I understand how integers and rationals are expressed/derived in ZFC. But what about the irrational numbers? Can they also be expressed? If not, are there other axiomatic set theories able to express them? As for Dedekind cuts, from my understanding (maybe wrong), any irrational in question must have a 1-1 explicit function to a natural number, in order for the method to work (As an example the square root of 2 has such a function).  It boils down to this: is there an isomorphism between the the set of irrationals (as "just" numbers) and a a set of pure sets?
In set theory, if natural numbers are represented by nested sets that include the empty set, how are the rest of the real numbers represented as sets?    Thanks for the answers. Several answers basically said for irrational numbers that  A Dedekind cut is a pair of sets of rational numbers $\{L, R\}$. The set of real numbers is defined to be the set of all Dedekind cuts, where a Dedekind cut is a pair of sets of rational numbers $\{L, R\}$ which have no elements in common, and where all the elements of $L$ are less than any element of $R$.  Each Dedekind cut is a real number. This is where I have a problem - surely that can’t be correct. The set $L$ is a set of all rationals, and there must be a rational in the set $L$ that is greater than all other rationals in that set, even if we have no method of determining it. And similarly, there must be a rational in the set $R$ that is less than all other rationals in that set, even if we have no method of determining it. If every irrational number has a corresponding set $L$, then each irrational number has some such corresponding largest element of that set $L$, and then each irrational number has some corresponding rational number. And that would mean that the irrational numbers are countable. So, with Dedekind cuts,  the only conclusion is that there must be irrational numbers $x$ which are either greater or lesser than some irrational cut $y$ of the rationals, and between $x$ and $y$ there is no rational number.  But that is impossible, so that the Dedekind cuts cannot be the correct representation of the real numbers.  Surely the problem with Dedekind cuts is in using sets of rationals that include all rationals up to a certain rational. But there is an alternative method of representing irrationals can be defined in terms of infinite sets of rational numbers. For example, in binary notation, the non-integer part of $\pi$ is $.00100100\ 00111111\ 01101010\ 10001$. You define a set by: if the nth digit is a $1$, then  the natural number $n$ is in the set. And then we have that, for the real numbers between $0$ and $1$, that the set of real numbers is simply the set of all subsets of natural numbers. Each subset corresponds to some real number between $0$ and $1$.  And in this way, all real numbers can be considered to be some set based only on nested sets of the empty set.  But I still haven’t got a satisfactory answer for how negative numbers can be represented in terms only of sets containing the empty set. Any ideas?
$x^2+y^2=z^n$: Find solutions without Pythagoras! I was presented with the following problem:      Prove that there exist solutions to $x^2+y^2=z^n$ for all $n$, with $x,y,z, n \in \mathbb{N}$   I showed that by taking any Pythagorean triple $x^2+y^2=z^2$ and multiplying by $z^{2(n-1)}$ we get $(z^{n-1}x)^2+(z^{n-1}y)^2=(z^2)^n$, which allows me to generate solutions easily for any value of $n$. I managed to find several similar questions on this site, such as  concerning the specific case $n=3$. I notice that all of these questions take a similar approach and start with a Pythagorean triple and use it to generate general solutions.  Is there a way to prove the statement (or better yet provide solutions to the equation) without first relying on Pythagoras? Can it be proven that : If $a_n&gt;0$ and $\sum a_n$ converges then $na_n \to 0$ (without assuming $a_n$ is decreasing)  Please note This question does not require $a_n$ to be decreasing as condition,  Is it possible to prove $na_n \to 0$ without requiring $a_n$ to be decreasing? I keep thinking that $a_n&gt;0$ and $\sum a_n$ converging then that implies that $a_n$ must be decreasing anyway(not monotonically decreasing necessarily )  Is it possible that $a_n&gt;0$ and $\sum a_n$ converges then $na_n \not \to 0$?  There are already answered questions that assume $a_n$ to be decreasing as a requirement: (this question does not assume $a_n$ is decreasing)       was the post that made me ask this question, a counter example of monotonoic decreasingness of $a_n$ was given in comments by the OP. How do I troubleshoot random rebooting? Why do residuals in linear regression always sum to zero when an intercept is included? I'm taking a course on regression models and one of the properties provided for linear regression is that the residuals always sum to zero when an intercept is included.   Can someone provide a good explanation for why this is the case? How to hide Authoring information,Revision information,Publishing options tabs in create/edit node for any content type I am trying to hide Authoring information, Revision information, Publishing options tabs in create/edit node for any content type. I tried different modules with no luck. Please give me suggestions how to achieve this. Thanks a lot has been solved, in XeTeX, by this code:  \def\kright#1{\leavevmode #1\kern-\XeTeXglyphbounds3 \the\XeTeXcharglyph`#1 } \def\kleft#1{\leavevmode \kern-\XeTeXglyphbounds1 \the\XeTeXcharglyph`#1 #1}   which allows me to align text more precisely to the ink by accessing the side-bearing measurements (\XeTeXglyphbounds1 and 3).  This XeTeX dependency is currently the only thing stopping me from moving to LuaTeX (mainly for microtype). Is an equivalent function available in LuaTeX, or planned?    After reading a bit more on microtype, could the equivalent effect be achieved by an appropriately-tailored set of character protrusions? That way, my move to LuaTeX might be a lot simpler. Strip HTML from Text JavaScript How to remove a watermark from a PDF file? I thought this would be a simple task, but it turned out the other way.  