Different sizes of infinity...?
- Thread starter Killjoy
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Okay. But that renders it somewhat tangential to the matter at hand, doesn't it. Measurement isn't the same as maths, as you admit.
There is quantitative value in the notion of Alephs, and cardinalities. It is also clearly defined what those quantities mean and relate to.
The numbers don't involve measurement. Numbers are mathematical objects representing a quantity relative to a unit (1). No measurement is involved. Just a statement. 5 doesn't require measurement. When we measure something in the real world we assign a number to that measurement to reflect the quantity relative to a unit of that measurement. So a 5cm length is 5 times bigger than the unit of 1cm. But the numbers themselves are not measured, and do not involve measurement.
So I'm not sure what you're trying to say, as it seems confused.
It's not about human experience, which is necessarily grounded in the real world, but in concepts that can move beyond our ability to directly experience them. The idea of the infinite is such a concept. And infinite sets, such as the set of positive integers, can be considered as a whole.
There is no last element. But, if you do not think it is complete, which element do you think is missing? If you can provide an answer to that, then you have grounds for not considering it complete. A single example of an element that is not included within the set, please?
Maths moves beyond reality and experience. It is not confined with that which is real. Heck, many philosophers would argue that numbers themselves are not real. Maths can certainly help model reality. That is what science is good at: modelling based on observations in order to predict and confirm when the prediction happens etc. But we can also model unreal things. We could model a space of 200-dimensions. We could model it, describe it, perform predictions on it as to how things would behave inside it etc. But it wouldn't be real. Maths is not bound by the need to match reality. It can deal in abstract concepts. Such as infinity.
To disagree on things you first need to understand the basics. And you don't seem to have that, yet. Wikipedia would help you in that regard.
Given sequence s0 = all 0's and its complement s1 = all 1's.
Left, both are parallel with no interference.
Center, both are parallel with no interference.
Right, s0 is diagonal and s1 is horizontal, and interfere at the diagonal.
Both orientations cannot exist in the same list.
Which sequence is correct?
Since the list initially can consist of horizontal or diagonal sequences, but not both,
the conflict can be resolved with the removal of Cantor's definition of E based on the diagonal as in the right form.
All the lists represents Cantor's transfinite set M.
If you don't see this, maybe you have a vision problem.
Left, both are parallel with no interference.
Center, both are parallel with no interference.
Right, s0 is diagonal and s1 is horizontal, and interfere at the diagonal.
Both orientations cannot exist in the same list.
Which sequence is correct?
Since the list initially can consist of horizontal or diagonal sequences, but not both,
the conflict can be resolved with the removal of Cantor's definition of E based on the diagonal as in the right form.
All the lists represents Cantor's transfinite set M.
If you don't see this, maybe you have a vision problem.
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This is extremely odd. You are inventing your own process, then using it to try to argue against something else.
The process is:
1: Make a list of sequences
2: Use a method to make another sequence
3: By understanding the method, figure out if that new sequence could be a member of the list
What you do is try to smush up all the steps together and make false comparisons.
You are the only person trying to do what's in your right-side picture.
The list is a list of sequences, so yes, seen in grid form they are "horizontal".
But again (and again and again), the diagonal is just used to select members of a new "horizontal" sequence.
The point is to see if the new sequence could be an already existing member of the list.
That a method involving some diagonal was used to generate that new sequence is totally irrelevant.
There is no conflict if following Cantor's method. It is only you stuck on the idea that the method of selecting elements for the new sequence, using the diagonal, is some issue.
When you are the only person seeing an issue, maybe the vision problem is yours.
Taking the left side picture, Cantor's method would get:
D = 01...
So then we get:
E = 10...
From that we know:
E cannot be row u=1 because it will be different where column v=1, and:
E cannot be row u=2 because it will be different where column v=2, and so on
And at that point we see your middle and right images have absolutely nothing to contribute.
Whatever they show simply isn't relevant.
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Aye. He's arguing a straw man. Whether this is deliberate on his part, or just a failure on his part to understand what Cantor was doing/saying, it seems a futile exercise from this point out.
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True that.
I've been wondering (before he mentioned "vision") if it's some kind of mathematical "synaesthesia". (I'm using that word very loosely.) He's seeing something in the diagonal that nobody else sees.
I do like trying to explain something, however, as it helps me understand better too. I was thinking of trying to suggest a different way to select the elements of E that doesn't use the nice simple diagonal so directly. Maybe a random number generator, with a history, so it doesn't repeat. Then thought of something based on modulus arithmetic, e.g. with 3's:
10101010...
01010101...
11001100...
00110011...
...
111... (E)
000... (D)
It made clear to me that as there are infinite possibilities for the modulus, this would give infinite E's that are not in the infinite list. Nice loop back to the start of this thread. (If anyone other than phyti thinks I'm wrong there, happy to be corrected.)
Had to retrieve this from a few years ago.
Cantor used a list to represent the infinite set M.
His definition of the diagonal D and its complement E could not coexist in the list.
He erroneously transferred that failure to the set M, which is independent of the list.
