Mathematics

2:47 AM

any one has a symbolic math software

1

Q: How can we solve the expression explicitly for $X$ in terms of $Y$?

I am thinking how to write $X$ explicitly in terms of $Y,A,B,C$? I have $AX^3 + X^2(B-1) + X(-C) + \alpha = Y$ I thought of using symbolic maths but could not find any. Any help is nice!

I am trying this but have no software, any help?

user21820

3:35 AM

Large Countable Ordinals & the South Dakota Tractor Museum.

N. Maneesh

3:47 AM

Can there exist a continuous function $f:\mathbb R \to \mathbb R^2$: $f(\mathbb R)=\{(x,y) \in \mathbb R^2: x^2+y^2\}$?

Can there exist a continuous function $f:\mathbb R \to \mathbb R^2$: $f(\mathbb R)=\{(x,y) \in \mathbb R^2: x^2+y^2=1\}$?

Since, $R$ is not compact. But unit circle is compact.

Martin Sleziak

4:04 AM

@N.Maneesh Why no simply $f(t)=(\cos t,\sin t)$?

BTW Math Meta Chat is not really a room for this type of question.

N. Maneesh

@MartinSleziak sorry!

Thank you.

Ajay Mishra

4:27 AM

If anyone there, I think the answers are pretty obvious and they would be B and C but..

Well, Is anyone there?

Secret

5:08 AM

@user21820 That brings out the important point on how we really have no way to distinguish between 1) a potentially infinite process and 2) a process that is finite but looks never-ending because it is much larger than our personal tolerance before we start labelling something as going forever

> Anyway, we started getting really curious about this South Dakota Tractor Museum — it sounded sort of funny. It took 250 kilometers of driving before we passed it. We wouldn’t normally care about a tractor museum, but there was really nothing else to think about while we were driving. The only thing to see were fields of grain, and these signs, which kept building up the suspense, saying things like

ONLY 100 MILES TO THE SOUTH DAKOTA TRACTOR MUSEUM!

user21820

I don't know what you're talking about...

Secret

@user21820 I mean, I am just guessing, there must be a reason why when he is talking about ordinals, he keep going back to the South Dakota Tractor Museum and keep commenting about keep on seeing those road signs

To an ordinary person who is driving in a long road, it feels no different between seeing 100 signs vs seeing 10000000000 road signs along the way, it still will feel like the road is going forever

That said, I actually don't know what motivate you to talk about that link back above all of a sudden

so my guess may be way off

user21820

5:25 AM

@Secret It's just a joke. If you read through the series you will know why.

Secret

Ah...

user21820

That link was just part 1.

It's not often that a mathematically correct article about ordinals is actually amusing.

Secret

true

Secret

8:39 AM

Hmm.. it does sounds like the joke is strongly correlated to which stage of the ordinal is in, for example "run out of gas" coincides with "ran out of ordinal notation when getting to $\epsilon_0$"

Also chapter 3 is not published back in 2017 when I first read this blog

Secret

9:47 AM

---
(Unrelated)
https://math.stanford.edu/~feferman/papers/predicativity.pdf

Apr 23 at 4:31, by Rithaniel

Here's a question for you: We know that no set of axioms will ever decide all statements, from Gödel's Incompleteness Theorems. However, do there exist statements that cannot be decided by any set of axioms except ones which contain one or more axioms dealing directly with that particular statement?

Impredicativity thus mean: An object $M$ cannot be defined without using $\phi(V)=M$ where $\phi$ is some definable function and $V$ is the universe of the foundation in question

In other words, an object is impredicative if it cannot be defined "from below"

9

A: How can we know we're not accidentally talking about non-standard integers?

We know that any formal system cannot completely pin down the natural numbers. Incidentally, I said exactly this here. Besides what I said in that post, I wish to elaborate on the following points: A generalized version of the Godel-Rosser incompleteness theorem shows convincingly that the...

and it gets worse, you can have systems where the proof that demonstrate an inconsistency within the system is too long to fit within some specified finite number of words

Thus the question of "predicative infinity" can be rephrased as "Given any foundation F, is the following predicative "There exists a formula $\phi$ not involving $V$ such that $\phi \vdash m$ where $m$ is infinite" logically equivalent to Godel Rosser incompleteness theorems"

> If you want to read more about this, it helps to know that a function from ordinals to ordinals that’s continuous and strictly increasing is called normal. ‘Normal’ is an adjective that mathematicians use when they haven’t had enough coffee in the morning and aren’t feeling creative—it means a thousand different things. In this case, a better term would be ‘differentiable’.

Starting to wonder is the set of all normal objects not normal

user193319

10:07 AM

If $E$ is a subfield of $F(X)$, and $E$ contains some "rational polynomial" $f(X)/g(X)$, where $f(X),g(X) \neq 0$, does it follow that $E$ contains a "regular polynomial"; i.e., something in $F[X]$?

Secret

$g(X)$ can be constant polynomial or $g(X)$ divides $f(X)$?

both situations should give an element of $F[X]$

user193319

10:30 AM

@Secret Why must $g(X)$ divide $f(X)$?

Secret

Not necessary, it really depends on what $E$ is. But given so little info on what $E,F$ is (in particular, $E$ is not necessarily obtained as a quotient of $F(X)$), I don't see why you can rule out rational polynomials where $g(X)$ divides $f(X)$

user193319

Well, $F$ is a field of characteristic $0$ and $E = F(X^2) \cap F(X^2 - X)$; but I'm hoping what I claimed holds more generally.

user193319

10:50 AM

I am suppose to show that $E = F(X^2) \cap F(X^2 - X) = F$, but I'm beginning to wonder whether this is false.

Secret

11:08 AM

I don't think polynomials involving $X^2, X^2-X$ are likely to be linearly dependent, but to check whether $ab=c$ where $a,b$ are polynomials in $F(X^2)$ and $b$ are polynomials in $F(X^2-X)$ I don't know how it can be checked

typo: $c$ for the final $b$

Secret

11:28 AM

i.e. you need to check the algebraic independence of the elements in $F(X^2)$ and $F(X^2-X)$ to check if the intersection is the whole thing

which for polynomials this can be done by computing the determinant of a matrix containing the polynomials as its entries, but I am not sure if something different is needed for general polynomial fields

user193319

@Secret Not sure if you're making this mistake, but $F[X]$ and $F(X)$ are different things: the former is the polynomial ring over $F$, the latter is the fraction of field of $F(X)$.

Secret

you are right, I was talking about $F[X], F[X^2-X]$, in that case I am not sure as I am not really good with field of fractions

user193319

Sorry, "...the fraction field of $F[X]$"

I can show that the intersection doesn't contain any "proper" polynomials using a little bit of Galois theory, but I don't know how to show that it doesn't contain rational polynomials.

user193319

12:10 PM

Well...perhaps we can figure that out later...What does $\text{Aut }(E/F)$ denote?

Is it the group of all automorphisms of $E$ fixing the subfield $F$?

Secret

uh I thought $E$ is the subfield of $F$ according to what you said above?

user193319

Yeah, I'm working on something completely different now...sorry for the confusion in notation.

Secret

12:25 PM

it's ok

user193319