greater than one of 9cm, and therefore the lesser is a part of the greater with
>m = ( 10/9 )
Once again, you don't understand what Euclid means by "measure".

>Please explain your thinking!
How about you actually read the book?

>Why I am the only in these threads who at least attempts to make complete and meaningful statements and everyone else, apparently, is tuned to maximum brevity, proof-by-non-existent-evidence quips?
It's not my fault if you can't read the book.

>I mean seriously. You can look right at the foot note for Def 1 and immediately see that "m" is not restricted to the naturals.
Maybe if you would read the book you'd understand.

>>10809169
As a formal construction, no, they didn't exist. Because the existence of real numbers requires a definition that seems to align with the purposes that such a set should fulfill. We want conditions on a sequence in the set, called the cauchy criterion, to yield a limit point regardless of which so called cauchy sequence we choose.
The real numbers IMO can be better described as a field with an order that satisfies positivity where every set with an upper bound has a least upper bound. It's a little more descriptive and doesn't rely on more nominal advanced theory. (not that the definition of cauchy sequence is advanced, just nomial and not self descriptive)

>>10809187
Show him the far superior residue number system.

>>10810304
>As a formal construction, no
That's like saying "As the 2020 model, the Ford Mustang didn't exist until 2019." It's like you're deliberately dancing around formal definitions that aren't formal constructions. Don't you think the reason no one ever asked "How do you construct R?," until after all the important fundamentals were established is because no one cared, and it's not important at all for the fundamentals of analysis, and then later after analysis' heyday they were grasping at straws of new questions to ask and someone said, "Hey! How do we define R as a construction?"

>We want conditions on a sequence in the set, called the cauchy criterion, to yield a limit point regardless of which so called cauchy sequence we choose.
I don't want that. Why do you want it? What's the shortcoming here: a number is a cut in a number line. Why is this definition insufficient? Sure, it doesn't answer "How do you construct a given number?," but who came up with the requirement? For instance, let M be the set of all transcendental numbers. This is a perfectly well-defined set, but do we know how to construct its elements? Maybe we do, I'm asking because I doubt it.

>The real numbers IMO can be better described as a field with an order that satisfies positivity where every set with an upper bound has a least upper bound.
Why is it better in your opinion that all of R satisfies that than that R contains some subset that satisfies this? Is the most general possible definition of R preferable when it contains subsets that do exactly what you want, for instance the R_0 neighborhood of the origin in my definition of R? What is gained by requiring those properties of all of R when the subset does EXACTLY the same thing? It think it is clear that there is nothing gained, and in fact there is very much lost when requiring all of R satisfy the field axioms. Why not just require that a subset of R (R_0) satisfies the field properties?

>>10810372
>What's the shortcoming here: a number is a cut in a number line. Why is this definition insufficient?
Not the guy you are replying to, but this definition is insufficient because it is circular. You just defined the real numbers in terms of the real number line, which I imagine is defined as... the real numbers again?

Unless, that is, you have a separate definition for the real number line in mind, but if so I haven't seen it.

>>10809936

Oh my! You are really smart and have a long and fruitful career path ahead of you.

>>10810411
>which I imagine is defined as... the real numbers again?
you imagine wrong

> you have a separate definition
Yes, I certainly do:

(1) A line is a 1D space extending infinitely far in both direction.

(2) A line is a number line iff it is equipped with a chart "x" and the Euclidean metric
d(x,y) = | y - x |

(3) A number line is the real number line if it is a number line equipped with a unique label containing the text string "real"

(4) A cut in a line separates one line into two pieces whose intersection is the empty set and whose union is the entire line.

(5) A number x is a real number iff it is a cut in the real number line.

