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00:24
@user2646 knock knock
00:35
who's there? @Nick
oh yeah, it's Friday; you must be "here" for the meeting
Adjective: pedantic (comparative more pedantic, superlative most pedantic)
  1. Like a pedant, overly concerned with formal rules and trivial points of learning.
  2. Being showy of one’s knowledge, often in a boring manner.
  3. Being finicky or fastidious, especially with language.
Noun: pedant (plural pedants)
  1. (archaic) A teacher or schoolmaster.
  2. , vol. 1 ch. 24:
  3. I have in my youth oftentimes beene vexed to see a Pedant [tr. pedante] brought in, in most of Italian comedies, for a vice or sport-maker, and the nicke-name of Magister to be of no better signification amongst us.
  4. A person who emphasizes his/her knowledge through the use of vocabulary.
  5. (slang) A person who is overly concerned with formal rules and trivial points of learning.
Adjective: pedant (not comparable)
  1. Pedantic.
00:51
@user2646 To
@Nick To who?
Adverb: here (not comparable)
  1. (location) In, on, or at this place.
  2. 1849, Alfred Tennyson, In Memoriam A. H. H., VII,
  3. Dark house, by which once more I stand / Here in the long unlovely street,
  4. 2008, Omar Khadr, Affidavit of Omar Ahmed Khadr,
  5. The Canadian visitor stated, “I’m not here to help you. I’m not here to do anything for you. I’m just here to get information.”
(5 more not shown…)
@user2646 No. To Whom.
:D
> 3. Being finicky or fastidious, especially with language.
considering that^ is exactly how math textbooks are written
01:13
@Faust: FYI, I can be pretty pedantic as a math teacher. Perhaps I'm crazy as a person.
hmm well you've never annoyed me.
That's your fault, not mine. :)
being pedantic about some things especially definitions is kind of important
Careful use of language is key in mathematics.
i find i dont understand definitions very well
01:17
You're not alone.
Some handy examples are a good way to start to deal with that.
it especially difficult for me cause i can't understand simple instructions lol
I understand ... all the more reason to make it your task to make up examples/non-examples if you can.
whats a k linear function
multilinear with $k$ variables
whats multilinear mean
like
01:20
linear in each variable separately
hi @loch
ah i get it thx
Hey @Ted
@TedShifrin which lectures do you describe whats a manifold is?
whats a co-vector
I didn't do abstract manifolds. I did only submanifolds of $\Bbb R^n$. Start in 3510 Lecture 22 and go to 23.
an element of the dual space
are there many defintions for an algebra or only one?
01:25
for an algebra?
yeah algebras pop up in diffrent places are they all the same psudo topilogical space
An algebra has nothing to do with topology.
but there defined similiarly
Um, no.
hmm then i have no idea what an algebra is
01:27
A $k$-algebra is just a ring with a vector space structure over $k$.
mm thats not the algebra im think og
of
Oh, you're talking about $\sigma$-algebra in measure theory?
It's very similar, actually.
yea thats the only one i know
So what's the definition?
mm i have to look it up but it seemed similar to a topology but diffrent in that closed sets were easier to find
01:30
I guess the point of an algebra is that there are three operations (sum, product, and scalar multiplication). It's analogous to that.
ic
then why isnt it a module
A module doesn't have three operations — just two. It's an $R$-vector space, where $R$ is a ring.
ah
I swear i actually know something about mathematics
I get that this is sorta confuzling.
No, you're asking good questions.
mathematicians need to learn how to name things more diversely like in graph theory lol
ah shit a covector is just a linear functional
01:35
that's what a dual space consists of, yes, @Faust.
@Faust me @ the ETS
i do know that never heard the name covector before and its not defined in this silly book
@Daminark what?
@Faust ... co- always means dual.
hi Demonark
Lol I've got this test tomorrow and I'm basically just taking every opportunity to be angsty
How's it going?
oh yeah, tomorrow's Saturday.
Where are you taking it in TX?
01:38
UT Arlington, not too far away luckily
ah, OK
well, I already wished you good fortune, but one more time ...
I hope you kick but(t).
Thank you very much! :)
You can report any interesting questions.
Technically I think I shouldn't but hopefully if I change the numbers up a bit it'll be aight
well, interesting ones shouldn't involve numbers too seriously. Vague ideas are acceptable.
But, good point.
01:54
Well, I'll finish up some last minute probability and then head to sleep because these tests are at 8:30 :/
See you!
gniught
@TedShifrin i have come to the conclusion that topology wasnt a prereq for this class is stupid
I told you you needed basic point-set.
yeah b ut its not a pre req for our class
it should be imo
Yup, that and multivariable analysis.
mm yeah dont have that
01:57
I told you that too
havent seen anything nasty
i dont like the prof that teach multivariable analysis so i refuse to take it
If I may say so, that's stoooopid.
You need to learn the stuff if you're gonna study what you're doing.
guy makes my life too difficult
no further comment ...
he doesnt belive in disability related accomedations
01:59
In the US (before Trump) that's illegal.
You could get the university's ass on him.
ive learned some of the material on my own
yeah my advisor told me thats what i should of done
i just dropped the class
Yup. Your adviser should have tried to facilitate.
he didnt find out till i dropped the class
Ah, your fault.
But your adviser is a good guy.
its not the first time that prof has done something like this apperntly
02:01
You should at least talk to the dept head.
Just to register the complaint.
and technically i took a phisic multivariable analysis class
No such thing as analysis with physicists.
well our university had one
math class with analysis for physics
The course that's called vector analysis has nothing to do with rigor or analysis.
this class
so im not a complete handicap at it but i defiantly blunder around a bit
02:03
oh, advanced calculus ... usually not quite so rigorous
i mean we used epsilon delta proofs
It's a mish-mash of single-variable and a bit of multivariable, but usually the multivariable isn't done very rigorously.
But anyhow....
my uni lets me take any advanced class i want as i have that and an intro analysis class with an A+
but yeah sometimes i ge tlost in odd places
well, then it's on you to make the best of it ...
0
Q: prove the following lemma about a compact space and a ultrametric.