This binary tree is a more accurate representation of M.
It branches into 2 subsets forever.
The ma is the mirror axis showing the symmetry. For each sequence there is a complementary sequence (red). The occur in pairs.
All sequences are independent of each other, 1-dimensional, and none intersect.
There are no missing elements.
Cantor used a list to represent the infinite set M.
His definition of the diagonal D and its complement E could not coexist in the list.
He erroneously transferred that failure to the set M, which is independent of the list.
This binary tree is a more accurate representation of M.
It branches into 2 subsets forever.
The ma is the mirror axis showing the symmetry. For each sequence there is a complementary sequence (red). The occur in pairs.
All sequences are independent of each other, 1-dimensional, and none intersect.
There are no missing elements.
You're missing the point (still).
Cantors' set wasn't the set of all possible infinite sequences of 0's and 1's. If he'd started with that set and proved a sequence wasn't in it, he'd have had a paradox.
What he started with was an infinite set of sequences of 0's and 1's. And found a way to generate a sequence that provably wasn't in it.
It's a bit like comparing the set of { natural numbers } and { natural numbers except 77 }. Both sets are infinite. But "infinite" doesn't equal "all possible".
(Selecting a depth, writing down that many sequences from your binary tree, applying Cantor's method: you'll get a sequence that isn't in that subset. Add 1 to the depth, you'll get the same result. Forever.)
Cantors' set wasn't the set of all possible infinite sequences of 0's and 1's. If he'd started with that set and proved a sequence wasn't in it, he'd have had a paradox.
What he started with was an infinite set of sequences of 0's and 1's. And found a way to generate a sequence that provably wasn't in it.
It's a bit like comparing the set of { natural numbers } and { natural numbers except 77 }. Both sets are infinite. But "infinite" doesn't equal "all possible".
(Selecting a depth, writing down that many sequences from your binary tree, applying Cantor's method: you'll get a sequence that isn't in that subset. Add 1 to the depth, you'll get the same result. Forever.)
Sarkus#162;
"No measurement is involved. Just a statement. 5 doesn't require measurement. When we measure something in the real world we assign a number to that measurement to reflect the quantity relative to a unit of that measurement."
Counting is the most basic form of measurement. It answers the question of 'how many ?'. It was a practical necessity for many human activities, census, business, travel, etc.
Many languages used pictorial symbols, Chinese, Japanese, Egyptian, etc.
The natural or counting numbers were represented by symbols which stood for sets of abstract things that had no properties. The numeral 5 stands for the set {1, 1, 1, 1, 1}. Any small set can be estimated visually from experience.
Large sets have to be counted, and there are varied ways of doing that. The goal is to match a set of things to a predefined set from N, the integers.
"There is no last element. But, if you do not think it is complete, which element do you think is missing? If you can provide an answer to that, then you have grounds for not considering it complete. A single example of an element that is not included within the set, please?"
There are no missing elements if a set contains all possible elements, but they may be potential elements.
The set N is only infinite because the operation of adding 1 to a suggested largest integer forms a larger integer. You have to produce a larger integer to prove it exists. (The constructivist view.)
The particle physicist has to present evidence of a hypothetical particle from an experiment, not because they think it should exist.
Before Newton people thought light moved instantaneously from here to there and there was a universal time. Astronomers proved them wrong.
"Maths is not bound by the need to match reality. It can deal in abstract concepts. Such as infinity."
Cantor said something similar,
"… the essence of mathematics lies entirely in its freedom".
Source: Ewald, W., From Kant to Hilbert, Oxford 1996.
Applied mathematics is essential in today's complex world.
Abstract mathematics is fine for fantasy.
"No measurement is involved. Just a statement. 5 doesn't require measurement. When we measure something in the real world we assign a number to that measurement to reflect the quantity relative to a unit of that measurement."
Counting is the most basic form of measurement. It answers the question of 'how many ?'. It was a practical necessity for many human activities, census, business, travel, etc.
Many languages used pictorial symbols, Chinese, Japanese, Egyptian, etc.
The natural or counting numbers were represented by symbols which stood for sets of abstract things that had no properties. The numeral 5 stands for the set {1, 1, 1, 1, 1}. Any small set can be estimated visually from experience.
Large sets have to be counted, and there are varied ways of doing that. The goal is to match a set of things to a predefined set from N, the integers.
"There is no last element. But, if you do not think it is complete, which element do you think is missing? If you can provide an answer to that, then you have grounds for not considering it complete. A single example of an element that is not included within the set, please?"
There are no missing elements if a set contains all possible elements, but they may be potential elements.
The set N is only infinite because the operation of adding 1 to a suggested largest integer forms a larger integer. You have to produce a larger integer to prove it exists. (The constructivist view.)
The particle physicist has to present evidence of a hypothetical particle from an experiment, not because they think it should exist.
Before Newton people thought light moved instantaneously from here to there and there was a universal time. Astronomers proved them wrong.
"Maths is not bound by the need to match reality. It can deal in abstract concepts. Such as infinity."