(6) Let the symbol "inf" be the limit as x tends to zero from the right of the quotient (1/x)

(7) In interval notation, a cut "x" in the real number line takes the form
(-inf,inf) = (-inf,x) U [x,\inf)

(8) In set notation, a cut "x" in the real number line defines R- and R+ such that
R = R- U R+
R- = { y | -inf < y < x }
R+ = { y | x ≤ y < inf }

>>10810234
>TRINITY___FOOTthing
what the fuck

>>10809936
its the subdividitiveidentity, stupid

>>10810480
That's where NXIVM-related persons (meaning N'Drangheta-related persons) cut a small slice in my toe, then used a long wire to put one more more subdermal electroshocker implants in my foot. They can activate them by remote control, and they torture me with these implants regularly, and they sealed the slice in my toe with meat glue to make it look a mere blister. When I noticed the wound on my toe, I noticed three zapper locations in a line (extending out of frame) and one more zapper location off the line, but still where it could be implanted easily from the cut in my toe and some tool to thread it under the skin.

I've seen another thread with someone else having the exact same wound on their toe, and my ID in that thread said like, "Nx1umD0" or something. It wouldn't surprise me to learn that this a NXIVM torture gimmick, I personally know the head slavemaster over all of N'Dranghta, (someone named Cavalcanti) of which Kieth Raniere and NXIVM are just little pissant henchpeople spreading the misery f human slavery to the extent that they can. (Yes, the reason NXIVM is so super-duper scary and secret is because it is a front for N'Drangheta, the powerful homosexual mafia.)

This is a torture procedure they are know for, and you can find similar stories and pictures of similar wounds on the internet. Although I was feeling the electroshock sensation at four different places in my foot when I first got these implants, more recently I have only felt the one electroshock rape implant delivering the electroshock int he center of the arch of my foot. This is the fourth location out of frame in the FOOT THING pic.

>>10810480
Look at this other thread:
>>>>/x/23032154
I wasn't the one who posted the picture there. Someone else who knows about implants and recognizes this wound on my foot is the one who put it there. I have implants in my anus too, also probably put there by the fags in N'Drangheta, which is just a big group of pussy ass fags whose families are all going to suffer before they die.

>>10810482
So zero does exist then! That's what I thought.

Last time I was talking shit about N'Drangheta and/or the "Loyal" White Knights of the Ku Klux Klan, someone was like, "Oh my noes! You've used a name which is precious to me." Motherfucker listen to me, if that name is precious to you while you've been shitting on my own name, then you are in a really bad position.

Incidentally, to the extent that I have mentioned corrupt APD officer Paul Merritt in this thread, and also pointed out his weird fondness for name dropping "Italy" I am quite sure he dropping that name by way of the "Loyal" White Knights of the Ku Klux Klan, with "loyalty" indicating the subsidiary status in the N'Drangehta terror/slavery/crime syndicate.

If you're hiding my posts, your suffering will last and last, and your children will not make it into my mercy when I use them against you instead.

>>10810475
>(1) A line is a 1D space extending infinitely far in both direction.
How is this defined? Euclidean spaces are traditionally defined in terms of real numbers. Do you have a definition hereof that does not involve the real numbers? If not, it is once again circular.


>(2) A line is a number line iff it is equipped with a chart "x" and the Euclidean metric d(x,y) = | y - x |
This seems very obviously circular. Or is the right hand side of that d function something that does not involve real numbers? If so, how?

>(4) A cut in a line separates one line into two pieces whose intersection is the empty set and whose union is the entire line.
Is ({ 17, 23 }, R \ { 17, 23 }) a valid cut? What about (the even integers, (all of R except for the even integers))? The definition you use here is what is known as a partition, which sounds like it isn't quite what you want.

>>10810557
>How is this defined?
Which of those 12 words do you feel like needs further definition? The definition I gave does not use real numbers so it is not circular.

>This seems very obviously circular.
Yes, I agree. Therefore use the following:

(1) A number line is a 1D space extending infinitely far in both direction.

(2) A number line is the real number line if it is a number line equipped with a unique label containing the text string "real"

(3) A cut in a line separates one line into two pieces whose intersection is the empty set and whose union is the entire line.

(4) A number x is a real number iff it is a cut in the real number line.

(5) Let the symbol "inf" be the limit as x tends to zero from the right of the quotient (1/x)

(6) In interval notation, a cut "x" in the real number line takes the form
(-inf,inf) = (-inf,x) U [x,\inf)

(7) In set notation, a cut "x" in the real number line defines R- and R+ such that
R = R- U R+
R- = { y | -inf < y < x }
R+ = { y | x ≤ y < inf }

>Is ({ 17, 23 }, R \ { 17, 23 }) a valid cut? What about (the even integers, (all of R except for the even integers))?
Yes those are valid cuts. All cuts are partitions, I agree with you. However, those things you mention are n-tuples and a number is a 1-tuple.
>(4) A number x is a real number iff it is a cut in the real number line.