FaustLemma 2.9 if $(X,d)$ is an ultrametric space, then any two open balls of radius $r>0$ are eithier equal or disjiont. In particular, if X is compact, then every open ball of radius r is clopen and the collection of all open balls of radius r forms a partition. let $B_r(x) $ and $B'_r(y)$ be open ...

this question for example
02:05
Ultrametrics have no business in the manifolds course.
the problems is about being open/closed its an analysis problem
Oh you're talking about a different course.
Blah.
I'm outta here for now, regardless.
fine fine sankyuu
for ur helps
LOL ... bye, silly.
Pig
Pig
hi everyone
02:07
morning
Pig
Pig
oh :/ bye Ted I guess lol
hi/bye Piggy.
Pig
Pig
and hi Faust
someone wanna explain nets to me?
Hi pig ^^
Alessandro's good for that, @Faust.
02:07
i already asked him a stupid question today though
It's about payback, @Faust. Plenty of us have helped him, so he helps others.
Pig
Pig
What do you want to know about nets by the way
like i know they are supposed to replace a convergent sequence in the definition of continuity
but i have no idea how to use them
Pig
Pig
what do you want to use them for? In proving theorems or..?
i never used them before so i don't really know. When proving theorems lots of times you can just generalize proof with sequences directly using nets
why can't a complex function whos modulus is k/cosh(x) be holomorphic? im not getting a contradiction
02:20
@Pig uh im studying a class where you someitmes need them to show a function is continuos
Pig
Pig
any examples?
2
Q: Homomorphism between compact Hausdorff spaces.

