## Continuum

### Re: Continuum

What does "divisible to the point" mean? If you divide a nonzero length in half you get a nonzero length, always. Never a zero-length point. Your claim appears false on its face without further context.

Secondly, even a non-continuum such as the rational numbers has the same property of arbitrary divisibility.

### Re: Continuum

If a line is made of points then it is reasonable so say that line cannot be constructed from something which has size zero, point, unless the number of points are boundless.wtf wrote: ↑Mon Aug 10, 2020 11:46 pmWhat does "divisible to the point" mean? If you divide a nonzero length in half you get a nonzero length, always. Never a zero-length point. Your claim appears false on its face without further context.

Secondly, even a non-continuum such as the rational numbers has the same property of arbitrary divisibility.

### Re: Continuum

What does boundless mean? The number of integers is "boundless" in the sense of being infinite; but countably many points can not possibly form a mathematical continuum. Boundlessness (by which I gather you mean infinite?) is necessary but not sufficient for a collection of points to form a continuum.

Secondly, what does your response have to do with my question? I wish you'd reread it and respond to what I asked: which was, what does "divisible to a point" mean?

Mathematically you're speaking nonsense. If you have some other context in which your remarks are to be understood, please supply that context.

### Re: Continuum

Without any end or beginning.

You are talking about infinitude. I am saying that there is no absolut.wtf wrote: ↑Tue Aug 11, 2020 7:48 pm The number of integers is "boundless" in the sense of being infinite; but countably many points can not possibly form a mathematical continuum. Boundlessness (by which I gather you mean infinite?) is necessary but not sufficient for a collection of points to form a continuum.

It means that line is made of points.

Ok, I hope that things is clear now.

### Re: Continuum

The integers have no end and no beginning. They cannot possibly form a continuum. Do you understand this point?

Even the rationals, which have no end and no beginning but do have a dense order, can not possibly form a continuum.

No absolut what? Vodka? What do you mean there is no absolut? Or absolute if that was a typo.

Ok. But no amount of divisions can turn a line into a point. Do you understand that? If you divide a line segment in two you always get a pair of line segments and never a point.

Frankly worse than before. You are surely not making any mathematical point nor have you provided any other context (metaphysics, speculative physics, etc.)

I'm afraid you made things even more confused than earlier. I'm not giving you a hard time, I'm trying to understand what you are trying to say.

I won't continue to badger you. I reread your OP. It says nothing and means nothing.

### Re: Continuum

ps -- I don't mean to sound critical. Let me tell you where I'm coming from.

I studied some math back in the day. Here's what I know.

The mathematical continuum is taken to be the real number line. It has uncountably many points. On it are the integers, the positive and negative and zero whole numbers. Those are only countable in number, where countable has its usual technical meaning as in elementary set theory.

It's true that the real line is made up of points. But the real number line is just a modern version of Euclid's ancient idea that a line is made up of points.

It should be remarked that none of this is to be construed in a physical sense or as pertaining to nature or the real world or the study of physics. Rather, these are abstract mathematical ideas: lines, points, the real numbers, mathematical infinity. No connection with reality is implied or intended.

So when you say a continuum can be divided to a point, I don't understand. Say I take the unit interval [0,1] consisting of all the real numbers between 0 and 1, inclusive. If you divide it in half you get two intervals of length 1/2. Divide them and you get fourths. Divide again and you get eights. You never get a point, you always get an interval.

Even if we conceptually drill down to a point, by taking the intersection of a nested tower of intervals, as can be done; what of it? We already knew the real line is made of points from the time of Euclid and as reaffirmed in the modern formulation.

It's in this context that I'm not understanding your remarks. I don't understand if you're making a point about the world, or about abstract math. If the latter, your words don't correspond to any math I know. So that's why I'm confused.

I studied some math back in the day. Here's what I know.

The mathematical continuum is taken to be the real number line. It has uncountably many points. On it are the integers, the positive and negative and zero whole numbers. Those are only countable in number, where countable has its usual technical meaning as in elementary set theory.

It's true that the real line is made up of points. But the real number line is just a modern version of Euclid's ancient idea that a line is made up of points.

It should be remarked that none of this is to be construed in a physical sense or as pertaining to nature or the real world or the study of physics. Rather, these are abstract mathematical ideas: lines, points, the real numbers, mathematical infinity. No connection with reality is implied or intended.

So when you say a continuum can be divided to a point, I don't understand. Say I take the unit interval [0,1] consisting of all the real numbers between 0 and 1, inclusive. If you divide it in half you get two intervals of length 1/2. Divide them and you get fourths. Divide again and you get eights. You never get a point, you always get an interval.

Even if we conceptually drill down to a point, by taking the intersection of a nested tower of intervals, as can be done; what of it? We already knew the real line is made of points from the time of Euclid and as reaffirmed in the modern formulation.

It's in this context that I'm not understanding your remarks. I don't understand if you're making a point about the world, or about abstract math. If the latter, your words don't correspond to any math I know. So that's why I'm confused.

- RCSaunders
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### Re: Continuum

Wow. You are going to make half of the world's philosophers' heads explode. All their lives they've been certain that a negative cannot be proved and along you come and assert, "the biggest integer does not exist?" If you can prove the biggest integer does not exist you will have proved all those philosophers wrong. Go for it!