>>10810583
>Which of those 12 words do you feel like needs further definition?
The "1D space" fragment.

>The definition I gave does not use real numbers so it is not circular.
No, for it to not be circular, all of its components need to also defined recursively without involving real numbers along the way.

>Yes, I agree.
You agree that your definition of the real numbers is circular? Well, that is what's wrong with it.

>>10810592
>The "1D space" fragment.
Use pic definition of 1D topological space then.

>for it to not be circular, all of its components need to also defined recursively without involving real numbers along the way.
done:

(1) A number line is a 1D topological space extending infinitely far in both direction.

(2) A number line is the real number line if it is a number line equipped with a unique label containing the text string "real"

(3) A cut in a line separates one line into two pieces whose intersection is the empty set and whose union is the entire line.

(4) A number x is a real number iff it is 1-tuple and it is a cut in the real number line.

(5) Let the symbol "inf" be the limit as x tends to zero from the right of the quotient (1/x)

(6) In interval notation, a cut "x" in the real number line takes the form
(-inf,inf) = (-inf,x) U [x,\inf)

(7) In set notation, a cut "x" in the real number line defines R- and R+ such that
R = R- U R+
R- = { y | -inf < y < x }
R+ = { y | x ≤ y < inf }

>>10810603
>1D
What does that mean ?
>extending infinitely far
what does that mean ?
>in both directions
what are they ?

Also your definition of cut is lacking, because it then (line - {point}, {point}) is a cut, which I don’t think is what you want

>>10810676
>which I don’t think is what you want
It is what I want and that works fine if you want to use absolutely retarded "line minus point" notation.

1D means it's a space whose elements can only lie to the left or right of the other elements, and nowhere else.

The two directions are the left and right.

Extending infinitely far means the space does not include endpoints.

>what does "what" mean?

>>10809992
tooker btfo

>>10810740
that whole post is nothing but that guy explaining his opinion.

>>10809129
The nice thing about the reals is that they can be treated as 1D vectors, whose magnitude can be viewed rigorously as the length of a line.

>>10809845
wrong
infinite sum of the real integers = -1/12 actually exists and comes up regularly in quantum physics.

>>10810758
I agree that this is very nice

>>10810723
I don't think you understand what I meant. For you from what I understand, a cut is (morally) the two parts of the line that lie on either side of a number.
What I am saying is that with your definition, cutting the plane into one point and its complement is still a cut.

>1D means it's a space whose elements can only lie to the left or right of the other elements, and nowhere else.
Unless you define what a space is, there is no way to know what left and right is. Is pic related not 1D ?

>>10810839
>cutting the plane
I meant line*

>>10810723
are you trying to give an actual rigorous definition? because if you are, then you really need to elaborate.

>1D means it's a space whose elements can only lie to the left or right of the other elements, and nowhere else.
define "lie to the left" and "lie to the right"
>Extending infinitely far means the space does not include endpoints.
define endpoints

so far Q qualifies as a number line according to your naive definition. so does the circle. so does the collection of several real number lines placed next to each other.

remember that the answer "it's fucking obvious what I mean" is forbidden. that's the thing with mathematics.

>>10809129
That is simple, but if you understand real cut, it's easier to understand real number, because you see it in action.

>>10810839
>What I am saying is that with your definition, cutting the plane into one point and its complement is still a cut.
As I said last time you raised this point, that works perfectly well for my purposes. If you want to use a definition of "cut" other than the one I gave, why even read my post at all?

>Unless you define what a space is,
I did define the space: it's a topological space

>Is pic related not 1D ?
it is not.

>>10810873
>define "lie to the left" and "lie to the right"
it means there exists an ordering of the elements of the set such that every element is either grater than or less than some other element of the set

>define endpoints
points which do not contain an open neighborhood still in the space

>so far Q qualifies as a number line according to your naive definition
change to :
(1) A number line is a connected 1D topological space extending infinitely far in both direction.

