Post by: Bill S on 07/04/2020 23:42:46
Someone posted this link (thanks). I’ve lost track of it, but wanted to comment.
My question is: If p = t, does this mean that all infinities, countable and uncountable, are the same size?
Post by: evan_au on 08/04/2020 11:45:57
Quote from: OP
If p = t, does this mean that all infinities, countable and uncountable, are the same size?No.
If I have understood the article correctly, p and t are two different ways of constructing an infinite set whose size is larger than the countable numbers.
- They both create subsets of the natural numbers, but using different rules.
- And it turns out that p and t have the same (infinite) number of elements
But the size of p and t are both bigger than the number of integers (which is a smaller infinity)
- So it does not show that all infinities are the same size
p=t is a very particular proof, which doesn't affect Cantor's general method of creating larger and larger infinities.
see: https://en.wikipedia.org/wiki/Cantor%27s_theorem
Post by: Bill S on 08/04/2020 14:00:40
Post by: evan_au on 08/04/2020 22:49:37
Quote from:
if p & t are larger than countable infinities; does this mean that they include uncountable infinities?p & t are uncountable infinities.
But there are uncountable infinities that are larger than p & t.
- One is the number of algebraic functions on a plane.
As the article stated, we know that some infinities are bigger than others, but we are not sure whether there are infinities that lie "between" the infinities we know about.
- It all comes down to counting: trying to make a 1-to-1 correspondence between two very different things, and succeeding (or else proving that there is no way that this can be achieved).
- As the tale of p & t illustrates, in many cases the answer is "we don't know", and it can stay that way for many decades
Post by: alancalverd on 08/04/2020 23:00:49
Post by: Bill S on 09/04/2020 14:42:24
Quote from: Evan
p & t are uncountable infinities…………- It all comes down to counting: trying to make a 1-to-1 correspondence between two very different things, and succeeding
I thought uncountable infinities could not be put in 1 to 1 relationships. Isn't that what makes them uncountable?
Post by: Bill S on 09/04/2020 15:06:16
Quote from: Alan
All infinities are the same size, but some are larger than others.
Possibly the most apposite comment yet.
Post by: evan_au on 09/04/2020 23:54:52
Quote from: Bill S
I thought uncountable infinities could not be put in 1 to 1 relationships. Isn't that what makes them uncountable?Uncountable infinities cannot be put into a 1 to 1 relationship with the natural numbers (ie the counting numbers).
However, uncountable infinities can be put into a 1 to 1 relationship with other uncountable infinities (if they represent the same uncountable infinity).
This is exactly what has happened with p and t, which are both uncountable infinities.
- You need a generic method for selecting the "partner", which works for all members of the set, even if that set has an uncountable number of members.
- Which is why it is hard!
Post by: jeffreyH on 10/04/2020 22:14:46
https://www.google.com/amp/s/phys.org/news/2020-01-indeterminist-physics-world.amp (https://www.google.com/amp/s/phys.org/news/2020-01-indeterminist-physics-world.amp)
Post by: Bill S on 10/04/2020 22:20:14
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- Which is why it is hard!
I’ve no problem believing that.
A problem I see with 1 to 1 correspondence is its limited application to infinities.
An example from the article:
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If there is exactly one driver for each car, with no empty cars and no drivers left behind, then you know that the number of cars equals the number of drivers (even if you don’t know what that number is).
The crucial point here is the comment in parenthesis. It says that the 1 to 1 placing gives clear information about the finite example, but tells us nothing beyond that.
Can we extrapolate to infinity?
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(even if you don’t know what that number is)
This indicates that any extrapolation beyond the example involves a number, but infinity is not a number. Therefore, the best we can say is that our extrapolation works as long as there is a number to which it can be applied.
From our finite perspective, we can see no “final” number, so for convenience, we say that we can extrapolate infinitely.
I have no problem with this, as long as we are clear as to the way in which we are using the word “infinity.