Cantor said something similar,
"… the essence of mathematics lies entirely in its freedom".
Source: Ewald, W., From Kant to Hilbert, Oxford 1996.
Applied mathematics is essential in today's complex world.
Abstract mathematics is fine for fantasy.
phyti hasn't been acting like himself for a while now. He used to be very clued-in on mathematics. But lately, he's sort of all over the shop, posting arguments that are incoherent.
I started noticing his odd behaviour when he was seemingly unable to understand the Monty Hall problem. That was not the on-the-ball phyti I've been used to seeing on sciforums in the past.
I wonder if something has happened to him. Mental breakdown? Senility setting in? Or is he sharing his account with somebody else - somebody who lacks his capacity to understand maths?
Whatever the explanation, he seems to have lost the ability to follow at least some types of logical, mathematical arguments.
It's very odd.
phyti:
Rather than just talking about you, it seems more polite to talk to you.
What's up with you, man? You seem to have lost your maths mojo, lately. Have you had some kind of accident? Is there some other reason? Or are you not aware of any change?
An element is either in a set or it isn't in the set, if the set is well defined. Right?
Do you not accept this? (It can be proven by induction.)
I'm not sure I understand your complaint.
Are you just saying you don't like "abstract mathematics", all of sudden? If so, that strikes me as out of character for you.
Why this sudden preference for the concrete over the abstract? What has happened to cause you to suddenly take offence at abstract mathematics (if that's what we're seeing from you here)?
All "pure" maths is abstract, isn't it?
Rather than just talking about you, it seems more polite to talk to you.
What's up with you, man? You seem to have lost your maths mojo, lately. Have you had some kind of accident? Is there some other reason? Or are you not aware of any change?
Sure. But a set is a mathematical abstraction. It is not a form of measurement.
Yes.
Not necessarily. It can certainly represent the cardinality of that set.
That's very vague. What do you mean by "estimating" a set?
Some of them can be counted. Others can't be. The terms "countable" and "uncountable" have specific meanings in mathematics.
You can certainly determine whether a set is countable or uncountable that way.
What's a "potential element"?
An element is either in a set or it isn't in the set, if the set is well defined. Right?
It's infinite because there's no largest integer.
The operation of adding 1 to an integer will always produce a larger integer.
Do you not accept this? (It can be proven by induction.)
The corresponding process in mathematics is called "providing a proof". It applies to theorems and such.
They proved that a particular theoretical model does not correctly predict what is observed in nature, certainly.
Fantasies are typically unconstrained by the desirability of logical proof. In comparison, logical proofs are available in many areas of mathematics - including for sets and natural numbers.
I'm not sure I understand your complaint.
Are you just saying you don't like "abstract mathematics", all of sudden? If so, that strikes me as out of character for you.
Why this sudden preference for the concrete over the abstract? What has happened to cause you to suddenly take offence at abstract mathematics (if that's what we're seeing from you here)?
All "pure" maths is abstract, isn't it?
James R;
'All "pure" maths is abstract, isn't it?"
Yes. It's a practical convention used for many purposes.
The issue with Cantor is so many accept his idea of transfinite numbers based on his supposed proof.
Response here is arguing the details of the problem even when presented in Cantor's own words.
Given his definitions, he creates his own contradiction.
The list allows him to exclude E the complement of D (diagonal).
The binary tree is a correct representation of the infinite set M.
The failure of the list does not imply a failure in the set M.
The tree as the set M does not allow a sequence D to insect its complement E.
The symmetry of the tree reveals the sequences occur in pairs.
If E is excluded so is D.
'All "pure" maths is abstract, isn't it?"
Yes. It's a practical convention used for many purposes.
The issue with Cantor is so many accept his idea of transfinite numbers based on his supposed proof.
Response here is arguing the details of the problem even when presented in Cantor's own words.
Given his definitions, he creates his own contradiction.
The list allows him to exclude E the complement of D (diagonal).
The binary tree is a correct representation of the infinite set M.
The failure of the list does not imply a failure in the set M.
The tree as the set M does not allow a sequence D to insect its complement E.
The symmetry of the tree reveals the sequences occur in pairs.
If E is excluded so is D.
pzkpfw#
"You're missing the point (still)."
1. "Cantors' set wasn't the set of all possible infinite sequences of 0's and 1's. If he'd started with that set and proved a sequence wasn't in it, he'd have had a paradox."
2. "What he started with was an infinite set of sequences of 0's and 1's. And found a way to generate a sequence that provably wasn't in it."
It's a bit like comparing the set of { natural numbers } and { natural numbers except 77 }. Both sets are infinite. But "infinite" doesn't equal "all possible".
3. (Selecting a depth, writing down that many sequences from your binary tree, applying Cantor's method: you'll get a sequence that isn't in that subset. Add 1 to the depth, you'll get the same result. Forever.)
1. He used m and w in his example. His text in blue.
Let M be the totality [Gesamtheit] of all elements E.
EI = (m, m, m, m, … ),
EII = (w, w, w, w, … ),
EIII = (m, w, m, w, … ).
