Paradox?

[Philosophy Explorer]

Here's something some may take for a paradox.

Take a line that's an inch long with no gaps. Take a square whose sides are an inch long, again with no gaps.
I assert that both objects contain the same number of points and this can be proven geometrically or by Cantor's method of one-to-one correspondence using Cartesian coordinates.

IOW, the number of points is independent of distance. Any comments?

PhilX

[wtf]

IOW, the number of points is independent of distance. Any comments?

Right. The number of points isn't even related to dimension. The unit interval is cardinally equivalent to the unit n-cube in Euclidean n-space. And that's cardinally equivalent to the entire infinite extent of Euclidean n-space.

Cantor himself was quite surprised by this. At first he thought the line was Aleph-1, the plane Aleph-2, and so forth. Turn out there's all the cardinality of the reals, what ever that is. We don't actually know which Aleph the reals are.

Another curiosity along those lines is that bijection doesn't respect length. For example the unit interval [0,1] is in bijective correspondence with the interval [0,2], yet the first has length 1 and the second has length 2. In fact the unit interval is cardinally equivalent to the entire real line.

Bijection is in fact a very weak equivalence. Two things that have the same cardinality can be very different mathematical objects.

[Philosophy Explorer]

Don't mean to contradict WTF. Here's what I've read from various sources:

Aleph 0 or Aleph null is the infinity of the natural numbers.

Aleph 1 is the infinity of the real numbers.

Aleph 2 is the infinity of the curves.

It's unknown or unproven that there's an infinity in-between Aleph 0 and Aleph 1.

Two infinite sets are the same size if one-to-one correspondence exists between the two sets (bijection).

Infinity is NOT a number.

PhilX

[Philosophy Explorer]

While lengthy and a bit technical, this Wikipedia article on Georg Cantor should help in understanding what infinity is about:

https://en.m.wikipedia.org/wiki/Georg_Cantor

PhilX

[wtf]

Don't mean to contradict WTF.

If you have facts that contradict me then fire away, maybe I'll learn something.

Here's what I've read from various sources:

You're going to need better sources.

Aleph 0 or Aleph null is the infinity of the natural numbers.

Tru dat.

Aleph 1 is the infinity of the real numbers.

No, that's the content of the Continuum hypothesis (CH). We know that CH is independent of the axioms of ZFC. And what's worse, it's independent of every sensible additional axiom anyone has ever thought of. No matter what naturalistic or reasonable axioms you try to add to set theory, there's a model where CH is true and another model where CH is false.

Nobody knows what the cardinality of the reals is. The best we can do is prove (easily, exercise for the reader) that the cardinality of the reals is 2^(Aleph-0). But nobody knows how big that might be in terms of the Alephs. That is, 2^(Aleph-0) might be Aleph-1, or it might be Aleph-47, or it might be Aleph-(Aleph-47) for all we know.

Interestingly both Gödel, who proved that CH is consistent with ZFC, and Cohen, who proved that CH is independent of ZFC, did not believe that 2^(Aleph-0) = Aleph-1.

There is a plausibility argument (though of course not a proof). For all finite n, 2^n is a lot larger than n. Why should this not be true in the transfinite realm as well? Putting it another way, the successor operation is far far weaker than exponentiation. You have to do a lot of successors to get from 2^4 to 2^5 for example. By analogy, exponentiation is a far more powerful operation than succession in the transfinite realm. That's actually Cohen's heuristic argument for the falsity of CH.

Aleph 2 is the infinity of the curves.

Again no, for the same reason that it depends on CH. The cardinality of all curves in the plane (or in n-space, makes no difference) is 2^2^(Aleph-0). See https://math.stackexchange.com/question ... ty-aleph-2.

It's likely that the persistent misunderstanding that this value is Aleph-2 came originally from a mistake in George Gamow's book One, Two, Three, Infinity as outlined in the Stackexchange link.

It's unknown or unproven that there's an infinity in-between Aleph 0 and Aleph 1.

It's easily proven that there is no such cardinal. Aleph-1 is the very next transfinite cardinal after Aleph-0. There is no cardinal between them. Again, this is a common misunderstanding.

What you are thinking of is CH, the question of whether there is a cardinal strictly between Aleph-0 and 2^(Aleph-0). That's unknown.

But what this means is that for all we know, 2^(Aleph-0) might be some humongous Aleph such as Aleph-4838374873874. That would leave lots of Alephs between those two.

But between Aleph-0 and Aleph-1, there are no cardinals, and that follows directly from the definition of the Alephs.

Two infinite sets are the same size if one-to-one correspondence exists between the two sets (bijection).