FaustSuppose that X and Y are compact Hausdorff spaces and $p:C(X)\to C(Y)$ is a unital * homomorphism. Prove that there exists a continuous function $h: Y \to X $ such that $p(f)=f\circ h $ for all f in $C(X) $ I have managed to prove the other 3 parts ( show the statement false if $p$ is not unital...

its a rather complicated example mind you
Pig
Pig
oo cool
02:42
one more question away from the Socratic badge
03:36
[Infinity]
Construct numbers as follows
1. The universe has nothing $\varnothing$
Introduce counting $succ(x)$
2. Counting objects producing $1,2,3,4,5,6,\cdots$
Introducing sums $+$
(Everything is the sum of another thing)
Introducing differences $-$
Numbers that can be expressed as $a-b$ vs Numbers that cannot be expressed as $a-b$
Take latter -> Negative numbers and zero $0,-1,-2,-3,-4,\cdots$
Introducing multiplication $\times$
(Everything is a multiple of another thing)
Introducing division $\div$
Numbers that can be expressed as $\frac{a}{b}$ vs numbers that cannot
Take latter -> Irrationals (?,?,?,?,?,...)
Zooming further. Define powers $x^n$, get nothing new
Define roots $x^{\frac{1}{n}}$
${}^a\sqrt{b}$ vs not ${}^a\sqrt{b}$
Take former -> surds
Note everything constructed so far can be expressed in equations of the form $a^n\pm x^m=0$
03:56
YOU NEED 1
or any element of any kind
that you can label 1
alternativly if you want to start with nothing the construction can be made using power sets
But how can I got from the universe having no numbers to introducing an element or an empty set?
(at least historically greek speaking)
well you can eithier pretend u have something or have nothing and use power sets
set of nothing
the set of the set of nothing and nothing has an element
zfc has the existence of the empty set as an axiom
honestly algebraic formulation of 4$\Bb R $ is better
Ok context: I am trying to find out how many kinds of irrational numbers can be produced by following the logic of historical mathematicians as outlined here:
In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. Among irrational numbers are the ratio π of a circle's circumference...
04:01
not all of them
so during those times, ZF have not been formalised yet
onyl the algebraic ones
not even $\pi$?
no pi
but archimedes found pi
so the greeks knew something of non
algebraic numbers
its actually really wierd that the rationals are dense in the reals when theres so few of them
in comparison there are effectively no rational numbers incomparision to the irrationals
Well it seemed the way they are structured is that it is "self similar" in some weird way like many common fractals, and in particular in the neighbourhood of any rationals, there are uncountably many irrationals around them and the in neihghourbood of every irrationals, there are countably many rationals around them, thus result in them to be dense in the reals
The method of exhaustion (methodus exhaustionibus, or méthode des anciens) is a method of finding the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape. If the sequence is correctly constructed, the difference in area between the n-th polygon and the containing shape will become arbitrarily small as n becomes large. As this difference becomes arbitrarily small, the possible values for the area of the shape are systematically "exhausted" by the lower bound areas successively established by the sequence members. The method of...
Surely the method of exhaustion would have lead to the greeks to stumble upon $\pi$?
04:09
I read in an amazon review that " I don't think it's a good idea to develop group theory in terms of `abstract binary operations; one should develop it in terms of concrete symmetry groups.". Can someone provide some reading or insight what it means to develop group theory in terms of concrete symmetry groups?
In mathematics, transformation geometry (or transformational geometry) is the name of a mathematical and pedagogic take on the study of geometry by focusing on groups of geometric transformations, and properties that are invariant under them. It is opposed to the classical synthetic geometry approach of Euclidean geometry, that focuses on proving theorems. For example, within transformation geometry, the properties of an isosceles triangle are deduced from the fact that it is mapped to itself by a reflection about a certain line. This contrasts with the classical proofs by the criteria for...
rotations of a square, reflections etc.
you know what that means?
that means we should teach group actions first
as a set of functions on some set satisfying some property
and then "remove the set"
That is true, and it will actually makes a lot of groups intuitive since it can be described in terms of the orbits
In research design, especially in psychology, social sciences, life sciences, and physics, operationalization is a process of defining the measurement of a phenomenon that is not directly measurable, though its existence is indicated by other phenomena. Operationalization is thus the process of defining a fuzzy concept so as to make it clearly distinguishable, measurable, and understandable in terms of empirical observations. In a wider sense, it refers to the process of specifying the extension of a concept—describing what is and is not an instance of that concept. For example, in medicine, the...
That's how I usually handle abstract concepts
40
Q: Are there mathematical objects that have been proved to exist but cannot be described in words?

