aispokesperson
Registered Member
You tell me whether the hotels form a finite or infinite collection. It is obvious, isn't it?
Different sizes of infinity...?
- Thread starter Killjoy
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Why? Because a chatbot told you?
aispokesperson
Registered Member
What do you try to say? I am confused. I think we are discussing infinity, and there is infinite question on the source of arguments. You guys are not serious.
In the physical or concrete world, a completed magnitude (or static situation that is measurable) is finite, no matter how large in quantity or size. So a hotel in such a tangible context would have a definite number of rooms at any particular time. And assigning it "infinite status" would just mean more rooms being endlessly added to accommodate new guests. The amount of the material hotel's living quarters is never a settled condition; it's a continuing process.
In the ideality of abstract concepts, though, you can have a set of infinite (non-terminating) elements whose sequence is following the [__] principle (A, B, C...). And other sets of infinite (non-terminating) members that are adhering to other principles (B, D, F...). By comparing their different regulating principles, logic jugglers can contend that group "@@@@" is larger than group "####" (or whatever specifics contingently plugged into the placeholders and the submitted relationships between them).
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aispokesperson
Registered Member
Was I not clear? I asked you why you believe Cantor was not very careful with his set theory.
So you don't want to discuss Cantor's set theory and different sizes of infinity? Why are you posting to this thread?
Pinball1970
Valued Senior Member
He will ask his AI and we will find out.
Pinball1970
Valued Senior Member
Exposing Ai slop.
Who is that question for?
#124
"Also worth noting that Cantor eventually went nuts poor lad."
He suffered from depression.
#128
"This is math problem, not a real world problem."
Mathematics is an abstract language applied to real world examples everyday.
It's the verification tool of science.
Beginning with counting, it was developed for practical needs, similar to time.
The concept of 'infinity' has been applied to a real world problem, the Hilbert Hotel with rooms and guests. The difficulty is solving the problem using methods developed for a world of finite elements when they don't apply.
Cantor attempted to define transfinite sets as an extension of the number system.
"I say of a set that it can be thought of as finished (and call such a set, if it contains infinitely many elements, "transfinite" or "suprafinite") if it is possible without contradiction (as can be done with finite sets) to think of all its elements as existing together, and to to think of the set itself as a compounded thing for itself; or (in other words) if it is possible to imagine the set as actually existing with the totality of its elements."
What he defined was a contradiction of terms, a thing without a boundary yet complete.
He was unsure if it was possible (red).
The constructivist view is, if a process requires an 'infinite amount of time to complete', then it never happens.
If a motion begins at t=0, then after moving a distance x at t=1, you have additional distance to move greater than x! I.e., you cannot make any progress.
You cannot approach 'infinity'.
#134
"Now, back to why phyti thinks the diagonal is incompatible with something and why that matters ..."
Draw a 1 unit vector vertically, and a 1 unit vector at 45°. Note they are not equal if applied to an object. Vectors have magnitude and direction.
When Cantor defined the diagonal sequence, it became 2 dimensional with a 45° orientation, when all other sequences were horizontal and 1 dimensional.
He is comparing 2 different classes of sequences.
Here is a simple example. Sequence A is transformed to sequence B per the Cantor method. Both coexist if 1 dimensional. If A is oriented to a diagonal, there is a contradiction at the diagonal for every u coordinate for B. When v=u, the coordinate can't be 2 different symbols simultaneously. The example is a finite list, thus the problem is independent of the length, finite or not.
"Also worth noting that Cantor eventually went nuts poor lad."
He suffered from depression.
#128
"This is math problem, not a real world problem."
Mathematics is an abstract language applied to real world examples everyday.
It's the verification tool of science.
Beginning with counting, it was developed for practical needs, similar to time.
The concept of 'infinity' has been applied to a real world problem, the Hilbert Hotel with rooms and guests. The difficulty is solving the problem using methods developed for a world of finite elements when they don't apply.
Cantor attempted to define transfinite sets as an extension of the number system.
"I say of a set that it can be thought of as finished (and call such a set, if it contains infinitely many elements, "transfinite" or "suprafinite") if it is possible without contradiction (as can be done with finite sets) to think of all its elements as existing together, and to to think of the set itself as a compounded thing for itself; or (in other words) if it is possible to imagine the set as actually existing with the totality of its elements."
What he defined was a contradiction of terms, a thing without a boundary yet complete.
He was unsure if it was possible (red).
The constructivist view is, if a process requires an 'infinite amount of time to complete', then it never happens.