I replaced {m, w} with {0, 1} since there is more contrast.
2. He used the diagonal as a template and formed its complement E0.
The complement being exchanging any 0 for 1 and any 1 for 0.
If E0 wasn't in the list, neither was D. They occur in pairs.
Then there is a series b1, b2, … bv,…, defined so that bv is also equal to m or w but is different from av,v. Thus, if av,v = m, then bv = w.
Then consider the element
E0 = (b1, b2, b3, …)
of M, then one sees straight away, that the equation
E0 = Eu
cannot be satisfied by any positive integer u, otherwise for that u and for all values of v.
bv = au,v
and so we would in particular have
bu = au,u
which through the definition of bv is impossible.
3. Cantor refers to the totality of all infinite sequences, sets with no last element. That's why his manipulation of the diagonal beginning at u1 and intersecting all horizontal sequences produces his desired result of a missing complement. But that is a shortcoming of the list, not the binary tree.
Beginning at M0, always moving up gets a string of 0's.
Beginning at M1, always moving down gets a string of 1's.
The remainder of sequences are mixtures of 0 and 1.
There are always 2 choices. The set M contains itself at each branch.
Orientation affects the list. There is only 1 (horizontal) orientation for the binary tree.
"You're missing the point (still)."
1. "Cantors' set wasn't the set of all possible infinite sequences of 0's and 1's. If he'd started with that set and proved a sequence wasn't in it, he'd have had a paradox."
2. "What he started with was an infinite set of sequences of 0's and 1's. And found a way to generate a sequence that provably wasn't in it."
It's a bit like comparing the set of { natural numbers } and { natural numbers except 77 }. Both sets are infinite. But "infinite" doesn't equal "all possible".
3. (Selecting a depth, writing down that many sequences from your binary tree, applying Cantor's method: you'll get a sequence that isn't in that subset. Add 1 to the depth, you'll get the same result. Forever.)
1. He used m and w in his example. His text in blue.
Let M be the totality [Gesamtheit] of all elements E.
EI = (m, m, m, m, … ),
EII = (w, w, w, w, … ),
EIII = (m, w, m, w, … ).
I replaced {m, w} with {0, 1} since there is more contrast.
2. He used the diagonal as a template and formed its complement E0.
The complement being exchanging any 0 for 1 and any 1 for 0.
If E0 wasn't in the list, neither was D. They occur in pairs.
Then there is a series b1, b2, … bv,…, defined so that bv is also equal to m or w but is different from av,v. Thus, if av,v = m, then bv = w.
Then consider the element
E0 = (b1, b2, b3, …)
of M, then one sees straight away, that the equation
E0 = Eu
cannot be satisfied by any positive integer u, otherwise for that u and for all values of v.
bv = au,v
and so we would in particular have
bu = au,u
which through the definition of bv is impossible.
3. Cantor refers to the totality of all infinite sequences, sets with no last element. That's why his manipulation of the diagonal beginning at u1 and intersecting all horizontal sequences produces his desired result of a missing complement. But that is a shortcoming of the list, not the binary tree.
Beginning at M0, always moving up gets a string of 0's.
Beginning at M1, always moving down gets a string of 1's.
The remainder of sequences are mixtures of 0 and 1.
There are always 2 choices. The set M contains itself at each branch.
Orientation affects the list. There is only 1 (horizontal) orientation for the binary tree.
Yes of course. I used them myself in post #48 if you recall. I used 0 and 1 here because that's the current notation we're using.
Yes; you, wikipedia, and me (have you tried the program in post #60 yet?)
I don't know where you get "neither was D" from. Nobody says this, and it isn't the case. (D might be, it might not be. Cantors' method has nothing to say about that.)
My second example in post #48 ( https://www.sciforums.com/threads/different-sizes-of-infinity.167343/post-3788188 ) even showed: the D from the second list is also a member of the second list. (It's also the E from the first list.)
That's no issue at all.
You don't really respond to what you numbered 2.
'It's a bit like comparing the set of { natural numbers } and { natural numbers except 77 }. Both sets are infinite. But "infinite" doesn't equal "all possible"'. What do you say to that? See also below ...
Orientation again! You are missing something extremely important. (Though I've seen this pointed out to you before, years ago.)
From: (for Cantor to English) https://jamesrmeyer.com/infinite/cantors-original-1891-proof
[ I now assert that such a set M does not have the magnitude of the series 1, 2, 3, …, v, …, . This follows from the following proposition:
If E1, E2, …, Ev is any infinite series of elements of the set M, then there always exists an element E0 of M, which cannot be the same element as any element Ev. ]
Blue: "of" - the list we're looking at is infinite, but it isn't M.
Red: some element of M. M is all of them - that element is in M. But the actual question is ...
Green: can we show that there is an element from M that isn't in the E1, E2, …, Ev (Blue) set?
The interesting thing being that E1, E2, …, Ev is infinite. Not containing "everything" is interesting. (Maybe less when you realise that { natural numbers except 77 } is infinite.)