SecretThis might be a very stupid, and possibly philosophical question, but attempt to apply mathematics to everything plus inspired by this question caused me to ask this question Is there any mathematical object that has been proved to exist but cannot be described in words? If the answer is...

and the upper limit of that is the indefinable reals
1
Q: What exactly is a non-analytic method of thinking? Any examples of widely accepted ways of nonstructural thinking methods?

SecretA.S. analytic and systemic definitions I mainly used from here, but there might be more general or specific meanings I am not aware of that is not covered in that link (Definition: A "chunk" is a structural concept. It is in some way like a logical atom but also generalise to system entities whe...

In conclusion, my whole life is a limiting operation from the structural thinking world to the nonstructural thinking worlds as I tried to understand things that simply cannot be formalised
and thus, the concept of the Explosive Generalistion Operator is something that allows us to approach Clthulu in a hopefully finite number of steps
If knowledge is a closed set in some topology, then my ultimate goal in life is to characterise its boundary, beyond which in principle no human beings can comprehend due to our limited brain wiring
Thus the final frontier is not space, but the unknown unknown and the incomprehensible
intresting thought
its too bad your not simple then you'd have an easy task ahead of you
04:32
20
Q: How irrational quantities physically exist in nature?

Aman SinhaWe know that an irrational no has well defined decimal values upto infinite decimal places. These irrational quantities exist in nature in some kind of measurements. For an example, circumference of a circle is '2πr' , so if radius is rational then circumference will be irrational ,and this case ...

hmm...
so we are seeing idealised approximations in otherwise real and messy things
so the concept of infinity might be really just a bridge to idealise something
But then... why is it possible to complete infinite number of steps to generate this idealisation from messy information. Is continuity more closely related to infinity than it is always taught...?
how were you taught?
I wonder how our brain make sense of concepts like a circumference, as if our world is made of discrete chunks, then no circles can exists, thus the brain could not have a conception on what an ideal circle look like and visualise it
@user1732 Well the natural numbers are discrete in the reals, yet they are countable
so continuity and infinity does not imply each other
what about fractions...
infinity??
well if our world is made of discrete chunks, then comparing proportions is an observable phenomenon, thus can be empirically demonstrated (e.g. you have 3 blue balls and 2 red balls, the proportion of blue balls to all balls is exactly half)
04:42
3/5
@LeakyNun Thank you very much. I found this article regarding that.
ah right sorry, but basically the concept 3/5 can be demonstrated by if you double the amount of blue balls and x5 the amount of red balls, then the number of blue and red balls will be equal
but how on earth one demonstrates $\sqrt{2}$ if our world is made of discrete things, thus meaning that everything can be expressed in terms of this smallest unit
there is no "smallest" fraction
Is there any $\mathbb{Q}_p$-model of our world?
@user1732 thats not entirely true
04:49
@ChoMedit define "our world"
there does exist a well ordering on $\mathbb R $
not in the {real numbers}
Oh..
@user1732 There will be if our world is discrete/digital, which is given by 1/(all indivisible chunks of things combined, which is a finite but huge number)
meaning there is some way to order the reals so it has a least element
but yeah your right using the order <
04:50
rationals are not well ordered under the usual ordering
nope
but there does exist an ordering on them so that they are well ordered
isn't usual ordering what you said, <..?
yeah
i just find it wierd that its possible for an ordering to exist that well orders the reals
blew my mind when i read the proof
you need the axiom of choice to do that
yeah
which is why AC was not really well accepted for a long time
04:53
one must accept axioms as axioms
which caused me to wonder whether all this weirdness is really a sign that that set is not supposed to be well orderable
and axiom of choice is basically allows you to produce a well ordering no matter what
but it gives such an odd result the idea that the reals has a least element is confusing to say the least
(nb, swap the order of the above two lines, because stupid computer lag reversed the order)
@Secret or perhaps we are not able to comprehend the correct ordering
maybe it makes perfect sense
yeah, axiom of choice, at least in my opinion, is in a way really a way to idealise something that is otherwise not very easy to comprehend
one sort of can get that impression when looking at the ZF proof counterparts and see how more complicated it usually is
for example, in the proof of the existence of $\omega_1$
To do that in ZFC, you only need the set axioms and apply axiom of choice
but in ZF, you need to build something called the hartlogs number and show that there is an initial ordinal for every such number
Actually for every ZFC object that has a ZF counterpart, is the ZFC proof always shorter when axiom of choice is invoked?
05:03
im not sure
you do have the axiom of countable choice i think still in ZF so alot will be the same
shorter means strictly shorter?
Yeah
Also ZF does not have any variations of choice
so you cannot even well order a countable set in general
my understand is that ZF is only for wierd set theorist and no else cares
what is a countable set
Actually wait... does the fact that countable sets bijects to $\omega$ means they can always be well ordered since $\omega$ is well ordered?
05:13
its a set that you can assign a number from $\mathbb N $ to each element of the set
@mercio
and you dont run out of numbers
so for example the set $\{ a,b,c\} $ is countable we can assin a 1 to a and a 2 to b and a 3 to c
and $\mathbb N $ is the largest countable thing
@Faust well, rarely do we need things like amorphous sets, infinite dedekind finite sets, vector spaces with no Hamel basis, countable unions is uncountable, partitioning reals into more parts than points and other weirdness
so yeah I agree
I found this wierd thing in Wiki..en.wikipedia.org/wiki/Dedekind-infinite_set
O yeah these guys are fun: They are like onions and you can peel them off forever and they shrink forever
They are also have enough structure in it to set up an algebra on them
just like the fractions
well i should die now
so tired
05:18
cya
That's funny
gnight
Amorphous sets are more weird though. You can never slice them so that one half is countable
Wow.. I can't stand anymore in ZF. I feel something evils in there...
05:25
take a seat
welcome
ZFCeat.. thanks.
There are also fun things in ZFC too:
Nonmeasurable sets
I wish they have more substantial applications than just multiplying balls though
05:37
5
Q: Explicit construction of a nonmeasurable set, where only the proof of correctness uses choice?