>so does the circle
I think this is fine.

>>10811610
>points which do not contain an open neighborhood still in the space
points which do not *have* an open neighborhood still in the space

>>10811610
>it means there exists an ordering of the elements of the set such that every element is either grater than or less than some other element of the set
it means there exists an ordering of the elements of the set *by a single parameter* such that every element is either greater than or less than some other element of the set

>>10811616
>points which do not *have* an open neighborhood still in the space
doesn't make sense. every point in a non-empty topological space has an open neighborhood "still in the space".

>>10811703
Then it makes perfect sense.

Real numbers are all of the values along a single, infinite dimension. Complex numbers required two dimensions.

>>10809420
Not even a solution, he asked what a real number is, you gave a definition and anon got mad because you gave a definition

>>10811735
>anon got mad because
Anon got mad because he (or she) has a whole team of PhDs trying to shit on what I piece together without even the equivalent of BS in math, and they consistently fail to do so, and all the "best an brightest" of the USA Great Satan "intelligence" community can't hold a candle to my ability.

Hey you! You contractors picking apart the minutiae of posts which he or she would never find on their own because they don't even have the equivalent of a BS in math either: I'm going to crucify your families and throw you in the steamer so you can begin to rot to death from you skin coming off as you see them all dying on their crosses.

Hey the rest of you turds that are against me! You guys are all going to get a lot worse than that.

>>10811717
it's irrelevant to the definition tho

>>10811754
>without even the equivalent of BS in math
Oh, don't worry, you have plenty of BS in math.

>>10811762
It's not irrelevant. If it's part of the definition of what a topological space is, then it is HIGHLY relevant. Do you mean to say it's redundant? If so, I agree. By adding the words "connected topological," now I can get rid of the reference to endpoints by getting rid of the reference to infinite extent:

(1) A number line is a 1D connected topological space!

(2) A number line is the real number line if it is a number line equipped with a unique label containing the text string "real"

(3) A cut in a line separates one line into two pieces whose intersection is the empty set and whose union is the entire line.

(4) A number x is a real number iff it is 1-tuple and it is a cut in the real number line.

(5) Let the symbol "inf" be the limit as x tends to zero from the right of the quotient (1/x)

(6) In interval notation, a cut "x" in the real number line takes the form
(-inf,inf) = (-inf,x) U [x,\inf)

(7) In set notation, a cut "x" in the real number line defines R- and R+ such that
R = R- U R+
R- = { y | -inf < y < x }
R+ = { y | x ≤ y < inf }

>>10811754
>Anon got mad because he (or she) has a whole team of PhDs trying to shit on what I piece together
You're an irrelevant nobody though.
Theres no conspiracy against you, just your own mental illness.
Take your medication and stop reading 4chan. It just seems to feed your delusions.

>>10809129
>What is the thing in itself
https://en.wikipedia.org/wiki/Thing-in-itself

>>10811610
>A number line is a connected 1D topological space extending infinitely far in both direction.
So the long line counts?

>>10811770
>It's not irrelevant. If it's part of the definition of what a topological space is, then it is HIGHLY relevant. Do you mean to say it's redundant? If so, I agree.
according to you, any point in any non-empty topological space qualifies as an endpoint.

>A number line is a 1D connected topological space!
so far this just means a connected space with a total ordering. (you haven't yet specified that the ordering needs to be compatible with the topology but that can be fixed). still your number line is not unique.

>>10811793
>according to you, any point in any non-empty topological space qualifies as an endpoint.
the opposite, no point qualifies as an endpoint. my bad

>>10810873
>are you trying to give an actual rigorous definition? because if you are, then you really need to elaborate.
See? The thing about this is that, in any rigorous treatment of anything, there is always more elaboration "required." You can always say, "If the globe in on the back of a turtle standing on a turtle standing on a turtle, then what is that third turtle standing on?" However, this is circumvented when a mathematician resorts to the principle of waving hands to say, "Come on, dude! You see what I mean there! Stop fucking with me!" Then, although it is true that "stop fucking with me" is not good mathematics, the counter party often relents at some point because everyone knows that the counter party is, indeed, correct to always point out that more elaboration is required, and nothing would ever get done at all if not for the counterparty's willingness to eventually stop pointing out that more elaboration is required.