From: (for common understanding) https://en.wikipedia.org/wiki/Cantor's_diagonal_argument
[ If s1, s2, ... , sn, ... is any enumeration of elements from T,[note 3] then an element s of T can be constructed that doesn't correspond to any sn in the enumeration. ]
If you check Note 3 it says (my bold) [ Cantor does not assume that every element of T is in this enumeration. ].
Blue: the enumeration is from T (i.e. M) but isn't T itself.
Red: but we want to see if some element of T is or isn't in the enumeration
Green: unlike your previous "orientation" complaints, the constructed element is a perfectly good member of T (i.e. M).
I am happy to agree that your binary tree represents the members of M (or T ...) - but that's not relevant.
Cantor wasn't trying to show there's an E that isn't in M.
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phyti:
If you can't show that Cantor was wrong, then I don't see any reason to believe you over him and all the expert mathematicians who have looked at his proofs.
He constructs a set with an infinite number of elements (e.g. an infinite list of numbers) that has cardinality equal to that of the natural numbers, and then shows that there is at least one element that does not appear in that list. He concludes, therefore, that there must be sets that have cardinality greater than that of the set of natural numbers.
In other words, there are distinguishable infinities. Different infinite sets can potentially have different cardinalities. Those cardinalities are the transfinite numbers.
Well, I'm happy to go with the experts on this one. The consensus among mathematicians is that Cantor's proofs of transfinite numbers are solid.
If you can't show that Cantor was wrong, then I don't see any reason to believe you over him and all the expert mathematicians who have looked at his proofs.
Exactly! That's the point. It's a proof by contradiction. Didn't you realise?
He constructs a set with an infinite number of elements (e.g. an infinite list of numbers) that has cardinality equal to that of the natural numbers, and then shows that there is at least one element that does not appear in that list. He concludes, therefore, that there must be sets that have cardinality greater than that of the set of natural numbers.
In other words, there are distinguishable infinities. Different infinite sets can potentially have different cardinalities. Those cardinalities are the transfinite numbers.
I don't understand what you're talking about. What are E and D?
What is M?
If you say so. What kind of "failure" are you referring to?
I don't know what that means.
Which sequences?
You seem to be talking about something different to what I have been talking to you about.
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pzkpfw;
I enlarged the Cantor text because it becomes difficult to read with subscripts.
Cantor:
Namely, let m and n be two different characters, and consider a set [Inbegriff] M of elements
E = (x1, x2, … , xv, …)
which depend on infinitely many coordinates x1, x2, … , xv, …, and where each of the coordinates is either m or w. Let M be the totality [Gesamtheit] of all elements E.
To the elements of M belong e.g. the following three:
EI = (m, m, m, m, … ),
EII = (w, w, w, w, … ),
EIII = (m, w, m, w, … ).
I maintain now that such a manifold [Mannigfaltigkeit] M does not have the power of the series 1, 2, 3, …, v, ….
This follows from the following proposition:
"If E1, E2, …, Ev, … is any simply infinite [einfach unendliche] series of elements of the manifold M, then there always exists an element E0 of M, which cannot be connected with any element Ev."
comment 1. M is the set of all E's, and has a cardinality different from N the set of all integers, 'totality' meaning complete.
Cantor:
For proof, let there be
E1 = (a1.1, a1.2, … , a1,v, …)
E2 = (a2.1, a2.2, … , a2,v, …)
Eu = (au.1, au.2, … , au,v, …)
………………………….
where the characters au,v are either m or w. Then there is a series b1, b2, … bv,…, defined so that bv is also equal to m or w but is different from av,v.
Thus, if av,v = m, then bv = w.
comment 2. The bv elements are compared to the diagonal elements avv and altered to differ. I label the diagonal sequence of a's D for my own convenience to identify the sequences involved.
Cantor:
Then consider the element
E0 = (b1, b2, b3, …)
of M, then one sees straight away, that the equation
E0 = Eu
cannot be satisfied by any positive integer u, otherwise for that u and for all values of v.
bv = au,v
and so we would in particular have
bu = au,u
which through the definition of bv is impossible. From this proposition it follows immediately that the totality of all elements of M cannot be put into the sequence [Reihenform]: E1, E2, …, Ev, … otherwise we would have the contradiction, that a thing [Ding] E0 would be both an element of M, but also not an element of M.
comment 3. Here is where Cantor defines the diagonal D as occupying a 2-dimensional (u,v) space. He forms a horizontal E0 with its v elements differing from the v elements of D. Then declares E0 can't be in the list. A conditional statement, (IF D as a diagonal sequence is present) which Cantor has provided.
Before the definition of D, u1 is a horizontal sequence. D can't be u1.
This is where Cantor equates 'E0 not in the list' to 'E0 not in the set M', even though they are two different and independent things.
------------------------------------------------------------------------
Cantor (not me) defines an infinite set of E sequences of all possible orderings of the symbols m and w. The binary tree is one way to represent the set M. It displays properties of the set not evident in the list, one being symmetry. In example #168, 010101... or mwmwmw...if reflected in the mirror axis ma interchanges the 2 symbols. If one exists the other must exist. E0 can be missing and D be present in the list but not in the set M. There are at least as many random lists as there are sequences. If each can declare a missing E via the diagonal argument, what's left? .