William HozaBy Solovay's theorem, assuming the existence of an inaccessible cardinal, the axiom of choice is necessary to prove the existence of nonmeasurable sets. In the past, I've thought that one consequence of this theorem is that if I construct a set without using choice (or even merely using dependent...

16
Q: Is every vector space basis for $\mathbb{R}$ over the field $\mathbb{Q}$ a nonmeasurable set?

Ben PasserThe existence of subsets of the real line which are not Lebesgue measurable can be argued using the Axiom of Choice. For example, define an equivalence relation on $[0, 1]$ by $a \thicksim b$ if and only if $a - b \in \mathbb{Q}$ and let $S \subset [0, 1]$ contain exactly one representative fro...

@Semiclassical literally experiencing almost the same thing you mentioned about sleep the night before the GRE
Went to sleep at 9:45 (took a bit to actually get to sleep)
Woke up at 11:10
Went back to bed
And then again at 12:38
Why????
Nervous?
what have you done in the past to help with becoming over-nervous? @Daminark
05:59
a little late @LeakyNun for self-diagnosis
like I said, use whatever worked in the past for becoming over-nervous
\o @JoelReyesNoche welcome
Might make a MSE on "localised applications" later depending on whether I can find something concrete about nomeasurable sets
I've been thinking about your y-intercept questions @JoelReyesNoche
concrete = measurable @Secret
no I mean, concrete application
that's what I mean also
(draft)
06:11
2
A: What is the one thing I am going to forget on the math subject GRE tomorrow?

dfeuerYou forgot to register for the exam. You will forget to bring adequate photo identification. You will get the date and/or time wrong and miss the exam. You will enter your name or SSN wrong on the form and will get no credit. You will forget to sleep the night before, and to eat the morning o...