From there you can extrapolate: Since the concept of God includes omnipotence, which if measured would have to be measured with the biggest possible integer, if the biggest possible integer does not exist, neither does God's omnipotence.

This is truly revolutionary.

### Re: Continuum

If you claim n is the largest integer, we know that n + 1 is also an integer that is larger. Hence there is no largest integer.RCSaunders wrote: ↑Sat Aug 15, 2020 8:05 pm=

Wow. You are going to make half of the world's philosophers' heads explode. All their lives they've been certain that a negative cannot be proved and along you come and assert, "the biggest integer does not exist?" If you can prove the biggest integer does not exist you will have proved all those philosophers wrong. Go for it!

How did you manage to write such nonsense, calling out the only thing the OP said that was correct!

- RCSaunders
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### Re: Continuum

I didn't say you were wrong. I was just pointing out the irony of what you wrote relative to so many claims made by others around Philosophy Now, such as theists, who like to use the argument that a negative cannot be proved. You see the statement, "absence of evidence is not evidence of absence," all the time, for example.wtf wrote: ↑Sat Aug 15, 2020 10:56 pmIf you claim n is the largest integer, we know that n + 1 is also an integer that is larger. Hence there is no largest integer.RCSaunders wrote: ↑Sat Aug 15, 2020 8:05 pm=

Wow. You are going to make half of the world's philosophers' heads explode. All their lives they've been certain that a negative cannot be proved and along you come and assert, "the biggest integer does not exist?" If you can prove the biggest integer does not exist you will have proved all those philosophers wrong. Go for it!

How did you manage to write such nonsense, calling out the only thing the OP said that was correct!

It was no meant that seriously.

### Re: Continuum

What did I write? Your post quoted the OP, not me.RCSaunders wrote: ↑Sun Aug 16, 2020 1:13 am I didn't say you were wrong. I was just pointing out the irony of what you wrote relative to so many claims made by others around Philosophy Now,

### Re: Continuum

What I am saying is that if real number exists then it follows that it is made of unbounded number of zero. Therefore, the claim of the biggest integer (what matematicians call absolute) leads to absurdity. The proof is as following: Consider that real number exists and is made of points. Consider that the largest integer number also exist, so-called Li. Now for each real number, X, we have X/Li>0 since if the X/L=0 then we could conclude that any for any X>Y the X/L=Y/L=0 which means that two different intervals are made of same number of points. This is absurd. Therefore, X/Li>0 which means that there are more points in any real interval than Li, this is contradictory since we assume that there is a larges integer number. Therefore, the largest integer number does not exits.wtf wrote: ↑Wed Aug 12, 2020 10:39 pmThe integers have no end and no beginning. They cannot possibly form a continuum. Do you understand this point?

Even the rationals, which have no end and no beginning but do have a dense order, can not possibly form a continuum.

No absolut what? Vodka? What do you mean there is no absolut? Or absolute if that was a typo.

Ok. But no amount of divisions can turn a line into a point. Do you understand that? If you divide a line segment in two you always get a pair of line segments and never a point.

Frankly worse than before. You are surely not making any mathematical point nor have you provided any other context (metaphysics, speculative physics, etc.)

I'm afraid you made things even more confused than earlier. I'm not giving you a hard time, I'm trying to understand what you are trying to say.

I won't continue to badger you. I reread your OP. It says nothing and means nothing.

Last edited by bahman on Sun Aug 16, 2020 5:21 am, edited 1 time in total.

### Re: Continuum

Here is the proof: Consider that real number exists and is made of points. Consider that the largest integer number also exist, so-called Li. Now for each real number, X, we have X/Li>0 since if X/L=0 then we could conclude that for any X>Y, X/L=Y/L=0 which means that two different intervals are made of the same number of points. This is absurd. Therefore, X/Li>0 which means that there are more points in any real interval than Li, this is contradictory since we assume that there is a larges integer number. Therefore, the largest integer number does not exist.RCSaunders wrote: ↑Sat Aug 15, 2020 8:05 pmWow. You are going to make half of the world's philosophers' heads explode. All their lives they've been certain that a negative cannot be proved and along you come and assert, "the biggest integer does not exist?" If you can prove the biggest integer does not exist you will have proved all those philosophers wrong. Go for it!

From there you can extrapolate: Since the concept of God includes omnipotence, which if measured would have to be measured with the biggest possible integer, if the biggest possible integer does not exist, neither does God's omnipotence.

This is truly revolutionary.

And I agree that this puzzles theists.

- RCSaunders
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### Re: Continuum

Sorry. Meant the OP.wtf wrote: ↑Sun Aug 16, 2020 2:10 amWhat did I write? Your post quoted the OP, not me.RCSaunders wrote: ↑Sun Aug 16, 2020 1:13 am I didn't say you were wrong. I was just pointing out the irony of what you wrote relative to so many claims made by others around Philosophy Now,

### Re: Continuum

Real numbers aren't made of points. They are points.

For every real number there exists a point.

So, you've just re-invented infinity and you've labelled it differently. What is Li + 1?