If you accept the binary tree, then every path originating in M0 and M1 traces a unique infinite sequence. If you form sequences for each v from top to bottom, they are in order with 0 preceding 1.
Post#60
If I understand it, the program produces a finite square array of symbols 0 and 1.
They will always have missing elements. A 4x4 array contains 4 of 16 possible sequences. You can compare all for duplication. Cantor had no method of detecting duplicates. On 2nd thought, that's not a problem. He would have spent all his time forming the 1st infinite sequence.
As for posts from the past, there have been many revisions.
I enlarged the Cantor text because it becomes difficult to read with subscripts.
Cantor:
Namely, let m and n be two different characters, and consider a set [Inbegriff] M of elements
E = (x1, x2, … , xv, …)
which depend on infinitely many coordinates x1, x2, … , xv, …, and where each of the coordinates is either m or w. Let M be the totality [Gesamtheit] of all elements E.
To the elements of M belong e.g. the following three:
EI = (m, m, m, m, … ),
EII = (w, w, w, w, … ),
EIII = (m, w, m, w, … ).
I maintain now that such a manifold [Mannigfaltigkeit] M does not have the power of the series 1, 2, 3, …, v, ….
This follows from the following proposition:
"If E1, E2, …, Ev, … is any simply infinite [einfach unendliche] series of elements of the manifold M, then there always exists an element E0 of M, which cannot be connected with any element Ev."
comment 1. M is the set of all E's, and has a cardinality different from N the set of all integers, 'totality' meaning complete.
Cantor:
For proof, let there be
E1 = (a1.1, a1.2, … , a1,v, …)
E2 = (a2.1, a2.2, … , a2,v, …)
Eu = (au.1, au.2, … , au,v, …)
………………………….
where the characters au,v are either m or w. Then there is a series b1, b2, … bv,…, defined so that bv is also equal to m or w but is different from av,v.
Thus, if av,v = m, then bv = w.
comment 2. The bv elements are compared to the diagonal elements avv and altered to differ. I label the diagonal sequence of a's D for my own convenience to identify the sequences involved.
Cantor:
Then consider the element
E0 = (b1, b2, b3, …)
of M, then one sees straight away, that the equation
E0 = Eu
cannot be satisfied by any positive integer u, otherwise for that u and for all values of v.
bv = au,v
and so we would in particular have
bu = au,u
which through the definition of bv is impossible. From this proposition it follows immediately that the totality of all elements of M cannot be put into the sequence [Reihenform]: E1, E2, …, Ev, … otherwise we would have the contradiction, that a thing [Ding] E0 would be both an element of M, but also not an element of M.
comment 3. Here is where Cantor defines the diagonal D as occupying a 2-dimensional (u,v) space. He forms a horizontal E0 with its v elements differing from the v elements of D. Then declares E0 can't be in the list. A conditional statement, (IF D as a diagonal sequence is present) which Cantor has provided.
Before the definition of D, u1 is a horizontal sequence. D can't be u1.
This is where Cantor equates 'E0 not in the list' to 'E0 not in the set M', even though they are two different and independent things.
------------------------------------------------------------------------
Cantor (not me) defines an infinite set of E sequences of all possible orderings of the symbols m and w. The binary tree is one way to represent the set M. It displays properties of the set not evident in the list, one being symmetry. In example #168, 010101... or mwmwmw...if reflected in the mirror axis ma interchanges the 2 symbols. If one exists the other must exist. E0 can be missing and D be present in the list but not in the set M. There are at least as many random lists as there are sequences. If each can declare a missing E via the diagonal argument, what's left? .
If you accept the binary tree, then every path originating in M0 and M1 traces a unique infinite sequence. If you form sequences for each v from top to bottom, they are in order with 0 preceding 1.
Post#60
If I understand it, the program produces a finite square array of symbols 0 and 1.
They will always have missing elements. A 4x4 array contains 4 of 16 possible sequences. You can compare all for duplication. Cantor had no method of detecting duplicates. On 2nd thought, that's not a problem. He would have spent all his time forming the 1st infinite sequence.
As for posts from the past, there have been many revisions.
phyti;
Looking back at your posts here and elsewhere, I think the fundamental disconnect between you and, well, everyone, is here ...
It's clear that M is all possible sequences. Cantor even gave you examples:
EI = (m, m, m, m, … ),
EII = (w, w, w, w, … ),
... which show the symmetry you are proud of in your binary tree.
But the method is applied to an infinite series from M: the series E1, E2, …, Ev, it's not applied to M.
Cantor then asks if all of M is in E1, E2, …, Ev - by seeing if he can make an E0 (which would be in M as M is all) that clearly isn't in E1, E2, …, Ev.
You later claim the reverse: [ This is where Cantor equates 'E0 not in the list' to 'E0 not in the set M' ]. That is absolutely not clear at all, and I do not see anybody thinking that other than you.