"Same size" is misleading, because bijection is a very week metric for equality of size. For example the interval of real numbers [0,1] has the same cardinality as [0,2] and the same cardinality as the entire set of real numbers. But their lengths are very different: 1, 2, and infinity, respectively.

Infinity is NOT a number.

Whatever that means. +infinity and -infinity are numbers in the extended real numbers. We use the extended real numbers to simplify the expression of various statements in calculus and measure theory.

Also, there's an extensive theory of transfinite cardinal and ordinal numbers. Aleph-0 is a number, as is Aleph-47.

So when you say, "infinity is not a number," that claim needs to be put into context.

Finally, I note that I explained these points to you over three years ago in these two threads:

https://forum.philosophynow.org/viewtop ... 26&t=15446

https://forum.philosophynow.org/viewtop ... 26&t=13711

[Philosophy Explorer]

Don't mean to contradict WTF.

If you have facts that contradict me then fire away, maybe I'll learn something.

Here's what I've read from various sources:

You're going to need better sources.

Aleph 0 or Aleph null is the infinity of the natural numbers.

Tru dat.

Aleph 1 is the infinity of the real numbers.

No, that's the content of the Continuum hypothesis (CH). We know that CH is independent of the axioms of ZFC. And what's worse, it's independent of every sensible additional axiom anyone has ever thought of. No matter what naturalistic or reasonable axioms you try to add to set theory, there's a model where CH is true and another model where CH is false.

Nobody knows what the cardinality of the reals is. The best we can do is prove (easily, exercise for the reader) that the cardinality of the reals is 2^(Aleph-0). But nobody knows how big that might be in terms of the Alephs. That is, 2^(Aleph-0) might be Aleph-1, or it might be Aleph-47, or it might be Aleph-(Aleph-47) for all we know.

Interestingly both Gödel, who proved that CH is consistent with ZFC, and Cohen, who proved that CH is independent of ZFC, did not believe that 2^(Aleph-0) = Aleph-1.

There is a plausibility argument (though of course not a proof). For all finite n, 2^n is a lot larger than n. Why should this not be true in the transfinite realm as well? Putting it another way, the successor operation is far far weaker than exponentiation. You have to do a lot of successors to get from 2^4 to 2^5 for example. By analogy, exponentiation is a far more powerful operation than succession in the transfinite realm. That's actually Cohen's heuristic argument for the falsity of CH.

Aleph 2 is the infinity of the curves.

Again no, for the same reason that it depends on CH. The cardinality of all curves in the plane (or in n-space, makes no difference) is 2^2^(Aleph-0). See https://math.stackexchange.com/question ... ty-aleph-2.

It's likely that the persistent misunderstanding that this value is Aleph-2 came originally from a mistake in George Gamow's book One, Two, Three, Infinity as outlined in the Stackexchange link.

It's unknown or unproven that there's an infinity in-between Aleph 0 and Aleph 1.

It's easily proven that there is no such cardinal. Aleph-1 is the very next transfinite cardinal after Aleph-0. There is no cardinal between them. Again, this is a common misunderstanding.

What you are thinking of is CH, the question of whether there is a cardinal strictly between Aleph-0 and 2^(Aleph-0). That's unknown.

But what this means is that for all we know, 2^(Aleph-0) might be some humongous Aleph such as Aleph-4838374873874. That would leave lots of Alephs between those two.

But between Aleph-0 and Aleph-1, there are no cardinals, and that follows directly from the definition of the Alephs.

Two infinite sets are the same size if one-to-one correspondence exists between the two sets (bijection).

"Same size" is misleading, because bijection is a very week metric for equality of size. For example the interval of real numbers [0,1] has the same cardinality as [0,2] and the same cardinality as the entire set of real numbers. But their lengths are very different: 1, 2, and infinity, respectively.

Infinity is NOT a number.

Whatever that means. +infinity and -infinity are numbers in the extended real numbers. We use the extended real numbers to simplify the expression of various statements in calculus and measure theory.

Also, there's an extensive theory of transfinite cardinal and ordinal numbers. Aleph-0 is a number, as is Aleph-47.

So when you say, "infinity is not a number," that claim needs to be put into context.

Finally, I note that I explained these points to you over three years ago in these two threads:

https://forum.philosophynow.org/viewtop ... 26&t=15446

https://forum.philosophynow.org/viewtop ... 26&t=13711

I have several disagreements with what you've said. For now I'll focus on Aleph one which is the size of the set of the real numbers (its cardinality) and can't be put into one-to-one correspondence with the natural numbers. There are several proofs of this, most notably Cantor's diagonal proof.

