I am familiar with Cantor's work. I understand what his ideas are. I was only showing that your definition was wrong. If we take the definition you gave and add the two facts I added, we could conclude all sets are infinite, which is obviously false. You have now kind of corrected the error and people reading the thread will not be confused by the flawed definition. We are all trying to learn. Right?
computerised random number generation & infinity
- Thread starter RainbowSingularity
- Start date
I am familiar with Cantor's work. I understand what his ideas are. I was only showing that your definition was wrong. If we take the definition you gave and add the two facts I added, we could conclude all sets are infinite, which is obviously false. You have now kind of corrected the error and people reading the thread will not be confused by the flawed definition. We are all trying to learn. Right?
The problem with Cantor and his Set Theory is actually quite simple. Set Theory uses the Axiom of Infinity. It is an unprovable assumption that infinite sets are existent mathematical objects. I am sure you can see why that could be a problem, or at least why some people might disagree. Many great mathematicians have disagreed with Cantor on this. Many great mathematicians have agreed with him. If you are going to do Set Theory today then you need to make this assumption that the Axiom of Infinity is true. But is it really true? It is a valid question IMO.
So it is just a fact that you can’t use Cantor or his Set Theory to make convincing proofs about whether infinite sets are a thing. And that is the subject of this whole thread IMO. So this part of the thread seems to be a dead end.
My own meager belief is that the Axiom of Infinity is in a relationship with Set Theory that is similar to that of the Parallel Postulate with Geometry. It is interesting that the Parallel Postulate also is entwined in the problematic concept of infinity. Once the Parallel Postulate was removed a new much richer geometry we call non-Euclidean Geometry was created and that allowed some great advances in mathematics and physics. Perhaps the same thing will happen with Set Theory if and when they jettison the Axion of Infinity.
I also want to agree with phyti in post 309 above. I am not pointing at anyone here, but too many people use the term infinity as if it were a number. It is not a number. I really wish the terms infinity and infinite could be retired. Often a better word would undefined or something similar. Or maybe just use the word’s original meaning which was ‘not finished’ or ‘never finishes’.
So it is just a fact that you can’t use Cantor or his Set Theory to make convincing proofs about whether infinite sets are a thing. And that is the subject of this whole thread IMO. So this part of the thread seems to be a dead end.
My own meager belief is that the Axiom of Infinity is in a relationship with Set Theory that is similar to that of the Parallel Postulate with Geometry. It is interesting that the Parallel Postulate also is entwined in the problematic concept of infinity. Once the Parallel Postulate was removed a new much richer geometry we call non-Euclidean Geometry was created and that allowed some great advances in mathematics and physics. Perhaps the same thing will happen with Set Theory if and when they jettison the Axion of Infinity.
I also want to agree with phyti in post 309 above. I am not pointing at anyone here, but too many people use the term infinity as if it were a number. It is not a number. I really wish the terms infinity and infinite could be retired. Often a better word would undefined or something similar. Or maybe just use the word’s original meaning which was ‘not finished’ or ‘never finishes’.
DaveC426913
Valued Senior Member
'without limit' is pretty common.
Speakpigeon
Valued Senior Member
The numeral 1000 is 1000 +1 steps away from zero.
The infinite, as a numeral, is an infinite number of +1 steps away from zero.
I don't see where the problem would be with that. It doesn't even seem to be out of quilter with our ordinary notion of actual infinity.
EB
RainbowSingularity
Valued Senior Member
while i dont claim to come close to comprehending the mathamatics involved, this thought in a form, crossed my mind some years ago as i was pondering number selection via a computer selecting numbers of a set range.
the computer in theory randomly selects from a known range
yet when looking at the results, some numbers are chosen more often.
some numbers seem rarely chosen.
though this is a statistics issue and i think i have had this explained in several diffrent formats over several years, it left me wondering about the creative unpredicability of the human mind.
the ability of AI to duplicate this as a process of "awarenes" or real-life-ness... as it were in the process of creating things.
