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# Gödel's second incompleteness theorem and Consistency.

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According to Gödel's second incompleteness theorem, no consistent axiomatic system which includes Peano arithmetic can prove its own consistency. As I understand it, this result contributed to spark a crisis in the foundations of mathematics. What I don't really understand is of what use would be proving that a system of axioms is indeed consistent. Indeed, let's assume that we were somehow able to prove that a system of axioms does not produce any contradictions. But in an inconsistent system every statement is true, so we would be able to prove consistency (by contradition). Therefore proving consistency is useful only if the system does not contain contradictions, making the endeavor entirely circular. So we should we care so much about being able to prove the consistency of the axioms? I did not take any course in logic, so I apologize if I misunderstood some results or made wrong assumptions.

PS

I'm aware that this question has been already asked here: Godel's Second Incompleteness and the Assumption of Consistency
but I didn't find the answers particularly illuminating.

## How to determine if a subspace of \$ mathbb{R}^n \$ has an integer basis

Let $$W$$ be a sub vector space of $$mathbb{R}^n$$. How can we determine if $$W$$ admits an integer basis?

This is equivalent to asking how to determine if $$W cap mathbb{Z}^n$$ spans $$W$$.

## Centering a string in another

I have a string of length `N` which I want to center in another string.

Thus setting some string of length N

``````old_string="***"
``````

I want a new string (length 55) with the old string centered in it.

``````new_string="     ***     "
``````

How can I do this in bash ?

Although in actual fact the trailing spaces are redundant, so I can-simply have

``````new_string="     ***"
``````

## What is the relationship between measurable or continuos cross-sections?

Let $$G$$ be a locally compact Polish (or compact) group acting continuously on a locally compact Polish (or compact) space $$X$$, and $$mu$$ a Borel measure on $$X$$. To be sure, continuity of the action means that the map $$(g, x) in G times X mapsto g cdot x in X$$ is continuous with respect to the product topology on $$G times X$$. Let $$X/G$$ denote the orbit space endowed with the quotient topology, and $$pi : X rightarrow X/G$$ denote the orbit map. A cross-section to the orbit map is a map $$s: X/G rightarrow X$$ satisfying $$s circ pi = 1_{X}$$. If $$s$$ and $$t$$ are measurable or continuous cross-sections to the orbit map, then it is known that their images $$s(X/G)$$ and $$t(X/G)$$ are measurable (closed in the case of a continuous cross-section with compact $$G$$ and $$X$$). What is the relationship between $$mu(s(X/G)$$ and $$mu(t(X/G))$$? Is it reasonable to expect $$mu(s(X/G)) = mu(t(X/G))$$?

PS: It is enough for me to consider the case of $$X = G$$, that is, $$X$$ is the underlying topological space of $$G$$, and the action of conjugation, and $$mu$$ Haar measure on $$G$$. Thanks.

## Chromebook dying unexpectedly

My school provides me with a Chromebook.

Today, I closed the lid, then opened it up, expecting to see the lock screen, but instead, I saw where I last left off. I thought it might be because I had used the super zoom (ctrl+alt+brightness) because that usually happens, but instead of working again, it just turned off, and won’t turn back on.

It is charging but should show the low battery error, which it does not.

## Find orthogonal vectors in relation to span

Consider $$R^3$$ as an inner product space in relation to scalarmultiplication. Find all vectors in the subspace

$$Spanbigg(bigg[begin{matrix}1\2\1 end{matrix}bigg],bigg[begin{matrix}3\4\1 end{matrix}bigg]bigg)subseteq R^3,$$

which are orthogonal to the vector $$bigg[begin{matrix}-1\1\1 end{matrix}bigg]$$

Any help will be appreciated. I tried finding the random vector $$v=bigg[begin{matrix}x_1\x_2\x_3 end{matrix}bigg]$$ in the plane which i found to be $${x_3cdotbigg[begin{matrix}1\-1\1 end{matrix}bigg]bigg}$$ but i dont know where to go from here.

## I derived a formula for [x!]’ is it correct?

The starting point was that $$Gamma'(x+1)=Gamma(x+1)psi(x+1)$$ where $$psi(x+1)=-gamma+H_{x}$$ . Hence $$[x!]’ = x!biggl[-gamma+sum_{k=1}^{x}frac{1}{k}biggl]$$ For example $$[4!]’ = 24[-gamma+1+1/2+1/3+1/4]$$ which gives 36.1462 that, put in the tangent equation, gives us exactly the tangent for x=4! Let me know if I made any mistakes in the derivation/generalization!

## command “dir s b” in Windows 10 cmd gives me “File Not Found” error

It is very strange that this file exists, even I copy and paste its address in run and the file opens

https://i.postimg.cc/j2RNkrrx/screenshot-32.png

## An inequality about the 2-Wasserstein distance

Let $$W_2(mu,nu)$$ denote the $$2$$-Wasserstein distance between two given probability measures $$mu$$ and $$nu$$ on $$mathbb R^n$$. For a probability measure $$mu$$ and $$f:mathbb R^nto mathbb R^n$$, let $$f_{#}mu=mucirc f^{-1}$$ denote the push-forward of $$mu$$ under $$f$$, i.e. $$(f_{#}mu)(B)=mu(f^{-1}(B))$$ for every Borel set $$B$$ in $$mathbb R^n$$. Why does the following inequality hold true?
$$W^2_2(f_{#}mu,g_{#}mu)leq int_{mathbb R^n}|f(x)-g(x)|^2,dmu(x)$$
for all $$mu$$-measurable functions $$f,g:mathbb R^ntomathbb R^n$$.

Some comment: the product measure $$f_{#}muotimes g_{#}mu$$ is a so-called transport plan and by definition of the Wasserstein distance
$$W^2_2(f_{#}mu,g_{#}mu)leq int_{mathbb R^ntimes mathbb R^n}|x-y|^2,d(f_{#}muotimes g_{#}mu)(x,y)=int_{mathbb R^ntimes mathbb R^n}|f(x)-g(y)|^2,dmu(x),dmu(y).$$

## Is it possible to set up a task that will trigger only when a file is copied or moved into a specific folder?

My overall goal is to get a notification every time a file is copied or moved into a specific folder. I would like to try and avoid using external tools if possible. In order to achieve that I’ve created a task in task manager. This task should run every time a user copies or moves a file into this folder. I’m using the XML script that you can see below for the trigger.

Here is what ChatGPT spit out after several tries and refining of the request:

``````<QueryList>
<Query Id="0" Path="Security">
<Select Path="Security">
*[System[(EventID=4663 or EventID=4660) and
(EventData/Data[@Name='ObjectType'] = 'File' or