The watermark is the very same (overlapping, but transparent) image on every single page. I created the PDF file myself (so no copyright worries here) using PDFCreator 0.9.8.  I have already tried my friend's Adobe Acrobat Pro, but it didn't work. It tries to remove it, but it can't. I tried to remove header/footer, etc., but the watermark just won't disappear.    How can I remove the watermark? Is there a way to get the count of rows in all tables in a MySQL database without running a SELECT count() on each table? UK visit visa: What documents prove genuine relationship with the visitor? What's the simplest SQL statement that will return the duplicate values for a given column and the count of their occurrences in an Oracle database table?  For example: I have a JOBS table with the column JOB_NUMBER.  How can I find out if I have any duplicate JOB_NUMBERs, and how many times they're duplicated? Powers of infinite cardinals in the generic extension Are there any tools for merging CSS? I have a couple CSS files with overlapping CSS selectors that I'd like to programmatically merge (as in not just appending one file to the end of the other). Is there any tool to do this online? or a Firefox extension perhaps? How do I give a villager trades with custom NBT tags General relativity is expressed in terms of differential geometry, which allows you to do interesting things with the coordinates: multiple coordinates may refer to a single point, eg. the  has whole lines at the top and the bottom that correspond to the North and South poles, respectively; or a single coordinate may refer to multiple points, for example by using inverse geometry the origin refers to all the points infinitely far away.  So, is the singularity just a single point in the curved spacetime, or can it be a more extended object, described by a single coordinate? Show that the ring of all rational numbers, which when written in simplest form has an odd denominator, is a principal ideal domain. Show that the ring of all rational numbers $m/n$ with $n$ an odd integer is a principal ideal domain.  We haven't really discussed principal ideal domains. I've heard that this is easy, but I just lack the basic knowledge of what a principal ideal domain is. How do I find out the number, type, resolution &amp; refresh rate of additional monitors I can attach to my MacBook or MacBook Pro?  This is an attempt at a canonical QA, as per the Meta QA -   to try prevent the myriad "same but different" questions we get on every possible combination of computers, cable &amp; monitors There are perfectly good resources out there that Ask Different should not have to duplicate the efforts of. Is there a slowest rate of divergence of a series? $$f(n)=\sum_{i=1}^n\frac{1}{i}$$ diverges slower than $$g(n)=\sum_{i=1}^n\frac{1}{\sqrt{i}}$$ , by which I mean $\lim_{n\rightarrow \infty}(g(n)-f(n))=\infty$. Similarly, $\ln(n)$ diverges as fast as $f(n)$, as $\lim_{n \rightarrow \infty}(f(n)-\ln(n))=\gamma$, so they 'diverge at the same speed'.   I think there are an infinite number of 'speeds of divergence' (for example, $\sum_{i=1}^n\frac{1}{i^k}$ diverge at different rates for different $k&lt;1$). However, is there a slowest speed of divergence?  That is, does there exist a divergent series, $s(n)$, such that for any other divergent series $S(n)$, the limit $\lim_{n \rightarrow \infty}(S(n)-s(n))=\infty$ or $=k$? If so, are there an infinite number of these slowest series? Is there any way I can specify a standard or custom numeric format string to always output the sign, be it +ve or -ve (although what it should do for zero, I'm not sure!) Move rigidbody using WASD keys but rotate it based on camera/mouse movement I am very new to unity and my scripts are based on some scripts found on the web.   I want to make a first person view so the camera is attached to my rigidbodys head. Then i want to move my rigidbody based on keyboard input (wasd) and rotate it using the mouse movement.  Until now, i used the following code for moving my rigidbody:  void Update() {      /* calculate rigidbody movement */     movement = new Vector3( Input.GetAxisRaw("Horizontal"), 0, Input.GetAxisRaw("Vertical")).normalized;      /* calculate mouse look to rotate the rigidbody*/     mouseDelta = new Vector2(Input.GetAxisRaw("Mouse X"), Input.GetAxisRaw("Mouse Y"));     mouseDelta = Vector2.Scale(mouseDelta, new Vector2(sensitivity * smoothing, sensitivity * smoothing));     smoothV.x = Mathf.Lerp(smoothV.x, mouseDelta.x, 1f / smoothing);     smoothV.y = Mathf.Lerp(smoothV.y, mouseDelta.y, 1f / smoothing);     mouseLook += smoothV;  }  void FixedUpdate() {         /* apply rigidbody movement from Update() */     transform.Translate( movement * moveSpeed * Time.deltaTime );             /* apply rigidbody rotation from Update() */     transform.localRotation = Quaternion.AngleAxis(mouseLook.x, transform.up);             /* apply camera rotation from Update() */     Camera.main.transform.localRotation = Quaternion.AngleAxis(-mouseLook.y, Vector3.right);    }   I learned today that i dont have to move a rigidbody's transform but now i am totally confused and it seems i have to restart my whole (first!) project.  So when i use MovePosition and MoveRotation, how do i rotate it based on the camera perspective/mouse movement?  I think the movement Vector is ok so i can simply rewrite the FixedUpdate to something like  rigidbody.MovePosition(movement * speed * Time.deltaTime)   but i am totally stuck on how to apply the camera rotation to my rigidbody to use it like  void FixedUpdate() {      /* move the rigidbody */                rigidbody.MovePosition(movement * moveSpeed * Time.deltaTime);     /* rotate the camera */     camera.transform.localRotation = Quaternion.AngleAxis(-mouseLook.y, Vector3.right);      /* but how can i rotate the rigidbody based on the cameras rotation??? */     rigidbody.MoveRotation(?????);  }   I hope its clear what i want, thank you!