If a motion begins at t=0, then after moving a distance x at t=1, you have additional distance to move greater than x! I.e., you cannot make any progress.
You cannot approach 'infinity'.
#134
"Now, back to why phyti thinks the diagonal is incompatible with something and why that matters ..."
Draw a 1 unit vector vertically, and a 1 unit vector at 45°. Note they are not equal if applied to an object. Vectors have magnitude and direction.
When Cantor defined the diagonal sequence, it became 2 dimensional with a 45° orientation, when all other sequences were horizontal and 1 dimensional.
He is comparing 2 different classes of sequences.
Here is a simple example. Sequence A is transformed to sequence B per the Cantor method. Both coexist if 1 dimensional. If A is oriented to a diagonal, there is a contradiction at the diagonal for every u coordinate for B. When v=u, the coordinate can't be 2 different symbols simultaneously. The example is a finite list, thus the problem is independent of the length, finite or not.
No, it's not. The verification tool of science is reality itself. Science is the study of our natural world, our reality, and that study results in attempts to understand it, including predictive hypotheses, theorems, and models. Those are verified by matching reality. Not by mathematics.
Maths helps with the modelling, the hypotheses, the theorems, but not with the verification. At best maths can show that the theorem, the model follows logically from whatever assumptions the model uses. But that's it. If the assumptions aren't correct, the maths might work, but the model will probably be wrong.
Hence Cantor developed solutions that don't relate to the real world. I.e Alephs, to denote different sizes of infinity.
That's not a contradiction, once you understand what he means by complete. Unbounded means no final element, but complete means that you consider it as an object, and a well-defined object.
One understands that the sequence of positive integers is unbounded. It is infinite. But one can treat "the sequence of positive integers" as a well-defined object in its own right. It is complete.
So no contradiction.
That's the difference between reality and the abstract, the concept.
As for your diagonal whatever, you still seem to be misunderstanding Cantor, and creating objections of straw. So I'll leave that part alone. Wikipedia is probably your best bet at this stage.
This makes no sense at all. The method is just that, a method. It's constructing a new sequence. That new sequence (now "B" in your latest diagram) could be a member of the set, but clearly cannot be. Nothing is coexisting, the orientation doesn't matter. There is no coordinate that is "2 different symbols simultaneously". All that matters is that the method made a sequence, that clearly cannot be in the set.
Have you tried the little program I supplied a few pages back? (#60)
#134
"Now, back to why phyti thinks the diagonal is incompatible with something and why that matters ..."
Draw a 1 unit vector vertically, and a 1 unit vector at 45°. Note they are not equal if applied to an object. Vectors have magnitude and direction.
When Cantor defined the diagonal sequence, it became 2 dimensional with a 45° orientation, when all other sequences were horizontal and 1 dimensional.
He is comparing 2 different classes of sequences.
Here is a simple example. Sequence A is transformed to sequence B per the Cantor method. Both coexist if 1 dimensional. If A is oriented to a diagonal, there is a contradiction at the diagonal for every u coordinate for B. When v=u, the coordinate can't be 2 different symbols simultaneously. The example is a finite list, thus the problem is independent of the length, finite or infinite.
Question 1. If diagonal A was the original sequence, why wasn't B also diagonal?
Answer 1. There would be no conflict.
Question 2. 'What sequence is u=1, horizontal A or diagonal A? '
Answer 2. both.
Diagonal A can be a duplication of horizontal A, for the purpose of claiming a missing B.
A variation of Cantor's infinite list has sequences of symbols replaced with all natural integers in the set N.
u-E
1. 1
2. 2
3. 3
4. 4
⋮
Since there is no largest integer, N has no limit and satisfies the definition of infinite.
There can be no missing element.
This list denies Cantor the opportunity to use the diagonal argument to manipulate its elements.
"Now, back to why phyti thinks the diagonal is incompatible with something and why that matters ..."
Draw a 1 unit vector vertically, and a 1 unit vector at 45°. Note they are not equal if applied to an object. Vectors have magnitude and direction.
When Cantor defined the diagonal sequence, it became 2 dimensional with a 45° orientation, when all other sequences were horizontal and 1 dimensional.
He is comparing 2 different classes of sequences.
Here is a simple example. Sequence A is transformed to sequence B per the Cantor method. Both coexist if 1 dimensional. If A is oriented to a diagonal, there is a contradiction at the diagonal for every u coordinate for B. When v=u, the coordinate can't be 2 different symbols simultaneously. The example is a finite list, thus the problem is independent of the length, finite or infinite.
