Lord Antares

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?

No. Each of the sets of even numbers and odd numbers are bijectable to the natural numbers. Right?

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?

/cut

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:

the set of rationals is infinite, and has a lower cardinality than the interval set in R.

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:

So in considering just the set of positive whole numbers we have found an infinity.

 

But we haven't included any of the fractional numbers in between, let alone those that can't be expressed as fractions.

 

So we are forced to the conclusion that more transfinite numbers are needed to place infinite sets into one-to-one correspondence.

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?

I've given you a lot more info than you asked for. The tl;dr is that

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.

/cut

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.

 

Please use the word set for your collection of numbers - this is the accepted correct term. Older names are aggregate or collection or sometimes class.

A 'group' is a very important particular type of set in mathematics and only some collections (sets) of numbers form groups.

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.

How are you doing so far?

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.

I cannot comprehend this topic at all.

Why exactly would someone photoshop your photo to a porn site? Are you considering uploading nude pics to facebook? Or are you thinking they would photoshop your head on a nude body?

If so, it is always clearly visible when it is photoshopped like that, so that would prove nothing. And who exactly would waste their time doing that? You can easily report them even if you care about that.

And if you have ''enemies'', why do you keep them on facebook?

As someone else said, of course this does happen occasionally but only because there are so many people in the world. Car accidents, rape, robbery and physical violence all happen more frequently than this type of thing.

Suppose you are faced with a mysterious machine that has 2 buttons on it.

Button A has a 100% chance of giving you $100

Button B has an X% chance of giving you $1.000.000

What is the minimum value of X that would make you choose button B?

I can hardly imagine anyone would go for the mathematical solution. For example, if you gave A = 100% for $100, and button B = 50% for $202 to a non-pragmatic math machine, it would choose B, while around 100% of humans would choose B.

P.S. Obviously, it is implied you can only press the button once, so no B spamming until it rains money. (Or A spamming, depending on the value of x )