the axiom of choice itself seems to be intuitively right. But some of its implications appear absurd. I am not sure if standard math, used as tool which maps to realitiy as closely as possible, should include it.
Stochastics: "To be honest we don't know if this is true, but its probability is still 1 because It's the limit of a sequence of events and their probability converges to 1. So we can throw a coin that lands on head with a probability of 1 to decide if this is true, or we just assume it's true"
Just a technical note, that's kinda important, the ship is _not_ "at rest" with respect to itself precisely because it _is_ accelerating with respect to itself. It would be "at rest" with respect to itself if it was not accelerating (i.e. inertial). The most noticeable consequence of this is that the rigid physical material of the ship feels a force (change in momentum) in the direction it is being accelerated. If it was not accelerating it would not feel a compressive force (and everything inside is weightless). Essentially, pushing on one end of a rigid object, propagates a wave of pressure through the material at the speed of sound, and continuously accelerates each bit of material next to it as the elastic forces within the material restores it's shape... eventually the far distant end of the object gets accelerated forward if there's nothing there to push back... In relativity... the two ends of the object have a relative velocity to each other. The end being pushed is moving faster than the far distant end... which is why the material is compressed... and the two ends of the object are in _different inertial reference frames_ from each other. That's why proper acceleration is sometimes called "absolute motion". It's _moving_ relative to _itself_. (Not "at rest" with respect to itself.)
Special relativity is an impressively self-consistent theory, the majority of "paradoxes" people find are simply because they apply classical intuition, the mixing of special relativistic and classical conclusions is what leads to the paradoxes
It can't be both invented and discovered. This is a serious problem about the nature of truth's attainability at all, and not a matter of 'just how you feel about it'.
"So this is a sort of philosophical question There's no right or wrong answer I think it just comes down to how you think about things!" You can always tell an American but you can't tell him much!😅 "It doesn't really matter. It could be one or the other. Depends on your taste." Oops! Bye-bye dumb boy!
So could we explain the expansion of the universe in the same way? From Reference S, the universe appears to be moving away from us, but from Reference I, the rest of the universe is moving at the same speed as us, it's just that we observe the rest of the universe as undergoing length contraction.
I feel like this video misses the point of why people care about this. It's not just that it's "a symbol". It's that it represents something, in this case, a limit. And the people, like myself, who disagree that it is 1, are disagreeing with the whooooole enterprise of limits. It's not just some small notational thing. This is an argument over whether to embrace infinitesimals, or not. Whether to accept infinite sequences as equaling finite numbers, or not. Whether it makes sense when talking about integers, which traditionally represent raw quantities, as if they can be represented EQUALLY WELL through an infinite series. And we deny that claim. Whatever the spirit of the unit "1" is, it simply is not "equal" to 0.999 repeating. There's no way to talk about a sequence when describing a singular, and only a singular object. If we want mathematics to be useful.
'It's that it represents something, in this case, a limit' In that case, literally every single decimal represents a limit. 'And the people, like myself, who disagree that it is 1, are disagreeing with the whooooole enterprise of limits' Then you should google what a limit of a sequence is. This is just your own ignorance speaking, and its suggestions should be ignored. 'This is an argument over whether to embrace infinitesimals, or not' There are no infinitesimals in the space of real numbers. If you want to work with a different system, like spaces of hyperreal numbers, then you can and nobody is preventing you from doing so. But be honest - you have no clue what an infinitesimal even is. 'Whether to accept infinite sequences as equaling finite numbers, or not' You obviously just can't tell apart what a series is, what its sum is, and what the sequence of its partial sums is. 'Whether it makes sense when talking about integers, which traditionally represent raw quantities, as if they can be represented EQUALLY WELL through an infinite series' The fact that you refuse to learn how series work and any sort of construction of a real numbers just makes you unprepared to confidently speak on the matter. 'Whatever the spirit of the unit "1" is, it simply is not "equal" to 0.999 repeating' 0.999... = 9/10+0.999.../10 => 0.999...*(1-1/10) = 9/10 => 0.999... = (9/10)/(1-1/10) = (9/10)/(9/10) = 1. Decimals '0.999...' and '1' mean the same thing and your tantrums don't change that fact. 'There's no way to talk about a sequence when describing a singular, and only a singular object' You have no clue what you are talking about.