Question 1. If diagonal A was the original sequence, why wasn't B also diagonal?
Answer 1. There would be no conflict.
Question 2. 'What sequence is u=1, horizontal A or diagonal A? '
Answer 2. both.
Diagonal A can be a duplication of horizontal A, for the purpose of claiming a missing B.
A variation of Cantor's infinite list has sequences of symbols replaced with all natural integers in the set N.
u-E
1. 1
2. 2
3. 3
4. 4
⋮
Since there is no largest integer, N has no limit and satisfies the definition of infinite.
There can be no missing element.
This list denies Cantor the opportunity to use the diagonal argument to manipulate its elements.
This simply doesn't matter. The diagonal/vector/45° is just being used to select elements to make a new sequence (and from that another).
The new A and B are then sequences of elements, no different to any of the rows in the list.
Row 3 is a total of 9 elements, 0's and 1's.
Row 7 is a total of 9 elements, 0's and 1's.
Row B is a total of 9 elements, 0's and 1's.
No difference!
There are not "2 different classes of sequences", that's plain silly. See above.
Imagine you showed that table, fully filled out (all the rows 1 to 9, with A and B included) but no evidence or description of how A and B were constructed. You ask that person to say whether A and B could be members of the list. Looking at A and B, why would they say "no"? They wouldn't, because there is no difference. If A or B had other symbols in them, or were a different length, then there'd be a difference.
The diagonal used to construct A and then B does not leave any permanent scar on them.
Nobody is putting two symbols in one place. v=u is used to select elements, to construct a new sequence. The table and it's cells are just to organise things and make the method clear.
My little program in post #60 uses an array (list) of strings (sequence of characters) to achieve the same result. Note that the constructed strings do not affect the original array or the strings in it.
No, because that would miss the point. The diagonal is used to select the elements for A. That's then used to make B. The point is to make a sequence of elements, just like all the rows 1,2, ... in the list.
There's no reason at all why B needs to be diagonal.
What??
Well, sure. But ... apples and oranges, it doesn't say anything about the diagonal method or what it showed.
Sarkus #152;
"The verification tool of science is reality itself."
[When experience matches theory, there is always some form of measurement that verifies the theory. A recent example was the successful trip to the moon and back.
It was based on a given amount of matter in motion, gravitational effect of the moon, etc., all involving past measurements.
I’ll restrict my statement to 'measurement is the verification tool of science', leaving mathematics for other purposes like research.
1. If you want to know the reality of 'the distance from Earth to moon', you measure the transmission time for a round trip radar signal.
2. When Einstein predicted deflection of star light by the sun, the 1919 eclipse was photographed, and the deflection was measured to verify it occurred.
3. For time dilation, the slowing rate of moving clocks was compared to clocks at rest, via measurements.
4. Statistics are used for many purposes that benefit the population.]
"Hence Cantor developed solutions that don't relate to the real world. I.e Alephs, to denote different sizes of infinity."
[He presented a symbol without any quantitative value, because there is none.
Hotels, rooms, guests are words representing real world objects. Mathematics is the manipulation of symbols, which typically represent numbers, which have all properties of the things they represent removed. The numbers involve measurement.]
"But one can treat "the sequence of positive integers" as a well-defined object in its own right. It is complete."
[Human experience has never involved anything infinite. No one can even imagine a universe 14 billion years old. 'Infinite' is used as a figure of speech when counting becomes inadequate.
If it is complete, what is the last element?]
Given 2 finite strings beginning with 0's, Sa ends with 0, Sb ends with 1.
Sa precedes Sb.
Given 2 infinite strings beginning with 0's, Sa has no end, Sb has no end.
The 2 strings can't be ordered.
"That's the difference between reality and the abstract, the concept."
[We live in reality, a world of experiences.]
"Wikipedia is probably your best bet at this stage."
[The Wikipedians have difficulty agreeing on things.]
"The verification tool of science is reality itself."
[When experience matches theory, there is always some form of measurement that verifies the theory. A recent example was the successful trip to the moon and back.
It was based on a given amount of matter in motion, gravitational effect of the moon, etc., all involving past measurements.
I’ll restrict my statement to 'measurement is the verification tool of science', leaving mathematics for other purposes like research.
1. If you want to know the reality of 'the distance from Earth to moon', you measure the transmission time for a round trip radar signal.
2. When Einstein predicted deflection of star light by the sun, the 1919 eclipse was photographed, and the deflection was measured to verify it occurred.