Often times the principle of waving hands manifests in axioms which are unproven statements assumed to be true. When the one party says, "Stop fucking with me, man!," then the other party eventually says, "Ok. Even though you didn't prove this stuff, I'll stop pointing out that it needs to be proven because I want to give you the stamp of approval but, really, if I don't want to give you the stamp of approval, then I would just say that you haven't proven your axioms and you work is garbage and meaningless."

So... the point is that the amount of elaboration "required" is nothing but function of the personal relationship between the first party and his critic. Since the USA critics all work for Satan, my work will NEVER reach the point where more elaboration isn't required. It might, hypothetically, stump some USA tards as to how to identify areas where more elaboration is required but really, a good mathematician will always be able to show areas where more elaboration is required in anything.

>>10811805
Did you ever consider the possibility that perhaps your explanations are not as clear to other people as they are to you?

Did you ever consider the possibility that perhaps there is in fact a commonly agreed-on standard on when something is and is not mathematically clear, and you just do not understand it?

For instance, consider pic related. Although, Pugh is able to prove that such a set exists, Pugh does not elaborate on his assumption that this set is the set of all real numbers. If Pugh was named Tooker and he wrote that, then everyone would say, "You need to elaborate on why you think that set is R. The empty set exists. Is the empty set R? My grandma exists. Is my grandma R?" And on and on. At some point it's about the mathematics, but eventually it always comes down to the critic saying, "Ok, I won't harp on this because this guy pretty much has something consistent there."

>>10811816
>At some point it's about the mathematics, but eventually it always comes down to the critic saying, "Ok, I won't harp on this because this guy pretty much has something consistent there."
I think you'll find that if you prove that you have something consistent in your work, people will start taking you more seriously as well.

>>10811792
>So the long line counts?
I don't know what the long line is.

>>10811793
>you haven't yet specified that the ordering needs to be compatible with the topology but that can be fixed
how would I make that statement?

> not unique.
doesn't the unique label "real" make it unique?

>Did you ever consider the possibility that perhaps your explanations are not as clear to other people as they are to you?
Yes, that's why I'm constantly seeking criticism and improving the language I use to give my explanations.

>Did you ever consider the possibility that perhaps there is in fact a commonly agreed-on standard on when something is and is not mathematically clear, and you just do not understand it?
Yes, absolutely. Since I do not even have the equivalent of an undergraduate degree in math, I certainly do not know what the commonly agreed upon standards are at the level of someone who has an undergraduate degree in math. However, when you all say "R is a complete ordered field," I am 100% certain that this just something you like to assume, and I am quite certain if someone besides me suggested the same alternative definition that I'm suggesting then you would be much more receptive to it.

Did you ever consider that no progress will ever be made beyond the currently agreed upon standards if no one ever suggests alternative standards? For instance, at some point in history, everyone agreed on the geometric Euclid standard of a real number. Then Dedekind suggested some other standard based on algebra, and then after that the standard changed. I think you are ignorant to demand that the standard should never change again, and it is my opinion that the geometric standard is better than algebraic standard because it contains everything useful about the algebraic standard in its subsets.

>>10811825
>if you prove that you have something consistent in your work
Please define "proof of consistency."
Please cite something "inconsistent" in my work without saying, "Look at the red herring and straw men I raised in 30 other threads."

>people will start taking you more seriously as well.
I think people who don't already take me seriously won't do so until I'm standing on a mountain of the corpses of their family members. I think there is no amount of "consistency" that I could demonstrate that would ever get anyone to say, "Ok, maybe we shouldn't use the field axioms to define R." Whether or not we should use these axioms is purely a matter of opinion, and by proxy a matter of ego for those who never accept my axioms. There will never be some consistency I can show beyond, "Use my axioms instead of your axioms," and the egos of the people who failed to solve RH will never let them concede that I solved if they have not done so by now.

It's an ego thing, not a consistency thing.