** If that's what you think, focus on that. **
---- side point 1
Frankly I don't think it helps to be so focussed on Cantors specific words. All that "[einfach unendliche]" stuff just makes for noise, and his (translated) language of that time is quite awkward. You'd be better off using a more modern presentation. Even the one from Wikipedia. There's been over a hundred years of study of this! It would also help in keeping all the notation consistent. m/w, 0/1, A/B, D/E, ... even words like enumeration, series, sequence, element, coordinate get all muddled with the flipping back and forth.
(Next on the forum: somebody argues against SR using an English translation of Einstein from 1920 ...)
---- side point 2
Using your binary tree, consider writing out the sequences. Start at M0 since it's at the top.
1st sequence of length 1:
0
First 2 sequences of length 2:
00
01
First 3 sequences of length 3:
000
001
010
First 4 sequences of length 4:
0000
0001
0010
0011
I hope you would agree that:
1: We could go on forever (time, paper, and ink ignored); there are infinite sequences starting with 0 at M0.
2: If so: clearly there are also infinite sequences that start with 1 at M1.
3: Finally: this means there are infinite sequences in your binary tree, that are not in the infinite sequence starting at M0. Agree?
As I wrote: "infinite" doesn't equal "all possible"'. You can define an infinite list from your tree, that doesn't mean it's all of the tree.
Back to the start of this post:
Cantor would be happy that a sequence starting with 1 is not in the list of infinite sequences starting with 0.
He would not try saying that any sequence starting with 1 is not in the binary tree.
Likewise, he was not saying that an E0 being absent from E1, E2, …, Ev means it is absent from M.
That would be like me saying a sequence starting with 1 isn't in your tree. That would be silly.
Looking back at your posts here and elsewhere, I think the fundamental disconnect between you and, well, everyone, is here ...
It's clear that M is all possible sequences. Cantor even gave you examples:
EI = (m, m, m, m, … ),
EII = (w, w, w, w, … ),
... which show the symmetry you are proud of in your binary tree.
But the method is applied to an infinite series from M: the series E1, E2, …, Ev, it's not applied to M.
Cantor then asks if all of M is in E1, E2, …, Ev - by seeing if he can make an E0 (which would be in M as M is all) that clearly isn't in E1, E2, …, Ev.
You later claim the reverse: [ This is where Cantor equates 'E0 not in the list' to 'E0 not in the set M' ]. That is absolutely not clear at all, and I do not see anybody thinking that other than you.
** If that's what you think, focus on that. **
---- side point 1
Frankly I don't think it helps to be so focussed on Cantors specific words. All that "[einfach unendliche]" stuff just makes for noise, and his (translated) language of that time is quite awkward. You'd be better off using a more modern presentation. Even the one from Wikipedia. There's been over a hundred years of study of this! It would also help in keeping all the notation consistent. m/w, 0/1, A/B, D/E, ... even words like enumeration, series, sequence, element, coordinate get all muddled with the flipping back and forth.
(Next on the forum: somebody argues against SR using an English translation of Einstein from 1920 ...)
---- side point 2
Using your binary tree, consider writing out the sequences. Start at M0 since it's at the top.
1st sequence of length 1:
0
First 2 sequences of length 2:
00
01
First 3 sequences of length 3:
000
001
010
First 4 sequences of length 4:
0000
0001
0010
0011
I hope you would agree that:
1: We could go on forever (time, paper, and ink ignored); there are infinite sequences starting with 0 at M0.
2: If so: clearly there are also infinite sequences that start with 1 at M1.
3: Finally: this means there are infinite sequences in your binary tree, that are not in the infinite sequence starting at M0. Agree?
As I wrote: "infinite" doesn't equal "all possible"'. You can define an infinite list from your tree, that doesn't mean it's all of the tree.
Back to the start of this post:
Cantor would be happy that a sequence starting with 1 is not in the list of infinite sequences starting with 0.
He would not try saying that any sequence starting with 1 is not in the binary tree.
Likewise, he was not saying that an E0 being absent from E1, E2, …, Ev means it is absent from M.
That would be like me saying a sequence starting with 1 isn't in your tree. That would be silly.
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James R;
I'm not ignoring you, just needed some extra time.
I started about 2006 at Physforum.
Forums seemed like a good idea, sharing information and learning new things. Over the years, I find people who can't comprehend what they read. It's no longer interesting having debates with people who won't or can't do a personal analysis of a subject for correctness, but blindly accept what they read based on an authority figure, as in the Cantor case. If only his contemporaries had stood their ground.
I discovered conveying the solution to of a problem to others is sometimes as difficult as finding the solution.
In the Savant case, the Savantists accepted her explanation based on her celebrity status, as a person with a high IQ (performed when she was 10).
The IQ subject came up 15 yrs later from a wiser 60 year old.
Her response, "attempts to measure it are useless".
vos Savant, Marilyn (July 17, 2005). "Ask Marilyn: Are Men Smarter Than Women?".