Post by: Bill S on 10/04/2020 22:23:57
Post by: evan_au on 10/04/2020 23:06:42
Quote from: Bill S
(even if you don’t know what that number is): It says that the 1 to 1 placing gives clear information about the finite example, but tells us nothing beyond that.Giving an example about cars and drivers is obviously a finite case.
The amazing thing about Cantor's work is that 1 to 1 correspondence also works for infinite cases (including uncountable infinities).
The trick is that you don't count the drivers and then count the cars, because you are:
- placing the drivers in a 1 to 1 relationship with the natural numbers
- then placing the cars in a 1 to 1 relationship with the natural numbers
- and then comparing the two natural numbers
- which obviously only works for finite natural numbers
- and which can only be extended as far as countable infinities
To cope with non-countable infinities, you must place them in 1 to 1 correspondence with other non-countable infinities
Quote from: link
"These [real] numbers are characterized by an infinite number of decimals that follow the dot," says Nicolas Gisin, ... "This implies that they contain an infinite amount of information."It is true that you could represent all the information in Wikipedia (at a certain point in time) as a real number.
But it is also true that some non-repeating real numbers can be represented very compactly. For example the value of pi can be calculated as:
π = 4(1-1/3+1/5-1/7...)
If a number can be represented by a compact formula like this, then it contains very little information.
- Even though there are websites that offer to find your name (or any other numerical or text string) in the digits of π
See: https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80
http://www.dr-mikes-math-games-for-kids.com/your-name-in-pi.html
Post by: jeffreyH on 11/04/2020 11:22:08
Looks an interesting link, Jeffrey, hopefully, I'll find a few minutes to read it later.
It brings indeterminacy into classical physics. This makes classical physics and quantum mechanics easier to marry together. This means it also applies to relativity. Pretty promising.
Post by: Bill S on 12/04/2020 16:42:12
Quote from: Evan
The trick is that you don't count the drivers and then count the cars, because you are:
- placing the drivers in a 1 to 1 relationship with the natural numbers
- then placing the cars in a 1 to 1 relationship with the natural numbers
- and then comparing the two natural numbers
- which obviously only works for finite natural numbers
- and which can only be extended as far as countable infinities
That makes good sense, even to my non-mathematical mind. Theoretically/mathematically, all good stuff, I would not wish to dispute its value, even were I qualified to do so. However, as I understand it, physical observation and experimental results tend to “trump” theory, and even mathematical “reality”. Extrapolating 1 to 1 relationships to some mathematical “infinity” requires only mathematical verification. However, if we accept that there is an expression of infinity that is not a number and that is not a physically reachable goal, then any attempt to extrapolate to that must involve assumptions that cannot be tested.
Post by: Bill S on 12/04/2020 20:20:35
Quote
https://www.google.com/amp/s/phys.org/news/2020-01-indeterminist-physics-world.amp
Still struggling for time, but want to have a go at understanding. Approach will have to be piecemeal. Comments on this as the essence of the first bit, please.
Classical physics is deterministic. It is characterised by precise equations that leave no room for randomness in the progression from BB to present. This does not necessarily equate to our observations of the world.
These equations describe the world as evolving in a precise, predetermined, way; using mathematics to represent the initial conditions as real numbers. These numbers are characterized by an infinite number of decimal places that follow the point. This implies that they contain an infinite amount of information. These numbers involve a series of random decimals, which, although not encounter in everyday life, are an accepted feature of classical maths and are commonly used in physics. This raises the question: how can a finite world, be described by numbers that are infinite and that feature an infinite amount of information?
Post by: evan_au on 12/04/2020 23:28:51
Quote from: Bill S
Classical physics is deterministic... It is characterised by precise equations that leave no room for randomnessWould you consider Boltzmann's gas equations as "Classical" or "Modern"?
They were based on statistical properties of particles in a gas.
