As I said, I can’t help you.

A koch snowflake has a finite area but an infinite boundary. Odd, that. Very nice. — Banno

The koch snowflake is just sending a finite boundary into infinity. It can't exist. -Mathematical infinity swallows itself and there's nothing that save it. — Gregory

A circle has infinite amount of tangents, yet in every place of it's circumference it has just one.

In Mathematics, infinity exists. Clear and simple. We just don't understand everything about it. Hence we have things like the Continuum Hypothesis. Yet our ignorance doesn't make it illogical and false. In Mathematics, it's as real as a finite number or the circle is. Or a Koch snowflake etc.

Yep.

Take all the points on a segment and line each one up one at a time to the whole numbers. — Gregory

You can't. Between any two points you select, there are infinitely many more points. There are only countably many whole numbers, but far more points in a segment...

Treat it as points, or as a continuum, but not both. — Banno

Then why does mathematics combine the two? Real numbers are points representing a continuous number line.

They don't. The continuum is not just a set of points.

. Between any two points you select, there are infinitely many more points — Banno

There is the same problem with the odd vs the whole numbers. With these, imagine them going off into the horizon. In how they explain this, they just pull all the odd numbers towards number 1 and say woopy! equal! Why is that a legit move? Why can't you do this move with the uncountable as well?

Why is that a legit move? Why can't you do this move with the uncountable as well? — Gregory

Where would you start?

First of all, do you know what's the difference between countable and uncountable here?

Basically, the "legit move" is that you can make a bijection with the set of natural numbers (1,2,3,4,5,...) and the set you are thinking about. This means that you basically can write the numbers you are talking about in a way that you get every of them and don't miss any number (if you would have infinite time a so on...). If you can't do this, then it's uncountable.

If I don't make my point clear, just go and look at this site: Countable and Uncountable Sets Remember to look at the proofs.

Where would you start — ssu

With any point bijected to 1, 2, and 3.

don't miss any number (if you would have infinite time a so on...). If you can't do this, then it's uncountable — ssu

Again, if you can start with 1, 2, and 3 and move the 2 to one and the 3 to 2 ect. you could also take a segment parallel to the whole numbers and move each point down to the left like you did before and assume it's all good at the other infinite end, like you did trying to prove the even numbers are equal to the whole numbers. Also, doesn't this violate the principle that the whole is greater than the part? Infinity doesn't make any sense. What am I missing?

What am I missing? — Gregory

Again, if you can start with 1, 2, and 3 and move the 2 to one and the 3 to 2 ect. you could also take a segment parallel to the whole numbers and move each point down to the left like you did before and assume it's all good at the other infinite end, like you did trying to prove the even numbers are equal to the whole numbers. Also, doesn't this violate the principle that the whole is greater than the part? — Gregory

Uh, nope.

OK, let's try another way. I assume (from the above) you know the idea of the Hilbert Hotel works. Please watch this video (only six minutes!), it sums up perfectly the uncountable infinite and Cantor's diagonal argument. And just why sometimes the Hilbert Hotel cannot accomodate every possibility of guests.



Basically this is what in the earlier link I gave you was told in theorem 1.20 of the uncountability of the reals. But for me the above video is more easier to understand.

The very first step of the video i question. If all the rooms are filled you can't move 1 to 2 and 3 to room 4 because all the infinite rooms are already filled. The problem with the diagonal argument for me is this first part about the odd numbers equalling the whole numbers: You are not using the same logic for the two types of infinities.

What do you say?

Nice video.

If all the rooms are filled you can't move 1 to 2 and 3 to room 4 because all the infinite rooms are already filled. — Gregory

If you were right then you could specify who does not get a room. In the first case, each individual is assigned to the room one more than the room they are in, and so every individual gets a new room. The person who was in room two is now in room three; the person who was in room three is now in room four; and so on. In the second case, each individual is assigned to the room twice the number of the room they are in. Again, each individual gets a room. In the third case, in which and infinity of new guests arrives, and the spreadsheet is used, each individual is still assigned a room. But for the party bus, the diagonal argument shows that there will always be an individual who does not get a room.

This is what happens when you try to assign a whole number to every point on a segment of the continuum. There are too many points on the segment to be counted.

But this is a different story to the one we started with. Achilles starts behind the tortoise, which has a small head start. By the time he reaches the tortoise’s starting point, the tortoise has moved forward a shorter distance. Achilles then reaches this new position, but the tortoise has moved again. This process repeats infinitely, but the distances form a geometric series that converges to a finite sum. The total time taken also converges to a finite limit. Achilles reaches the tortoise’s position in a finite time and then surpasses it. The paradox arises only if one mistakenly assumes that infinite steps must require infinite time, which they do not.

This process repeats infinitely, but the distances form a geometric series that converges to a finite sum. The total time taken also converges to a finite limit. — Banno

Your statement explains the paradox in an accurate way.

Another simple way to look at this paradox is to see that:

"if you only look at times BEFORE Achilles reaches the tortoise, then it will appear as if Achilles never reaches the tortoise".

"if you only look at times BEFORE Achilles reaches the tortoise, then it will appear as if Achilles never reaches the tortoise". — Agree-to-Disagree

There are an infinity of intervals before Achilles passes the tortoise, each one half the time of the previous, and so with a finite sum. The process of Achilles passing the tortoise therefore takes a finite time.

End of story, really.

The very first step of the video i question. If all the rooms are filled you can't move 1 to 2 and 3 to room 4 because all the infinite rooms are already filled. — Gregory

That would be true if there would be a finite number of rooms. Then the person in the last room would find there's no room for him or her. But it's an infinite hotel. There is no last room.