The subject here was closed and I was accused of lying.
phyti said:
Marilyn Savant is the person who claimed she could beat the odds of the game, as a result of her misconceptions of probability.
Her conclusion,
"When you switch, you win 2/3 of the time and lose 1/3, but when you don't switch, you only win 1/3 of the time and lose 2/3."
Response:
This is a lie, and a deliberate smokescreen from phyti. Marilyn vos Savant correctly explained the odds of the game - along with about 10,000 other people who were able to correctly analyse it. It's not her fault that, for whatever reason, phyti either lacks the mathematical skill or else is unable to be honest enough to acknowledge the correct analysis.
If phyti is hung up on something about the personalities involved, due to some unstated prejudice, he could easily write a simple computer program of his own to play the Monty Hall game repeatedly and examine the results. They will duplicate the kinds of results DaveC obtained and presented to phyti in this thread, of course, and which phyti has deliberately and repeatedly and dishonestly ignored, like a troll.
Either phyti is scared to run the simulation (or actually play the game for himself) or else he actually knows what the results will be and is too dishonest to admit that they don't support his incorrect analysis. One more nail in the coffin.
The criticism is nonsense.
A computer program will perform the instructions given based on her explanation. Why would anyone expect anything different. The program does not prove the correctness of her response, only the ability of the programmer.
Her comment 1990.
"Of the letters from the general public, 92% are against my answer, and and of the letters from universities, 65% are against my answer. Overall, nine out of ten readers completely disagree with my reply."
Were all those professional people wrong?
There are no experts, only some people with more experience than others (check the definition)
Truth will never be decided by an opinion poll. (Quora, Reddit, Stack Exchange, etc.)
I'm not ignoring you, just needed some extra time.
I started about 2006 at Physforum.
Forums seemed like a good idea, sharing information and learning new things. Over the years, I find people who can't comprehend what they read. It's no longer interesting having debates with people who won't or can't do a personal analysis of a subject for correctness, but blindly accept what they read based on an authority figure, as in the Cantor case. If only his contemporaries had stood their ground.
I discovered conveying the solution to of a problem to others is sometimes as difficult as finding the solution.
In the Savant case, the Savantists accepted her explanation based on her celebrity status, as a person with a high IQ (performed when she was 10).
The IQ subject came up 15 yrs later from a wiser 60 year old.
Her response, "attempts to measure it are useless".
vos Savant, Marilyn (July 17, 2005). "Ask Marilyn: Are Men Smarter Than Women?".
The subject here was closed and I was accused of lying.
phyti said:
Marilyn Savant is the person who claimed she could beat the odds of the game, as a result of her misconceptions of probability.
Her conclusion,
"When you switch, you win 2/3 of the time and lose 1/3, but when you don't switch, you only win 1/3 of the time and lose 2/3."
Response:
This is a lie, and a deliberate smokescreen from phyti. Marilyn vos Savant correctly explained the odds of the game - along with about 10,000 other people who were able to correctly analyse it. It's not her fault that, for whatever reason, phyti either lacks the mathematical skill or else is unable to be honest enough to acknowledge the correct analysis.
If phyti is hung up on something about the personalities involved, due to some unstated prejudice, he could easily write a simple computer program of his own to play the Monty Hall game repeatedly and examine the results. They will duplicate the kinds of results DaveC obtained and presented to phyti in this thread, of course, and which phyti has deliberately and repeatedly and dishonestly ignored, like a troll.
Either phyti is scared to run the simulation (or actually play the game for himself) or else he actually knows what the results will be and is too dishonest to admit that they don't support his incorrect analysis. One more nail in the coffin.
The criticism is nonsense.
A computer program will perform the instructions given based on her explanation. Why would anyone expect anything different. The program does not prove the correctness of her response, only the ability of the programmer.
Her comment 1990.
"Of the letters from the general public, 92% are against my answer, and and of the letters from universities, 65% are against my answer. Overall, nine out of ten readers completely disagree with my reply."
Were all those professional people wrong?
There are no experts, only some people with more experience than others (check the definition)
Truth will never be decided by an opinion poll. (Quora, Reddit, Stack Exchange, etc.)
You are wrong about the Monty Hall problem too, phyti, but it's not good to mix two topics into one thread so I'm not going there (here).
However, in relation to this thread it's interesting that your focus is so fixed on Marilyn vos Savant in the preceding post. After all, the result has been confirmed by many people, even Mythbusters who actually did the experiment.
The parallel in this thread is your need to show Cantor was wrong, ignoring over a hundred years of study and refinement of the language and descriptions (and no mathematical disproof). There's no "In the Savant case, the Savantists accepted her explanation based on her celebrity status" in either case.
However, in relation to this thread it's interesting that your focus is so fixed on Marilyn vos Savant in the preceding post. After all, the result has been confirmed by many people, even Mythbusters who actually did the experiment.
The parallel in this thread is your need to show Cantor was wrong, ignoring over a hundred years of study and refinement of the language and descriptions (and no mathematical disproof). There's no "In the Savant case, the Savantists accepted her explanation based on her celebrity status" in either case.
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