so it seemed to me that there may be something in the abstract concept of true random selection being used to create AI (processes)life, that may contradict its self.
very simply it left me asking
does the human mind have an infiniteoption of responses to any given point in time(creativity making things, inventing things etc).
equally, does this concept of "creativenes of invention" become in some way tied to a potential of random range that may be attributed to a concept of infinity ?
please excuse grammar and typos ive been up for 23 hours so im a little off 100%
I would dispute this. While 1000 and 1001 are distinct integer values, infinity is not, because infinity = infinity+1 = m*infinity+n, where m and n are any natural numbers you choose. Infinity does not seem to be a value but is instead a a member of class of numbers that have the property infinite.
Aristotle coined the terms actual infinity and potential infinity. Is that what you are referring to here?
https://en.wikipedia.org/wiki/Actual_infinity
Aristotle was not happy with actual infinity and supposedly disproved its existence.
Cantor also used actual infinity in his work, but it was as an (unproved) axiom called Cantor’s Axiom: All infinite sets are actually infinite.
Interestingly, Cantor said there are 3 types of infinity:
Some refer to them as Cosmological, Physical and Mathematical, which sounds a little little less kooky that way.
As I've explained: imagine a clock with an extra-hand that is set one-minutes fast. While the clock ticks, the extra-hand can never be caught, while still only reading finite numbers. This raises the question: is infinity actually a finite number (such as the symbol used in mathematics.)
Speakpigeon
Valued Senior Member
I'm not sure how that's a problem.
0 ⨯ m = 0 for any m (I would include m = the infinite). Is that a problem?
m + 0 = m for any m. Problem?
m/0 is undefined for any m. Problem?
m/1 = m, so m divided an infinite number of times by 1 is still m.
I think you'd need to find pairs of operations with results that would be obviously contradictory. Once we accommodated ourselves to zero as a number, we should be able to welcome weirder things.
And think of transcendental numbers like π, is it not the case that they must have a number of decimal digits?
Aristotle is my second best hero after Descartes. For he founded formal logic and it was more that 2400 years ago. That should feel awesome to anyone with some sense of proportion. And no one has yet improved on the substance of his method of logic.
Seems rather very convenient to me. Sort of like a just so story.
Sounds less kooky but also shamelessly cooked up. Preaching for his own parish, as we would say in French.
I think we're finite beings with a finite amount of time in front of us. Doing things like 1 + 1 won't get you to infinity. So, I'd say infinity is conceivable and be happy to leave it at that.
Cantor is my best scoundrel. Some of his work seems wrong to me. Still, he seems to have been overly bright.
EB
I noticed that you did not include subtraction.
infinity - infinity = infinity
Take the set of natural numbers and remove all the odd numbers. Now remove all the even numbers. The size of the set it still infinite. What are these infinite number of elements still in the set. Whatever they are, they aren’t odd or even because we removed those numbers. Is one of them infinity and the rest something else we have not defined? Are they all infinity? If so are they all distinct from each other? Or did we remove infinity (some say infinity is even)? If we removed infinity then what are these objects? What is your opinion?
Note that this is similar to the Hilbert Hotel paradox.
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Speakpigeon
Valued Senior Member
Hey, don't ask me to write a compendium on the question in just one post!
What you're doing here is a bit unfair. Obviously, as I said, we definitely have to accommodate the infinite into the "natural" numbers. It shouldn't be an impossible task. I can't give you my best answer off the top of my head. I guess I need to eat something first.
Still, given that you appear knowledgeable about mathematics, isn't that a problem that's already been solved? I would at least expect so, not least by Cantor himself. If not, I guess the French expression will be that I'm falling off the wardrobe, i.e. I'm realising just now what everybody has known for years.
So, you tell me first what the situation is and I'll try to find an honest answer if even Cantor couldn't get it right again!
Sure, and like hotels, mathematical concepts need good management. You can't just plod in with all you luggage and expect to be served a hot meal.
Now, talking of a hot meal, it's my turn to have one.