>>10811805
>It's an ego thing, not a consistency thing.
>So... the point is that the amount of elaboration "required" is nothing but function of the personal relationship between the first party and his critic.
Trust me that I see your point, but you're wrong and I'll tell you where. You think that it all depends only on an subjective agreement between. But actually there IS a standard of rigour which is universally agreed upon and that's set theory (I'm talking about real analysis now, don't start with categories please). I accept all three standard definitions (cauchy sequences, dedekind cuts, complete ordered field) not because they're standard, but because they're precise. And here "precise" means that if I really really wanted, you could unwind these definitions into a language which uses nothing else but set theory. It would be tedious, but possible. And then I would say "okay, I'll stop fucking with you, I don't need you to elaborate on what a SET is." Whether you like it or not, this is where the threshold hold is. You're right that there's an agreement going on, but it is more like "I could write all of this in detail if I wanted, but I won't bother" and "I'm convinced that you could do it so I will believe your statement". You're right that it's always about some kind of agreement, but it's FAR more grounded than you think.

>>10811847
Honestly Tooker, there's a part of me that wants to believe there's a method to your madness. If you actually put together the time machine you say you're going to I'll certainly be impressed.

>>10811950
Nice red herring. There is no standard of rigor which goes into choosing one set of axioms of another. The geometric axiomatic definition of R contains a subset which does everything required of the real field. Choosing to be able to "unwind the definitions" into set theory as a requirement is merely another opinion when it's equally valid to be able to "unwind" the definitions into Euclidean geometry.

Indeed, each variety of set theory has its own unproven axioms. To the best of my knowledge, if you want to argue why the axioms of a given variant are probably true, you will use Euclidean geometry. Therefore, it's just another opinion when you "need" to be able to unwind into intermediate set theory instead of simply being able to unwind directly into basal geometry.

Furthermore, you are ignoring the fact the geometric definition of R which I use contains a subset R_0 such that
R' = {R_0, +, x}

is the real field in common usage today! You are ignoring my point that there is absolutely nothing gained by requiring all of R to satisfy the field axioms in
R'' = {R, +, x}

when R' does everything that R'' would do if you chose to axiomatize R that way.

To make a futile analogy that can be easily cast aside with another futile analogy, it's like you've only loaded a C compiler on your computer, and you're saying, "No other languages besides C exist. I see what you're saying that it was just my arbitrary choice to load a C compiler, but actually my choice is far more grounded than you think because C is object-oriented, it allows you to create your own libraries, and it supports modular architecture." The fact is, it doesn't matter how great C is. At the end of the day, it was nothing but your opinion to load the C compiler, EVEN IF you chose that compiler because all of your colleagues also write C. In fact, there are other languages, and even non-object oriented languages, that are perfectly valid frameworks for writing computer programs.

>>10811990
>If you actually put together the time machine
Did you not see when the last page of my website turned into the first page of the Bible: the first page of the first Book of Moses?

Is this not obviously my face? And this is no random picture of me. This is my medal from the highest accoolade I would ever win in jiu jitsu: 2009 men's brown belt adult ultra-heavyweight 2nd place. Not only that, this was this first picture in about 20 years where I didn't cock my eyebrow.

Pic related, if you don't believe me I don't expect to convince you by making posts about how obvious it is.

For all of you who think I'm crazy to believe that the government is fucking with me: who do you think is inserting the typos which you can see miraculously appear in this video?
>https://www.liveleak.com/view?t=u7Q6o_1559880838
Pic related, if you have shoop software with which to overlay the images, you can see that the lines on this pre-2012 victory face meme are plainly the lines on my face from this still shot of my 2012 video.

Furthermore, I would not be surprised in the least to learn that the government is censoring this video by making my evidence of their crimes classified because it shows their "sources and methods." If so, I will feed all of their children to the rapists and then the monsters, and I will make them eat shit while I mock them and electrocute their anuses and genitals with cattle-prod level electroshock equipment. In fact, if I'm on the fact internet and you all are on the fake internet here with me, then all of you are going to get that treatment. Hopefully I'm not on the fake internet, but if they felt like they needed to insert typos into my posts to mess up my presence on the real internet, then it wouldn't surprise me at all to learn that they censor me totally today. If so, there is no limit to the malice I will invest in the rape and torture I put them into.