If there would be a last room, guess what? The hotel would have finite amount of rooms.

Your problem is that you simply don't understand the concept of infinity, so it's quite futile for me to show that these are mathematical proofs, not just my opinions. You see, calculus exists. Infinity is a very useful and logical concept and is used a lot in mathematics. Modern set theory has infinity as an axiom.

I totally understand it's really puzzling. A lot of the brightest minds in the history of math found this very puzzling. Galileo Galilei was one of the first people to point this. (See Galileo's Paradox)

But this is a different story to the one we started with. — Banno

Yeah, but I guess everyone should understand the connection that infinity and an infinitesimal has. (Or limits)

you were right then you could specify who does not get a room. In the first case, each individual is assigned to the room one more than the room they are in, and so every individual gets a new room. The person who was in room two is now in room three; the person who was in room three is now in room four; and so on. In the second case, each individual is assigned to the room twice the number of the room they are in. Again, each individual gets a room. In the third case, in which and infinity of new guests arrives, and the spreadsheet is used, each individual is still assigned a room. But for the party bus, the diagonal argument shows that there will always be an individual who does not get a room — Banno

This is just repeating the video. What infinity are we talking about with the first hotel situation. We haven't established what infinities are so maybe you can't move guy 3 to room 4 because theybare all filled. Imagine planks going infinitely into the horizon. Two sets. With the odd vs the wholes you are pulling the odd numbers back to line up with the whole numbers and that is geometrically crazy. Just as it is to say all the points on the edge of a cube are the same as the points in the cube. None of it makes sense

This process repeats infinitely, but the distances form a geometric series that converges to a finite sum. — Banno

False. It's infinite and finite at the same time in the exact same respect

True enough.

Your problem is that you simply don't understand the concept of infinity — ssu

More than that, there seems also to be a resistance to learning about infinity - hence flimsy response "If all the rooms are filled you can't move 1 to 2 and 3 to room 4 because all the infinite rooms are already filled". Notice how the OP, which has a relatively simple answer, was exploded into quantum nonsense and "dimensions and contrivances" with such glee, within a few posts of the OP?

Folk don't want an answer...

So what is it they want?

Your problem is that you simply don't understand the concept of infinity — ssu

Then answer my arguments

So what is it they want — Banno

For you to honestly address the issue

Here it is again: A refusal to recognise the answer when it is set out before them.

It reeks of some sort of anti-intellectualism, or at least anti-expertise.

Explains a lot of recent politics, too.

You see, calculus exists — ssu

Why couldn't its foundations be wrong?

Then state my argument, or at least ONE of them, in your own words

They don't. The continuum is not just a set of points. — Banno

So are saying that there is more to the continuous number line than the points which are the real numbers? Can you explain that?

...there is more to the continuous number line than the points which are the real numbers — Metaphysician Undercover

Yep. It is also connected and complete; it has a topological structure. Of course, not all the issues are ironed out and answered. If you want more you will need to talk to a mathematician.

Then state my argument, or at least ONE of them, in your own words — Gregory

I can't, becasue they are incoherent. Take

We haven't established what infinities are — Gregory

Yeah, we have, at least enough to be getting on with. For every number there is a next number.

Why couldn't (calculus's) foundations be wrong? — Gregory

They cannot be wrong, any more than 4+4+2=10 can be wrong. But it can be misunderstood.

Then answer my arguments — Gregory

I did answer your arguments. Just in the last response I wrote you. Infinite is different from the finite. If you start from a finite situation, now wonder you have problems to understand the infinite.

Why couldn't its foundations be wrong? — Gregory

Hopefully you do notice that calculus is very, very useful in physics or economics etc. It does answer correctly to many real world problems, that can be calculated by using calculus.

For me, one way is to look at the history of mathematics, how new ideas have been responded to, how from one thing we have gotten to another. This way, the theorems aren't taken just as a given.

If you look at the history of calculus, you see the obvious foundational problems it has had. From Newton himself and Leibniz. Yet if your argument is that infinity doesn't exist, then basically calculus wouldn't exist.

Yeah, we have, at least enough to be getting on with. For every number there is a next number — Banno

No you don't know how a countable infinity relates to uncountable and their qualities before the argument starts.

I can't, becasue they are incoherent — Banno

You couldn't try? You're dishonest

They cannot be wrong — Banno

But you say:

Of course, not all the issues are ironed out and answered — Banno

So you have only probable assurance that Zeno's paradoxes have an answer?

Yet if your argument is that infinity doesn't exist, then basically calculus wouldn't exist — ssu

Exactly. It's useful, not true. Like Gabriel's horn. Obviously false. I've presented at least 5 cogent arguments against infinity on this thread and you didn't indicate that you understood any of them

ep. It is also connected and complete; it has a topological structure. Of course, not all the issues are ironed out and answered. If you want more you will need to talk to a mathematician. — Banno

Are you saying that topology adds something to the line, which is more than just the real numbers? What more could there be?

No you don't know how a countable infinity relates to uncountable and their qualities before the argument starts. — Gregory

Yeah, we do. We learn how to count, then notice that whatever number we chose, there is a bigger number. Or most of us do, around the age of seven or eight. Then some see Hilbert's Hotel and the diagonal argument and go "Holly shite! there are numbers that cannot be counted..."

You couldn't try — Gregory

But I did try, in the post to which you are responding. You can't seem to recognise that the responses you are receiving actually answer your questions. It's odd. But it's not about maths, it's about you.