EB
Your view that the final set is empty seems perfectly reasonable to me. I should have been more specific about what I was talking about here. I was addressing the idea that infinity - infinity = infinity. Sorry that was not clear. Totally my fault. And here I will explain better. I hope.
1. The count of numbers in the set of natural numbers is infinity.
2. The count of numbers in the set of odd natural numbers is infinity.
3. So if we remove all of the odd numbers from the natural numbers, one would think that what is left is the set of even numbers, which has the count of elements at infinity.
4. So lets check, infinity - infinity = infinity. Looks OK.
5. Now remove the set of even natural numbers which also has the count infinity.
6. What is the count of the remaining set? Well, infinity - infinity = infinity. So the remaining set still has an infinite number of objects in it.
What are these objects? Well, we started by saying they were all natural numbers. And we removed all the odds and evens, so what is left are the natural numbers that are neither odd nor even. And there are mysteriously an infinite number of these things leftover because the count should be infinity, if you beleive that infinity - infinity is infinity. Sounds pretty weird to me. I found this proof that all natural numbers are either odd of even. I am going to assume it is correct. I don't have time to parse it but it seems reasonable from my scan. http://math.colorado.edu/~jonathan.wise/teaching/math2001-spring-2016/proof09.pdf
My "proof" is about as far from rigorous as a person can get. I will state outright that I think it is complete nonsense. My original plan was start with this mess and begin to clean it up and point out the many errors in it. I would try and insert a little more rigor and remove the errors in a piecewise method. And at the end I was going to use my reasoning to show:
1. infinity is not a number.
2. infinity - infinity = infinity is nonsense.
3. Answer the OP's question about "computerised-random-number-generation-infinity".
Unfortunately, on rethinking the whole thing, I have vastly underestimated the amount of posts and time that would take. I am trying to think of a better way to proceed. I think I can do it. The question is if I have the time and writing skill to get my points across.
SUMMARY: I think you are right. What is left should be the empty set.
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You are correct. It was unfair. But I planned to admit this later and show how everything in my post #22 was incorrect. I was acting the part of a person that believes infinity is a number (which I don't). There are tons of glaring errors. I was hoping that by asking you where it was wrong I could be like Socrates and engage you in a dialog. That seldom works on the internet.
So let me ask you to point out some the errors. Should be easy. Or don't. That is allowed. Up to you.
I will try and post more later. No time right now.
I was doing some research and as it turns out infinity - infinity is what is termed an indeterminate form. It does not equal infinity, or zero, or anything else. It can be any of those but it depends on circumstances as far as I can tell. The difference is literally anything. Anyway I have to beg off in this discussion. I found that doing internet searches ALWAYS returns dubious answers. Most said that infinity - infinity = infinity. Some said zero. I am now pretty sure the answer is indeterminate. So that whole paradox I wrote about is wrong. Wunnerful.
I did searches on many subjects related to my posts. In many cases I got contradictory answers even from university profs. For an example I would mention the sum of the natural numbers being equal to -1/12. I got some pretty authoritative info confirming this. And a few questionable sources denying with very convincing reasoning. I guess my message here would be: never believe anything you read on the internet, including this. Ignore seeming authoritative sources. Everyone is on equal footing. Fallacy of appeal to authority is a very real thing. Ignore "experts". I guess we all knew this already. I just did not understand how bad it was in this subject. I had plans to continue on with other approaches to the questions but whats the point?
On the positive side, this frees up time. And I am out of here -------->
I did searches on many subjects related to my posts. In many cases I got contradictory answers even from university profs. For an example I would mention the sum of the natural numbers being equal to -1/12. I got some pretty authoritative info confirming this. And a few questionable sources denying with very convincing reasoning. I guess my message here would be: never believe anything you read on the internet, including this. Ignore seeming authoritative sources. Everyone is on equal footing. Fallacy of appeal to authority is a very real thing. Ignore "experts". I guess we all knew this already. I just did not understand how bad it was in this subject. I had plans to continue on with other approaches to the questions but whats the point?