@@thetaomegatheta No, not every decimal represents a limit, only those with infinite amounts of decimals, like 0.9999, or 1/3. 1, or 0.33 (not repeating) aren't limits under my view, they're just numbers. And no, I don't need to google shit just because I disagree. If you want me to agree, provide an argument, don't just tell me to google. And then you say I don't even know what an infinitesimal is. Look, I get it, you wanna project your own ignorance onto me. But if neither of us supposedly know what we're talking about, why even bother replying to my comment? You clearly have no interest in genuinely debating the topic, or convincing me of anything. You just wanna disparage my (correct) opinion. This is the classic "if you disagree it must be because you don't understand" type of argument, and it's frankly, getting old.
'No, not every decimal represents a limit' Yes, really. The decimal '1' refers to the limit of the sequence of partial sums of the series 1+0+0+0+..., for example. Not coincidentally, that limit is the same as that of the sequence of partial sums of the series 9/10+9/100+9/1000+... 'only those with infinite amounts of decimals' This is obviously incorrect, and you are just trying to save face by inventing new rules that aren't actually there. '1, or 0.33 (not repeating) aren't limits under my view' The decimal '1' refers to the limit of the sequence (1, 0, 0, 0,...). The decimal '0.333...' refers to the limit of the sequence (0.3, 0.33, 0.33, 0.33,...) 'they're just numbers' So is 0.999... 'And no, I don't need to google shit just because I disagree' Yes, you do, because you have no understanding of this topic. If you don't want to investigate something, you have no ground to speak confidently about it. In particular, you have no ground to claim that literally every single mathematician in the world is wrong about a basic topic that falls under their area of expertise and is covered early in calculus courses and textbooks. 'If you want me to agree, provide an argument' I will be posting a few proofs of the fact that 0.999... = 1 that I wrote for RU-vid after this reply. Notably, I already gave you one proof, which you opted to ignore. 'don't just tell me to google' If you have no understanding of what you are talking about, I sure as hell have a reason to refer you to investigate relevant topics. 'And then you say I don't even know what an infinitesimal is' Because you clearly don't. 'Look, I get it, you wanna project your own ignorance onto me' I have a degree in math. You are making very obvious mistakes and are confidently advertising your inability to do basic math. 'But if neither of us supposedly know what we're talking about' Again, I have a degree in math. I had to prove my ability to actual mathematicians, i.e. to experts on this and many other topics. 'You just wanna disparage my (correct) opinion' Haha. This is cute. 'This is the classic "if you disagree it must be because you don't understand" type of argument' I literally provided proof that 0.999... = (9/10)/(1-1/10) = 1, which you are yet to find any flaws is.
1) Consider relation R between Cauchy sequences of rational numbers: for any two Cauchy sequences of rational numbers a=(a_1, a_2, a_3,...) and b=(b_1, b_2, b_3,...) the relation aRb holds iff lim(a_n-b_n)=0. Any given real number is an equivalence class of such sequences with respect to R. Any given digital representation corresponds to a Cauchy sequence of rational numbers, for example, 0.999... corresponds to (0.9, 0.99, 0.999,...), and 1 corresponds to (1, 1, 1,...). Let's check if (0.9, 0.99, 0.999,...)R(1, 1, 1,...): lim(1-sum(9/10^k) for k from 1 to n) as n->inf = lim(1/10^n) as n->inf = 0. That means that (0.9, 0.99, 0.999,...)R(1, 1, 1,...) and 0.999... = 1.
