Lord Antares

I said that it has no effect on Newton's law and that it holds true. The attraction of TWO masses will change if they get further apart as a result of the expansion of the universe, but that is in unison with Newton's law because r has increased between the two. I am saying that, as the universe expands, the force of gravitation of a mass has more and more reach every second. Simply put, it has more of the universe to cover. As the gravitational force is proportional to the mass of an object, and therefore limited by it, does that mean than it is constantly weakening as it needs to exert the same amount of force to a larger area? Or is the mass of everything increasing? This is what I'm asking. OR are you saying that there need not be an increase in gravitational force for a larger space because it takes no force to bend space, only to pull objects together? EDIT: It presents no questions to Newton's law so don't look there. It presents questions to general relativity.

Newton's law of universal gravitation states that F= g (m1 x m2)/r², i.e. a particle attracts every other particle in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. In Newton's view, there need to be two masses present for gravitation to take effect. It has been proven by general relativity that gravitation occurs with just one mass present, as it is not a force which acts upon objects per se, but rather on the space in between them. This all makes sense and Newton's law still holds true, as any two masses will behave in accordance to his equation. However, I have just one problem with Einstein's depiction of gravity. (problem as in ''I don't understand it'', not as in ''I'm trying to refute it'', to be clear). If it is true that Newton's inverse square law extends indefinitely and the universe is constantly expanding, then one of the two statements must be true: 1) The gravitational force of every object is weakening, because it needs to extend the same amount of force over a larger area. The overall amount of gravitational force exerted by a mass stays the same, but is decreased within any given distance. 2) The mass of every object grows proportionally to the rate of expansion of the universe. This is the only way an object could exert the same amount of force over a distance, but is bizzare to consider. These are the only two options I can think of. Neither of these would refute Newton's law in reality, because as the universe expands, the objects get further and further apart, and so the weakening in gravitational attraction would simply be explained by the increase in r. Actually, I am not sure how the second case would affect Newton's law. I am trying to think about it, but this option is far-fetched anyway. This problem only occurs when you talk about general relativity's concept of gravity. What do you think about this? What am I missing here? All replies are appreciated.

That's an invalid question. It cannot be just due to biological properties because that depends on how much time has passed. Every prolongued process depends on time. It's similar to asking ''does the earth's rotation around the sun depend on time?''. Of course it does.

I believe studiot was making the same point. I would think it wouldn't be hard to guess as you are the only who have replied in the last several posts

Correct again. Disregarding that error, was that the point you two were trying to make?

Oooooh, I think I finally understand. For even numbers, you can pair each new even number with the new whole number, i.e. (2,4,6,8,) paired with (1,2,3,4,) and go into infinity. But you can't do this for rational and real numbers because which number would you pair it with, since there is always a smaller one than the one you consider pairing. Is this the point? This simple fact somehow eluded me. @studiot - yes, the answer is always the same. You can't do this for my A and B intervals. That expands upon wtf's example in a helpful way because I didn't see what he was getting at.

∞ But that supports what I said, no? EDIT: That is one abomination of an infinity sign.

I don't know if this is related to this server upgrade but similarly to the poster above, I am unable to upload or link via URL an image as my avatar.

This is what I've been saying all along. If we go back to considering my intervals A and B, A contains all real numbers in between 3 and 4. B contains all real numbers in between 3 and 5. If you removed the range of A from B, you will still be left with an infinity. How can then the cardinality of these two intervals be the same if what you said is true? Why would it be different in your example if both use the same exact logic? THIS is what doesn't make sense to me. Interval B is the higher infinity (comparable to R in your example) and A is the lesser infinity (comparable to Q in your example). Also, if you change my A and B to be rational numbers (i.e. a set instead of an interval, right?), nothing should change logically, since the same kind of infinity is present, no?

Thank you. This clarifies a few things, primarily of terminology but not of the logical issue I want to know about. For a moment I thought this was just a misunderstanding and that it's very simple UNTIL I read this: Can you explain why it has a lower cardinality than the interval in R? This is what I'm trying to find out. Both are infinite. If you're saying it has a lower cardinality because not every element of R can be expressed by any element of Q, that does not make any sense to me. A quick google says ''While the cardinality of a finite set is just the number of its elements, extending the notion to infinite sets usually starts with defining the notion of comparison of arbitrary (in particular infinite) sets.'' So that might, in fact, be what you mean to say. It's a matter of definition of cardinality then, isn't it? Thinking about the things in this link https://en.wikipedia.org/wiki/Cardinality makes my brain hurt. It contains so many paradoxes when it comes to infinite sets and intervals and things which I wouldn't say are inherently logical but in a way also are.

Yes but how? Or rather, why doesn't the same apply to intervals? I assume you are referring to this: Why aren't we forced to the same conclusion for these A and B intervals? Also, how do sets compare with intervals in this regard?