On the positive side, this frees up time. And I am out of here -------->
RainbowSingularity
Valued Senior Member
at last a challenge i stand a reasonable chance in
paradox + infinity ?
moving forward
my question/thread was not posted in malice or lascivious intent.
the question is as it is.
without opening up a completely new can of worms while the other is still on the plate,
when people speak of space being 'potentialy' infinite, i guess the word use is purely theoretical, like saying "really big".
because to have no end, one would need to define the length ?(mathamatically speaking?)
thats can of worms number 2.
coming back around to the start slightly...
(which was where i was kinda heading some years back) considering the statistical results of random number generation show the results as not being a fair distribution of random values of all values,
does this mean that the actual random value needs to be adjusted on the frequency of the more frequently picked random numbers ?
do some numbers inherantly have a lesser value of randomness ? (can of worms number 1.1)
Speakpigeon
Valued Senior Member
That's not easy, no. All proofs are based on assumptions and I would have to check whether some of them could be changed. This goes for the proof you link about all natural numbers being even or odd. Well, I accept that infinity is not a natural number, that's what I meant by the need to accommodate infinity into our idea of numbers. So, roughly, you remove all natural number and you're left with infinity. But that's probably a bit too simplistic. The point is that I would have to look at assumptions and it's nothing like easy.
And I'm busy, too.
EB
RainbowSingularity;
very simply it left me asking
does the human mind have an infiniteoption of responses to any given point in time(creativity making things, inventing things etc).
equally, does this concept of "creativenes of invention" become in some way tied to a potential of random range that may be attributed to a concept of infinity ?
--------
The human mind has no experience with anything 'infinite'. It's only a mental construct, with no physical representation, like a dimensionless point, a circle, an orbit, trajectory, etc. I could define a 5-sided square, and write a paper about it, but that doesn't make it real. I have a dog named Blue, that can add small numbers. I'd like to show you his math ability, but unfortunately he's invisible.
Conclusion, everything thinkable is not realizable.
-------
How would you verify something to be ‘without limit’
very simply it left me asking
does the human mind have an infiniteoption of responses to any given point in time(creativity making things, inventing things etc).
equally, does this concept of "creativenes of invention" become in some way tied to a potential of random range that may be attributed to a concept of infinity ?
--------
The human mind has no experience with anything 'infinite'. It's only a mental construct, with no physical representation, like a dimensionless point, a circle, an orbit, trajectory, etc. I could define a 5-sided square, and write a paper about it, but that doesn't make it real. I have a dog named Blue, that can add small numbers. I'd like to show you his math ability, but unfortunately he's invisible.
Conclusion, everything thinkable is not realizable.
-------
How would you verify something to be ‘without limit’
The word 'finite', based on its latin origin, means 'with an end' or 'limited', and in a mathematical sense, 'countable' or 'measurable'.
The prefix 'in' is the negation of a word, thus 'infinite' means 'without an end or unlimited'.
It's also the ability to distinguish an object by its boundaries that allows it to be measurable. An 'infinitely' long, one-ended stick, is therefore not measurable.
-----------
G. Cantor quote;
… Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other, and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established. In particular, in the introduction of new numbers, it is only obligated to give definitions of them which will bestow such a determinacy and, in certain circumstances, such a relationship to the other numbers that they can in any given instance be precisely distinguished. As soon as a number satisfies all these conditions, it can and must be regarded in mathematics as existent and real.
"… the essence of mathematics lies entirely in its freedom".
Ewald, W., From Kant to Hilbert, Oxford 1996
-----------In agreement with his quote, let's maintain consistency.
His 1-1 correspondence is only true sometimes depending on the sets.
The statement " there are as many even integers as integers",
is typically demonstrated using a 'one to one' correspondence as shown.
N: 1 2 3 4 5 6 ...
E: 2 4 6 8 10 12 ...
This is contradicted by:
1. Random sampling of integers results in an average of 50% even, 50% odd.
2. If N=D+E then N-E contains odd elements but E-E is empty.
A true example of a 1-1 correspondence, "there are as many even integers as odd integers "
D: 1 3 5 7 9 11 ...