Pic related, if you don't believe me I don't expect to convince you by making posts about how obvious it is.

>>10812129
>I would not be surprised in the least to learn that the government is censoring this video by making my evidence of their crimes classified because it shows their "sources and methods."
Why wouldn't the evil conspiracy that can controll time and space just kill you instead of this operation to make you look like a schizo.
Have you considered that your paranoid delusions are a sign of mental illness?

>>10812097
what if you are putting stuff in your own anus from the future?
also how do you plan on building a time machine?

>>10812383
Given that I have video evidence of the conspiracy in action, what makes you choose to frame my extreme chronic paranoia in terms of delusions? I tell you what else, when "Susan Rice" was on TV that day shortly after the 9/11 Benghazi attacks talking about "who changed the talking points" it was my mother Helene in a disguise telling lies about how she is the leader of the conspiracy against me. She is evil, and you all on this website that hate me are her servants, and you think you are doing the right thing because she is high above you in the grand scheme of things, but you are not doing the right thing because she is my enemy and I am above her in the grand scheme of things, and I am above her ex-husband too who many of you serve, and he is not really even very high in the grand scheme of things at all, and I believe he is my enemy too.

>evil conspiracy that can control time and space
Since I suggest that I am ultimately the one who controls time and space, have you considered that your reading comprehension is very low when it is only the people who control facebook (and/or my computer) that insert typos into my posts? Them polluting my posts is in accordance with the way they spice everything I do with their own shit? I would not be surprised at all to learn that toothpaste in bathroom right now has semen in it. They always do this to me. Everything I touch, they poison it and then snicker amongst themselves when I eat their poisons, and then say it's my fault that life sucks because I don't like eating their poisons.

>>10812406
>what if you are putting stuff in your own anus from the future?
Then I would be very surprised. However, if I am the one making my life a living hell, then what is Satan out there doing? What is his role? You think I'm here and Satan is off in Burma somewhere telling the jungle people to eat their children and just letting me exist in peace without fucking with me? Do you really think that I am more likely the source of my misery than that my enemy is the source?

What if you all are working for Satan? I am here with you now giving commands that you disobey. Why do you obey those other commands when you only assume you know who they come from?


>also how do you plan on building a time machine?
I would work with some electrical engineers to develop the dynamical mechanism I discovered between electromagnetism and the geometry of spacetime
>Geometric Cosmology
>http://www.vixra.org/abs/1301.0032
with specific attention to the construction of closed timelike curves via electrical antennae.

It says God sent Jesus to suffer so he could save the world. That is God allowing himself as a younger man to suffer because it is so important to me that I set things on the right track. My suffering now is not evidence of my impotence, it's evidence of how much I care about what I'm trying to do. Pic related, try to imagine one more frame to the right there.

>>10812609
>one more frame to the right there.
>mfw

http://www.abovetopsecret.com/forum/thread966329/pg1
http://www.abovetopsecret.com/forum/thread1240030/pg1
https://www.reddit.com/r/IAmA/comments/ukbz6/i_am_the_anonymous_physicist_featured_in_the/

>>10812595
>>Geometric Cosmology
>>http://www.vixra.org/abs/1301.0032
oh i see, ty

i am not an engineer, however i am interested in this time traveling thing you are talking about. how does one become an apprentice of the tooker?

>>10813659
>how does one become an apprentice of the tooker?
Send me an email with a proposal.
Also, buy my book and tell me which was your favorite part.

>>10813672
nice book

This is my favorite book now

>>10812092
>There is no standard of rigor which goes into choosing one set of axioms of another.
there is. it's a matter of agreement, I'm not disputing that, but that doesn't make it any less true. Euclid's postulates are not "axiomatic math" as it is done today, we treat it as if they were okay only because Hilbert and Tarski formulated them in first order logic. if you cannot formulate your theory in some universally accepted foundation of mathematics (ZFC, first or second order logic, category theory, homotopy type theory..) then your theory is useless and every mathematician will tell you that. never ever in mathematics will you find an "axiom" as naive as "XXX is a 1D space stretching into infinity in both direction" without these words being explicitly defined in the language of topological spaces. because that's how math was done centuries ago. it led to ambiguities and troubles and that's why the foundations came into existence in the first place.

also, I think that the actual "geometric" definition would be "real line is a connected second countable hausdorff densely ordered set without maximal and minimal element", but you can see very easily that this definition is equivalent to the "ordered field with LUB property" definition.