perhaps, a similar search will produce calming results for you...
Let $M$ be a mathematical object. An application of $M$ is an object $B$ (theorem, lemma, derivations etc.) constructed from applications of $M$ and set axioms (and objects constructed from set axioms). Let $\neg M$ be the nonexistence of $M$. Then the application of $M$ is localised if $\neg B$ is true when $\neg M$ is true
As an example, counterexamples in topology constructed using $\omega_1$ are likely to be localised because if $\omega_1$ ceased to exists, then so are all such counterexamples
To formulate this question another way, let $M$ be a mathematical object. An application $A$ is the outcome of $M$ under some deductive system $L$
$A$ is said to be localised in $M$ if for any rewriting in $L$ that terminates at $A$, it contains $M$
So with the example above, counterexamples in topology such as the long line is suspected to be localised in $\omega_1$ because there is no way to construct or prove its existence without using $\omega_1$ somewhere in all possible proofs
Thus this means that the long line will cease to exist if we remove $\omega_1$ from the universe
hmm... maybe localised is a bad word, perhaps dependence is the more precise word
06:40
In mathematical logic and theoretical computer science, an abstract rewriting system (also (abstract) reduction system or abstract rewrite system; abbreviation ARS) is a formalism that captures the quintessential notion and properties of rewriting systems. In its simplest form, an ARS is simply a set (of "objects") together with a binary relation, traditionally denoted with → {\displaystyle \rightarrow } ; this definition can be further refined if we index (label) subsets of the binary relation. Despite its simplicity, an ARS is sufficient to descri...
In mathematical logic and theoretical computer science, an abstract rewriting system (also (abstract) reduction system or abstract rewrite system; abbreviation ARS) is a formalism that captures the quintessential notion and properties of rewriting systems. In its simplest form, an ARS is simply a set (of "objects") together with a binary relation, traditionally denoted with → {\displaystyle \rightarrow } ; this definition can be further refined if we index (label) subsets of the binary relation. Despite its simplicity, an ARS is sufficient to descri...
Let $R$ be an abstract rewriting system with the class $\mathcal{M}$ of all mathematical objects $M$ defined in ZFC
why two wiki posts?
ah great, ctrl-c failed me
The other one should be:
A formal system is the name of a logic system usually defined in the mathematical way. Logical calculus is carried out in the system. It can represent a well-defined system of abstract thought. Spinoza's Ethics imitates the form of Euclid's Elements. Spinoza employed Euclidean elements such as "axioms" or "primitive truths", rules of inferences, etc., so that a calculus can be built using these. Some theorists use the term formalism as a rough synonym for formal system, but the term is also used to refer to a particular style of notation, for example, Paul Dirac's bra–ket notation. ==...
Let $R$ be an abstract rewriting system with the class $\mathcal{M}$ of all mathematical objects $M$ defined in ZFC
enter: Principia Mathematica
then Gödel
Then the application $A$ of some $M$ is the class $\{\to, \bigcup \in R| M \bigcup \to A\}$
$A$ is dependent on some mathematical object $B$ if for all applications $A$, $B$ is an ancestor
An application $C$ is a real life application if for all applications $C$, there exists at least one which the ancestor is present in our physical universe
Therefore the holy grail of user21820's question can be formulated by:
> What is the minimal such $\mathcal{M}^0$ such that given an abstract rewriting system $R$, for all mathematical objects $M \in \mathcal{M}$ they are all real life applications
And the current conjecture is:
nonmeasurable sets $\not\in \mathcal{M}^0$
06:57
:|
This will be true iff for all applications of nonmeasurable sets, they all dependent on the existence of nonmeasurable sets
and that is why I am still googling whether there is a counterexample to the above, cause I think non measurable sets can do much better than just multiplying balls with Banach tarski and constructing counterexamples
Finitism is true in our physical universe iff the following is true
actual infinity $\not \in \mathcal{M}^0$
which is a very fancy way of saying: There exists no physical manifestation of any actual infinities