E: 2 4 6 8 10 12 ...
The problem with the first example is the inclusion of even integers in both N and E. The even integers have links with two different odd integers and the odd have one link with an even.
The prefix 'in' is the negation of a word, thus 'infinite' means 'without an end or unlimited'.
It's also the ability to distinguish an object by its boundaries that allows it to be measurable. An 'infinitely' long, one-ended stick, is therefore not measurable.
-----------
G. Cantor quote;
… Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other, and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established. In particular, in the introduction of new numbers, it is only obligated to give definitions of them which will bestow such a determinacy and, in certain circumstances, such a relationship to the other numbers that they can in any given instance be precisely distinguished. As soon as a number satisfies all these conditions, it can and must be regarded in mathematics as existent and real.
"… the essence of mathematics lies entirely in its freedom".
Ewald, W., From Kant to Hilbert, Oxford 1996
-----------In agreement with his quote, let's maintain consistency.
His 1-1 correspondence is only true sometimes depending on the sets.
The statement " there are as many even integers as integers",
is typically demonstrated using a 'one to one' correspondence as shown.
N: 1 2 3 4 5 6 ...
E: 2 4 6 8 10 12 ...
This is contradicted by:
1. Random sampling of integers results in an average of 50% even, 50% odd.
2. If N=D+E then N-E contains odd elements but E-E is empty.
A true example of a 1-1 correspondence, "there are as many even integers as odd integers "
D: 1 3 5 7 9 11 ...
E: 2 4 6 8 10 12 ...
The problem with the first example is the inclusion of even integers in both N and E. The even integers have links with two different odd integers and the odd have one link with an even.
RainbowSingularity
Valued Senior Member
probability values
is there an infinite probability of a statisticaly balanced result ?
err-go = no(as you mention above)
i am attempting to avoid at great length the need to include probaility tables of results.
is randomness an equal property of uncertainty ?(my suspicion is no)
is random number generation equally uncertain as pure random results ?(my suspicion is no)
a random result has no frequency of statistical value above a single outcome of yes or no(as you mention above err-go 50% etc)
yet... physical random number generation via variant types(Galton box etc) shows clearly that some numbers come up more than others.
how does this result of increased frequency reflect a real value of 50/50 randomness when the mathamatical result is not 50/50 ?
so i asked myself...
it seems that given the results of some numbers coming up more often, they b their result attain a higher than 50/50 probability.
but there is no supposed equation that shows this to be real as a mathamatics equation.
does a computer process all numbers at the same interval of 0/1 ?
Given that you may have 50 numbers to choose from.
each time it choses a number from 1 to 50, does each value have a corresponding value of all other numbers equally ?
my suspicion is no, which leads to my question about the processing of creative design of computer randomness Vs human randomness
thus... does a computer pick a random number int he same way a human picks a random number?
(i accept there may be many different ways that humans pick supposed random numbers and shown by some methods these numbers are not random, yet some are).
i get the distinct feeling my question is far too complicated for most to deal with & find myself attempting to re-invent the wheel for those readers who simply do not comprehend it, so i will leave it there.
RainbowSingularity;
Assuming a random unbiased system of generating natural numbers, the system will have no memory, thus each drawing will be like the first one. I.e., time is not a factor.
Therefore a lottery drawing of the same number, 10 weeks in succession, would be a ‘rare’ event. This perception is removed by the previous statement. The probabilities of an event can be calculated by computer, but the ‘when’ cannot.
If all the results don’t have the same distribution, the period of sampling is too short, or like the weather person, you expect the temperature to always be ‘average’.
Assuming a random unbiased system of generating natural numbers, the system will have no memory, thus each drawing will be like the first one. I.e., time is not a factor.
Therefore a lottery drawing of the same number, 10 weeks in succession, would be a ‘rare’ event. This perception is removed by the previous statement. The probabilities of an event can be calculated by computer, but the ‘when’ cannot.
If all the results don’t have the same distribution, the period of sampling is too short, or like the weather person, you expect the temperature to always be ‘average’.