See: https://en.wikipedia.org/wiki/Boltzmann_equation
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real numbers. These numbers are characterized by an infinite number of decimal places that follow the point. This implies that they contain an infinite amount of informationIf real properties of nature are statistical, then all the digits which change at +/-1 standard deviation are nonsense (lost in the noise).
- This limits the amount of information in a number
- The Heisenberg Uncertainty Principle places just such a constraint on how accurately you can measure things.
If you tried to measure pi as the ratio of the circumference and diameter of a real circle, you would run into quantum properties of atoms before you got to 10 digits
- And if you tried to use a larger circle (light-years across), you would run into distortions in spacetime due to stars and planets, again limiting your accuracy
- And even when an infinite number of digits are exact (as in the Pure Mathematics definition of pi), this actually carries very little information (as shown by the Liebnitz series above).
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if we accept that there is an expression of infinity that is not a number and that is not a physically reachable goal, then any attempt to extrapolate to that must involve assumptions that cannot be tested.Let's take an example that does involve natural numbers, because we learn how to place them in sequence by the time we turn 10 years old. It's a thought experiment called "Hilbert's Hotel".
It is a hotel used by an intergalactic mathematics conference, and it has an infinite number of rooms, numbered 1, 2, 3, .... etc.
- All the (countable) infinite number of mathematicians are settled into their (countable) infinite number of rooms - no mathematicians left without a room, and no rooms without a mathematician.
- So the number of mathematicians equals the number of rooms (1 to 1 correspondence).
- Then there is a late arrival - another mathematician arrives
- The front desk gets on the speaker system and says "Could the guest in room 1 please move into room 2, the guest in room 2 move into room 3, and in general, could the guest in room n please move into room n+1".
- This leaves room 1 vacant, and the late arrival moves into room 1
- This shows that ∞=∞+1
- An inventive reception desk can even handle the case where an infinite number of mathematicians arrive late...
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cannot be tested.According to your concern, this cannot be tested.
- However, for every room number you give me, I can tell you which room they must move to
- So the test passes - it is testable!
- That is the point of a mathematical formula - you can make an infinite number of true statements in a very compact form, like "n→n+1".
- And prove that they are true (to a Pure Mathematician)
- This does not prove that the universe is infinite, or that it contains an infinite number of atoms
- Applied Mathematicians accept these finite limits as constraints on the solution provided by a Pure Mathematician.
See: https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel
Post by: alancalverd on 13/04/2020 02:38:07
Post by: hamdani yusuf on 13/04/2020 10:54:53
Quote
In their new work, Malliaris and Shelah resolve a related 70-year-old question about whether one infinity (call it p) is smaller than another infinity (call it t). They proved the two are in fact equal, much to the surprise of mathematicians.Let's say that the number of natural number = ~
Let's say that the number of real number between 0 and 1 = ~
Hence the number of real number = ~ * ~
= ~
Post by: evan_au on 13/04/2020 12:04:31
Quote from: hamdani yusuf
Let's say that the number of natural number = ~Yes, you can say it.
Let's say that the number of real number between 0 and 1 = ~
But Malliaris and Shelah are mathematicians, and (presumably) can prove it.
Just saying it does not make it so...
See: https://en.wikipedia.org/wiki/Transfinite_number
Post by: Bill S on 13/04/2020 12:33:15
Hilbert’s Hotel is an excellent example of the point I was trying to make about 1 to 1 correspondence. We’ve been there before, #191.
https://www.thenakedscientists.com/forum/index.php?topic=48702.msg419946#msg419946
Post by: hamdani yusuf on 13/04/2020 14:37:25
Which point of my argumentation above do you find to be incorrect? Do you know how to improve it?Quote from: hamdani yusufLet's say that the number of natural number = ~Yes, you can say it.
Let's say that the number of real number between 0 and 1 = ~
But Malliaris and Shelah are mathematicians, and (presumably) can prove it.