Why would you think that? You've given me exactly what I was looking for and earned an upvote in the process I had the same ideas as this George person but wanted to check what educated people think. Oops, terminology strikes again. I meant to say ''aren't then all intervals of any real numbers the same cardinality'', instead of sets, which you answer to be true. However, I'm glad I mixed this up because it brings up an even more interesting question: By which logic can infinite sets of real numbers have different cardinality, as opposed to intervals of reals? They both deal with the same issue and that is different-sized infinities. This wikipedia article states that the cardinality of real numbers is higher than the cardinality of integers. The cardinality of both is infinite. The cardinality of both my A and B intervals is also infinite, but apparently equal. How can this be? The only difference between the two is that the set of all reals vs. all integers have the same range as opposed to my intervals where the B interval has double the range, but why should that matter; there is no difference since the increment in all reals is infinitely smaller than the increment of all integers. If anything, it should point to the opposite.

Although this answer was more along the lines of what I was looking for, it is obvious that it would give the same cardinality because both results are infinite. I was still thinking since the range is double, it should affect the result but you are probably correct - every infinity is infinite in size and therefore nothing can be higher than it just by the nature of it. But then aren't all sets of real numbers the same size (cardinality)? And isn't it impossible to list any one of them? Also, it is interesting to think that set B has double the amount of members than A if you include only whole numbers. Also, if you include numbers up to one decimal point. And two, and three and however many you like except for infinite at which point B becomes equal to A. It's just a bit odd.

I undestand. I take no offense, it is only rational for you to try to understand exactly what I meant. I simply meant to say: the amount of real (rational?) numbers is infinite between any two numbers. For example, 3.9 is a real number. So is 3.99 and 3.999, 3.9999, 3.999932, 3.99999999 etc.

No harm done, it was just a misunderstanding. I understand that correct terminology is very important in math and I did consider the possiblity that you were just trying to teach me the correct usage, but I wasn't sure. I am sure your knowledge of math is better than mine (assuming yours is solid), but in my opinion, this is stricly a logical matter of discussion and the benefit of knowing math here is being able to verify or form thoughts more precisely and also being able to deduce more correctly due to knowing accepted terminology and what it means, but it shouldn't be hard for me to follow what you are saying if I use only logic, if you know what I mean. @wtf - yes, I used the term infinitesimal wrongly. That's not what I meant. I simply meant any real number which there is an infinity of between any given range of numbers because they can get exceedingly small the more you count them.

I am sorry. English is not my first language, no need to be condescending. Maybe you are not trying to be, but that's what I got out of how the post is formatted. Also, what I meant by infinitesimals are all the possible smallest increments between each number which there are an infinity of. Maybe they're called increments or something else; again, it's just a language thing. I can use English well for general purposes, but these specific things never came up since I discussed them in my native language. Again, you're confusing me. Is that supposed to be condenscending towards me? What's the meaning behind this question? I knew these things you said in your post logically, but you never answered my question, unless your answer was ''it's a paradox, there is no answer''.

Let's say the collection of numbers A contains all number higher than 3 and lower than 4, including infinitesimals. This group is infinite. Let's say that group B contains all numbers higher than 3 and lower than 5. This group is also infinite. My question is if group B contains more numbers than group A? Although B is a more ''extensive'' infinity, i.e. technically includes double the amount of numbers than infinity A, either one of them are infinite and therefore nothing can have a higher value than any one of them. I would imagine this has been asked a trillion times over the course of history but I don't know of a definite conclusion.

Thank you for this. I undestand now. Out of curiosity, could you as well just measure the degree with a protractor and resume calculations? Also, why divide by 90 instead of 180?

? 1/1 000 000 x 100 = 0.0001 You divided by hundred twice (i.e. divided by 10 000)

1 in 10 000 does not equal 0.01 but yeah, that's another interesting way to phrase the question.

As a chess player, this is ridiculous lol, I would never think of that solution. How is one supposed to know you are also allowed to promote to the opposing team's pieces? Also, it specifically needs to be a black knight as any other piece could either interfere with the rook discovered check (queen, rook, pawn) or take the rook (bishop, queen, pawn)

Thank you, Janus, for the explanation. I was sure you only need to measure the degree value of the width of the coin and divide 180 by the result but you managed to confuse me with atan @Acme - So this is the actual result when you account for real-life physics, instead of just mathematics? That's lower than I thought compared to the 5+% result Janus gave just for the impact odds.

Well that's exactly what I said in the first post: the mathematical answer is X = 0.0001 and above, but I wouldn't choose button B at such odds. That's why I gave the example of button A being much more desirable than B in the event of A = 100% for $100, B = 50% for $202, even though the mathematical choice here would be B. So I was asking at what odds would YOU opt for button B. It's not a puzzle, it's just a personal question.

Not neccessarily. Simply put, everyone would choose B if X = 50%. Everyone would choose B if X = 20%. Just find the value of X below which you would press button A and above which you would press button B.

Simple. Go 1 second into the future and resume life normally.