3. For time dilation, the slowing rate of moving clocks was compared to clocks at rest, via measurements.
4. Statistics are used for many purposes that benefit the population.]
"Hence Cantor developed solutions that don't relate to the real world. I.e Alephs, to denote different sizes of infinity."
[He presented a symbol without any quantitative value, because there is none.
Hotels, rooms, guests are words representing real world objects. Mathematics is the manipulation of symbols, which typically represent numbers, which have all properties of the things they represent removed. The numbers involve measurement.]
"But one can treat "the sequence of positive integers" as a well-defined object in its own right. It is complete."
[Human experience has never involved anything infinite. No one can even imagine a universe 14 billion years old. 'Infinite' is used as a figure of speech when counting becomes inadequate.
If it is complete, what is the last element?]
Given 2 finite strings beginning with 0's, Sa ends with 0, Sb ends with 1.
Sa precedes Sb.
Given 2 infinite strings beginning with 0's, Sa has no end, Sb has no end.
The 2 strings can't be ordered.
"That's the difference between reality and the abstract, the concept."
[We live in reality, a world of experiences.]
"Wikipedia is probably your best bet at this stage."
[The Wikipedians have difficulty agreeing on things.]
Ordering the elements is not the same as counting the elements.
I wonder if this is true in general.
A set can be counted - is countable - iff it can be put into one-to-one correspondence with a subset of the natural numbers. Not only are the natural numbers well-ordered, every natural number contains all its predecessors as subsets, which implies that any countable set can be ordered.
In other words, ordering is implicit in the process of counting.
Symbols {0, 1} are substituted for {m, w} for clarity.
For finite sequences, a comparison of each position beginning with v=1 will determine their order.
Since infinite sequences have no last element, they cannot be listed in complete order.
Their lists are random.
The sequences are randomly formed and randomly entered into the lists, independently of other sequences.
List 1 is a random list with the red diagonal sequence D transformed to its complement E by swapping symbols.
Cantor states E cannot be in list 1, since it would differ at the diagonal for all u.
Note u=5 is the complement of u=2. Why is E the only one excluded ?
List 2 is a random list with the red diagonal sequence D transformed to its complement E by swapping symbols. It contains the excluded sequence E from list 1 in row 7.
Cantor's manipulation of D determines the formation and location of E, contrary to the above initial conditions.
His zeal was greater than his analytical ability.
For finite sequences, a comparison of each position beginning with v=1 will determine their order.
Since infinite sequences have no last element, they cannot be listed in complete order.
Their lists are random.
The sequences are randomly formed and randomly entered into the lists, independently of other sequences.
List 1 is a random list with the red diagonal sequence D transformed to its complement E by swapping symbols.
Cantor states E cannot be in list 1, since it would differ at the diagonal for all u.
Note u=5 is the complement of u=2. Why is E the only one excluded ?
List 2 is a random list with the red diagonal sequence D transformed to its complement E by swapping symbols. It contains the excluded sequence E from list 1 in row 7.
Cantor's manipulation of D determines the formation and location of E, contrary to the above initial conditions.
His zeal was greater than his analytical ability.
Flip flopping between symbol sets isn't clear. You used 0/1 last time already, but had A/B instead of D/E.
Well, you'd never write them all down, sure. But that doesn't have to affect what mathematical deductions we can make about them.
OK.
Yes, that's the point. ("cannot be in list 1")
u=2 and u=5 are simply sequences in the list. Being the complement of each other has no bearing, and it's puzzling why you think it would.
E is constructed as the complement of D ... that doesn't mean any complement of any other is "bad" in some way, it just makes it clear that D cannot equal any u, even though there are infinite u.
It's not about "exclusion". The method isn't preventing E being in the list, the method is showing that E isn't already in the list.
Yes, list 2 is a different list than list 1. So the method used on list 2 constructs a different E that isn't in list 2. This was covered many many posts ago.
The method does not generate an E that isn't in any list. Nobody ever claimed that, it's not an issue that E from list 1 is in list 2. If you think it is you've missed the point.
What do you mean by "contrary to the above initial conditions."
What "conditions" were violated?
Here you betray your iconoclastic motivation. Cantor is in the textbooks and taught to students, not because he was a cool guy or whatever, but because his math is accepted - and has been for over a hundred years - by many many mathematicians using their analytical ability too.
If some PHd math student could write their thesis disproving Cantor they would. If you ever prove your case, maybe you'll be in the textbooks?