Just saying it does not make it so...
See: https://en.wikipedia.org/wiki/Transfinite_number
Arguments from authority is a form of logical fallacy, although it's often useful as shortcut when we don't want to spend some time to think through the problem.
Post by: Bill S on 13/04/2020 17:39:30
Quote from: Alan
Need to differentiate between randomness (could be here, could be there, at this moment) and indeterminacy (it's everywhere at the same time, just more likely to interact in these areas).
That’s a distinction worth remembering.
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But neither particularly relates to infinities.
Probably not a bad thing. Infinities can open the door to all kinds of crackpots. I have to say, though, that I have learned quite a lot by climbing on my “infinite soapbox”.
Post by: Bill S on 13/04/2020 17:59:52
Quote from: Wiki
"Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite."
So, transfinite numbers are infinite numbers, but are not necessarily infinite, which is probably a good thing, if infinity is not a number!
This might mean that “transfinite” could be used as a number, with impunity; but replacing “infinity” with “transfinity” could lead to some raised eyebrows.
Post by: evan_au on 14/04/2020 00:04:32
Quote from: hamdani yusuf
Arguments from authority is a form of logical fallacy"Arguments from a non-expert who just made it up on the spot" is also a logical fallacy.
Post by: hamdani yusuf on 14/04/2020 01:20:46
There are non-zero probability for non-experts to make correct guess. You can tell by scrutinizing their arguments. So, let me hear your opinion about them.Quote from: hamdani yusufArguments from authority is a form of logical fallacy"Arguments from a non-expert who just made it up on the spot" is also a logical fallacy.
My point is that my result agrees with those mathematicians, although it's lacking the element of surprise.
Post by: evan_au on 14/04/2020 12:53:30
Quote from: hamdani yusuf
My point is that my result agrees with those mathematiciansI'm not an expert either - but I was interested in the snippet of Cantor's work I studied at university, and , since then I've looked into it a bit when it came up.
The mathematicians who proved that "p=t" were comparing two sets which group the natural numbers in two different ways, and counting the number of elements in each set. As I understand the story, it was already know that p and t were uncountable infinities. In the recent work, they found that p and t were the same uncountable infinity.
Quote from: hamdani yusuf
Let's say that the number of natural number = ~The number of natural numbers is a countable infinity.
Let's say that the number of real number between 0 and 1 = ~
Hence the number of real number = ~ * ~ = ~
The number of fractions between 0 and 1 is a countable infinity.
This suggests that the two quantities you mention are not the same, and you should not use the same symbol "~" for them.
As one non-expert to another, in what way does your "result agree with those mathematicians"?
Neither "natural numbers", or "real numbers between 0 and 1" is a grouping of the natural numbers (as are p and t).
- As one non-expert to another, in what way does your result have anything to do with the work of those mathematicians?
*Thanks for providing the proof below, Bored Chemist!
Post by: chiralSPO on 14/04/2020 15:36:45
Screen Shot 2020-04-14 at 10.42.13 AM.png (15.17 kB . 394x306 - viewed 4877 times)In other words: if you think there are more slopes than x values in the setup described above, choose any real slope, and I will find you the only x value that gives it. Likewise, if for some bizarre reason you think that there should be more x values than slopes, I challenge you to find an x value in the domain that does not uniquely correspond to a slope.
Post by: hamdani yusuf on 15/04/2020 03:37:38
in what way does your "result agree with those mathematicians"?p=t
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The problem was first identified over a century ago. At the time, mathematicians knew that “the real numbers are bigger than the natural numbers, but not how much bigger. Is it the next biggest size, or is there a size in between?”https://www.quantamagazine.org/mathematicians-measure-infinities-find-theyre-equal-20170912/
The number of natural numbers is a countable infinity.Maybe the term countable infinity is an oxymoron, just like married bachelor.
The number of fractions between 0 and 1 is a countable infinity.