PhilX

[Eodnhoj7]

Here's something some may take for a paradox.

Take a line that's an inch long with no gaps. Take a square whose sides are an inch long, again with no gaps.
I assert that both objects contain the same number of points and this can be proven geometrically or by Cantor's method of one-to-one correspondence using Cartesian coordinates.

IOW, the number of points is independent of distance. Any comments?

PhilX

Interesting observation...I will present an argument eventually when I have the time.

[wtf]

I have several disagreements with what you've said. For now I'll focus on Aleph one which is the size of the set of the real numbers (its cardinality) and can't be put into one-to-one correspondence with the natural numbers. There are several proofs of this, most notably Cantor's diagonal proof.

PhilX

Phil,

Please give this a good read and let me know if you have any questions.

https://en.wikipedia.org/wiki/Continuum_hypothesis

It's clear that the cardinality of the reals is 2^(Aleph-0), and CH is the claim that this is equal to Aleph-1. But CH is independent of the rest of the axioms, and nobody knows whether it's true; or even if asking if it's true is a meaningful question.

Cantor proved that the cardinality of the reals was strictly greater than that of the natural numbers. He did not, and could not, prove that this cardinality is Aleph-1, for the reason that this assertion is independent of the rest of the axioms of set theory.

In fact it was Cantor himself who first posed the question of CH and was unable to prove it. That historical fact alone shows that Cantor could not have possibly proved that the cardinality of the reals is Aleph-1.

[Arising_uk]

Take a line that's an inch long with no gaps. Take a square whose sides are an inch long, again with no gaps.
I assert that both objects contain the same number of points ...

I thought you said there were no gaps so what's your point?

[Philosophy Explorer]

Take a line that's an inch long with no gaps. Take a square whose sides are an inch long, again with no gaps.
I assert that both objects contain the same number of points ...

I thought you said there were no gaps so what's your point?

That's exactly my point. No gaps means it must be filled, in this case, with points. I maintain that regardless of the size or distance, both objects contain the same set size with respect to their points.

PhilX

[Arising_uk]

That's exactly my point. No gaps means it must be filled, in this case, with points. I maintain that regardless of the size or distance, both objects contain the same set size with respect to their points.

If there are points then there are gaps? How are you defining a "point"?

[Philosophy Explorer]

That's exactly my point. No gaps means it must be filled, in this case, with points. I maintain that regardless of the size or distance, both objects contain the same set size with respect to their points.

If there are points then there are gaps? How are you defining a "point"?

Don't need gaps to have points. In Cartesian geometry, points are defined by coordinates. Since one-to-one correspondence is shown between these points (as shown by Georg Cantor), then these two sets have the same cardinal size.

PhilX

[Arising_uk]

Don't need gaps to have points. In Cartesian geometry, points are defined by coordinates. ...

But if you have points you have gaps? Although I guess the shape would make a difference, can you have square points?

Since one-to-one correspondence is shown between these points (as shown by Georg Cantor), then these two sets have the same cardinal size.

PhilX

Mathematics is beyond me so I'll leave this to you and wtf.

[Philosophy Explorer]

Don't need gaps to have points. In Cartesian geometry, points are defined by coordinates. ...

But if you have points you have gaps? Although I guess the shape would make a difference, can you have square points?

Since one-to-one correspondence is shown between these points (as shown by Georg Cantor), then these two sets have the same cardinal size.

PhilX

Mathematics is beyond me so I'll leave this to you and wtf.

Points are dimensionless. But they don't need gaps - points can lie next to one another on a line with no gaps, but with distinct identities (this involves the notion of limits that calculus handles well).

If math is beyond you, what are you doing in this category? Are you saying every branch of math is beyond your understanding? What are you looking for?

PhilX

[Arising_uk]

Points are dimensionless. ...

If point are dimensionless then you've already answered your own post so what paradox?

But they don't need gaps - points can lie next to one another on a line with no gaps, but with distinct identities (this involves the notion of limits that calculus handles well).

Surely if something is 'dimensionless' it can't lie anywhere?

If math is beyond you, what are you doing in this category? ...

I think the title of this category needs an inclusive or and not a conjunction but I could say the same to you as most of what you post is not Philosophy of Mathematics but Mathematics and should not be here but on a Maths forum. which also applies to most of your posts in the Phil of Science category.

Are you saying every branch of math is beyond your understanding? ...

Depends what you mean by understanding? But pretty much all Maths beyond basic Algebra is outside of my grasp or interest.

What are you looking for?

Logic and to read some Philosophy of Maths.