I partly agree with your analogy. the way I see it it's more like my real numbers are written in C, someone else's real numbers are written in Python, but yours are not even in a binary code.

Why is this thread not bumping?

>>10813720
>then your theory is useless and every mathematician will tell you that.
They would be all be wrong if they did tell me that. Do you know what "useless" means? Are you able to distinguish between "cannot be used for everything" and cannot be used at all?

Here's how I can explain that you are completely infatuated with the small of your own farts. It's another analogy, sorry, but it's a good one. Assume someone had solved the Riemann hypothesis in 1865. The math they would have used to do it would be precisely what you have described as "useless" and yet it would have the "use" of solving the RH.

I can also show the same thing without an analogy. Riemann didn't use any of that stuff you mention, and he was able to "use" math to define what is still called "the most important unsolved problem in mathematics." Do you see how my rebuttal to you claim of uselessness completely negates it?

What about the math Riemann used to create Riemannian geometry? Obviously that math was not "useless" because Riemann "used" it to define something we still use today in exactly the way he formulated it.

What about the math Euler used to build the Euler formula? If it was "useless," then how did he "use" it to build a formula that we still use today in the exact same way he formulated it?

You are a narcissist. Furthermore, in physics we "use" lots of math that is not substantiated with any kind of rigor, and it correctly predicts the results of experiments! How is this "useless?" Riddle me that please, you whose opinion of yourself is a bit too high.

>>10813720
>but yours are not even in a binary code.
The way I can tell you are full of shit is by the first lecture I had on my first day of real analysis when I was an undergrad. The prof came in and passed out the syllabus, and then he started teaching. He started asking, "What's a real number?" After several students' guesses, he proudly proclaimed, "A number is a cut in the real number line and that's it! Nothing more! It's that and only that!"

This was a full professor with a PhD, grants, and tenure at a major USA research university. It's too hard for me to believe that I had some crank prof telling me his own pet theory about what numbers are. Maybe he was a nutjob lunatic, but my opinion is that he did know what he was talking about, and he did not deem to teach us real analysis from his controversial crank opinion with which every other mathematician in the world would emphatically disagree as you all do disagree with me now when I repeat it.

Now you all just want to set up any standard which will allow you to say I'm wrong, when if you had never heard of me and that prof would have given you the same speech then you would have been like, "Well, yeah. Pretty much."

Remember last week when you guys were like, "You only found new trivial zeros," and then I showed that they were not trivial, and now you all reverted to claiming I didn't find any zeros because I have proven they are non-trivial?

>>10813803
>Remember last week

Furthermore, if I find out you new nay-sayers on here who are purpose-built to only say I'm wrong, the ones who backtrack when I prove you wrong, if I find out you are the profs at this school I'm about to go to, and I find out that you're trolling me by pretending to work with me when you have no intention to do anything besides shit on me, the I will torture you and all your loved ones in the most malicious fashion.

I'm not here to write a PhD thesis. I'm here to write a shitty undergrad research article like gets published on arXiv 50 times a day with shit far stupider than anything I ever thought of.

>On the influence of the Illuminati in astronomical adaptive optics
>https://arxiv.org/abs/1203.6708

>>10813827
So think carefully. If I ask you to work with me and you say no, then I will not that against you. However, if you are my enemy's agent and you say yes when I ask you to work with me, then I will hold it against you and I will make you suffer. Be warned.

So you know... if you think I'm stupid and you want to criticize me, that's one thing. If I find out you work for my enemy, or if you are in league with my enemy, or anything like that, then it will better for you to tell me that you don't want to work for me. Honest criticism, however, will earn you blessings and maybe a co-author credit or two if you are into that sort of thing.

Are you independent? This is an important question for you to be honest with youself about.