2) If x is some real number, |x|<=1, and r = p/10^n, where p is integer, and n is natural, we have: If x = r+x*1/10^n, then x*(1-1/10^n) = r x = r/(1-1/10^n) As said previously, r is rational, 1-1/10^n is a sum of two rational numbers, meaning that it's rational (because p_1/q_1+p_2/q_2 = (p_1*q_2+p_2*q_1)/(q_1*q_2), and a product of two integers is an integer, and a sum of two integers is an integer, meaning that p_1*q_2+p_2*q_1 = p_3a+p_3b = p_3, q_1*q_2 = q_3), and the division of a rational number by a rational number is also rational (because p_1/q_1/(p_2/q_2) = p_1*q_2/(q_1*p_2)), i.e. r/(1-10^n) is rational, and, because x = r/(1-10^n), so is x. If |x| was greater than 1, then we could do either of the substitutions x_1 = x/(floor(|x|)+1) or x_2 = x-floor(|x|) and prove that x_1 and/or x_2 is a rational number, from which it would follow that x is rational. That means that every real number x that can be represented with repeating decimals, i.e. x = p/10^n+x*1/10^n, where p is integer, and n is natural, is a rational number. And, of course, 0.999... = 9*1/10+0.999...*1/10, so 0.999... has to be rational, and 0.999... = 9*1/10/(1-1/10) = 9/10/(9/10) = 90/90 = 9/9 = 1.
I would love to see more proofs that explain the “obvious” concepts from elementary to advanced mathematics. Like does anyone really understand how or why the quadratic equation works?
@@FrostDirt yes I agree their is a proof. I meant that we take so many ideas and treat them as axioms when in fact profs exists and we all learn to memorize the formula without understanding how we derived that formula.
Thank you for Not using FOIL! I like to tell my students it is double distribution. I also warn them that FOIL fails once you have a trinomial or greater.
If I make this argument with the second derivative. Meaning: a diff equation y''=-(\omega^2)y, with initial condition y(0)=0 and phi(t)=cos(\omega t)+sin(\omega t) I would conclude that e^(i\omega t) is equal to phi(t), which is not true...
The two immediate issues here are: 1) What you wrote is not an initial value problem. A second degree initial value problem needs two initial values, you gave one (integration is a "lossy" operation; a second order differential equation will produce two arbitrary constants in its solution.) 2.) The argument references Picard-Lindelöf, which is about first order initial value problems. If you want to make an argument using existence and uniqueness through 2nd order DEs, you need to apply the theorem that covers those and that its conditions are met (they are more restrictive).
As a chem student learning about inorganic chemistry, it just dawned on me that point group operations like rotation and reflection are isomorphic functions. Like you perform rotation about a 180 deg angle, and the properties of the molecule after the rotation is still preserved. The only thing that changed are the matrix representations of the atomic orbitals. This is soo mindblowing.
Have you used character tables or learned about group representation theory in your studies? These particular fields are strongly linked. The fact that you are able to see a connection like that at such a foundational level is very impressive.
@@EpicMathTime Yessss! You determine the point group, represent them as matrices, and every time you perform an operation, the molecular structure is still the same (i.e. geometry, properties, etc) but you reduce the irreducible representations of the molecule. It helps us predict the translational, vibrational, and rotational motions of the molecule.
Because automorphisms respect the group operation, so for an automorphism f on group G, identity element e, and any group element x in G, f(e) • f(x) = f(e•x) = f(x). Automorphisms are surjective, so this tells you that f(e) • y = y for any y in G, hence f(e) is "an" identity by definition. Recall that the group identity is unique, hence f(e) = e.
If you want to think about multiplication as repeated addition, everything is offset by one, IE n * 2 means to "repeat the addition" _one_ time (n+n) while n*1 means to repeat the additon no times (n, just sitting there), while n*0 cannot even be thought of as repeating addition at all. That offset is what reconciles this, I believe.
I think it is easier to understand the meaning of the √ symbol by checking what happens in actual calculations rather than definitions and rules. If √ was a positive and negative symbol, then √9 would be +3 or -3, √4 would be +2 or -2. So what about √9+√4 in simple numerical calculations? (+3)+(+2)=(+5), but also (-3)+(+2)=(-1), (+3)+(-2)=(+1), (-3) +(-2)=(-5) come out as a result, which is confusing. Moreover, when it comes to calculating more complex expressions, it is clear that things get out of hand. Therefore, it is more natural to think of the √ as a symbol representing only the positive side of the squared number in √. [PS] In the video, the curve of y=√x, y=x², x=y² is drawn on the XY plane, so it is easy to understand. Currently, it is a calculator with a graph function, and it can be easily drawn, so in addition to the curve of the video, if you also draw the curve of y=-√x, x=√y, x=-√y ,y=x²-9, y=(x-3)², y=(x+3)², you will fully understand the meanings of √9=3 and √ symbol.