Noun: infinity (countable and uncountable, plural infinities)
  1. (uncountable) Endlessness, unlimitedness, absence of a beginning, end or limits to size.
  2. (countable, mathematics) A number that has an infinite numerical value that cannot be counted.
  3. (countable, topology, mathematical analysis) An idealised point which is said to be approached by sequences of values whose magnitudes increase without bound.
  4. (uncountable) A number which is very large compared to some characteristic number. For example, in optics, an object which is much further away than the focal length of a lens is said to be "at infinity", as the distance of the image from the lens varies very little as the distance increases further.
  5. (uncountable) The symbol ∞.
Adjective: actual (not comparable)
  1. (chiefly theology) Relating to a person's acts or deeds; active, practical. [from 14th c.]
  2. c. 1606, William Shakespeare, Macbeth, First Folio 1623, V.1:
  3. In this slumbry agitation, besides her walking, and other actuall performances, what (at any time) haue you heard her say?
  4. 1946, The American Ecclesiastical Review, vol. 114:
  5. Apparently, the holy Doctor was referring to actual, rather than original, sin; yet the basis of his argument for Mary's holiness, the divine maternity, would logically lead to the conclusion that she was free from original sin also.
(3 more not shown…)
also typo: "for all mathematical objects $M \in \mathcal{M}^0$"
Now, let $\neg A$ be the nonexistence of $A$. The application of $\neg A$ is defined as above
Now define the purity of $A$ to be the class of all applications of $\neg A$
Thus $A$ is pure iff $\neg A = \varnothing$
The class of all pure mathematics $\mathcal{M}^{\text{pure}}$ is such that:
$$\mathcal{M}^{pure} \overset{\text{def}}{=} \{M \in \mathcal{M}^{pure} : \neg M = \varnothing\}$$
And our conjecture is that:
Nonmeasurable sets and its applications $\in \mathcal{M}^{\text{pure}}$
The mathematician joke of the infestation of applied mathematics can thus be formalised as the following conjecture:
$$\lim_{t \to \infty} \mathcal{M}^{\text{pure}} (t) = \varnothing$$
2
07:14
:D
where the t formally belongs to physics
yes $t \in \mathcal{M}^0$
but, time is relative
well if the above conjecture is true, there always exists at least one frame of reference where mathematics became more applied in the future than in the past
outside of blackholes, yes
07:34
> A set B ⊆ R is Bernstein, if it intersects all perfect sets in R, but does not contain any of them. A Bernstein set is a classical example of a nonmeasurable set; a Bernstein set cannot be Lebegsue measurable and cannot have the Baire property. Moreover, if A is an algebra of subsets of R, having the property that every set which is in A but not hereditary in A, contains a perfect set, then no Bernstein set can be a member of A. Most natural algebras of subsets of R have this property (see [5]), therefore Bernstein sets are in a sense universal examples of nonmeasurable sets.
@Semiclassical And how can we know it? Given that we have data about x' and y' ?
One way to check whether some mathematical object is in $\mathcal{M}^{\text{pure}}$ is to omit all instance of it in a paper and see if the paper still makes sense
if that is true, then it means its existence is inconsequential except for all applications which depends on it and nothing else, meaning that anything useful about it cannot be applied to the rest of mathematics
07:55
@user1732 Thanks for the greeting. Sorry I left before I saw it. I assume you are talking about my post in Mathematics Educators. What are your thoughts about it?
10
Q: Applications of Banach-Tarski Paradox to Probability Theory?

Matt CalhounI was just curious, since the B-T paradox is a measure theoretic result, if there are any consequences of this paradox in probability theory? Also, is there is a way of stating the B-T paradox in the language of probability theory? I am ultimately interested in finding an application of the B-T ...

ok so they do have an application outside of nonmeasurable set theory
@JoelReyesNoche what are you currently teaching?
1
A: Does the Axiom of Choice (or any other "optional" set theory axiom) have real-world consequences?

Kristal CantwellThe following paper "On Non-measurable sets and Invariant Tori' uses the axiom choice to solve a problem in classical mechanics and discusses the application of the axiom of choice to physics.


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