computerised random number generation & infinity

There are no infinite numbers.
You cannot 'reach' infinity. The statement is a contradiction of terms.

 

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Apologies. 1-(1÷2^n)

 

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Which the word NO means in my post

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Yeah, I did the same double-take.

Upon reflection, I concluded that phyti wasn't refuting you - he was refuting the poster to which you responded. i.e.: phyti was actually actually concurring with you.

 

To complicate for Huey Dewey and Louie to work out

I'm on holiday so they stopped working

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From Michael 345 Post 4

The computer need not hold all the pseudo-random numbers generated. It need only be required to generate them one at a time on demand.

From DaveC426913 Post So, in this case, it is the fraction (precision of the floating point fraction) that will be limited by the computer's memory.

Assuming a program does not use multiple memory locations to hold a single long number, a pseudo Random number generator is limited by the word size of the computer, not the memory size.​

 

OK
Even my 3 neuron brain Heuy D

OK
Even my 3 neurone brain Huey Dewey and Louie can do that

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This, of course, raises the question of whether it is possible - even in principle - to generate truly random numbers. After all, that will require some sort of algorithm.

 

From DaveC426913 Post 23

See my Post 10

 

Quantum computers could do it. In theory.

 

Sometimes my posts are too sparse for the reader to get the meaning, so this may help.
OP:
RainbowSingularity;

Micheal 345;

Agree, with exception of last 2 sentences.
-------An elaboration of your response
The human brain has a finite memory, thus the same limitations of a computer.
To 'reach' or 'approach' "infinity' is a contradiction.
You cannot get closer to something that is a not accessible.
An analogy would be 'approaching the horizon'.
------The basic issue
After experiencing forum debates (about 12 yrs) on supposedly settled issues, I questioned the concepts taught by 'experts' via teachers and publications. "Infinity" is the most annoying and misinterpreted concept.
It's a math issue resulting from poor grammatical structure.
The mathematicians who defined limits should have consulted a dictionary.
'Infinity' defined as "without limit" cannot be used as a quantifier, i.e. it's immeasurable. It's not just about numbers, but a property of any collection of elements. Since it was represented by a symbol, many want to treat it as a number.

The Peano method of integer formation by adding a defined unit to a set of symbols is an abstraction. The symbols are representations of things, therefore the method does not imply a physical reality to an unlimited set of things.
By definition, an 'infinite' list has no end, therefore it is independent of time.

My suggestion for a definition of 'random' is 'unpredictable'.

 

Sure, that's certainly true today. Computers are artificial and so, for the future, it will depend on what will be considered a computer.

If infinity means "without end", "access" here means counting from any finite number to infinity and "counting" means counting like we count integers, then, yes, counting to infinity is a logical contradiction. There's no end to count up to.

It's not a math issue. It's up to people who argue about the infinite to specify what they mean, something most people don't do.
And it's not a "grammatical" issue but an etymological issue and a logical one.

I'm sure they did.
Dictionaries are not prescriptive but descriptive. They describe how words are used, they don't prescribe how they should be used. In effect, we are free to make up our own meaning and see if other people use it. That's what mathematicians do and it's what everybody does. And it's been like this since the beginning of words. Dictionaries came much later.
And you can look up a dictionary today and see that infinity in the mathematical sense is there:

That's not very well formulated but we can understand what mathematicians mean and that's OK. There's certainly nothing illogical in the idea of an infinite limit in this sense.

Words are abstractions, too. No word we use could possibly entail the reality of what the word is supposed to refer to. That's true of all our ideas about the material world. Now, I don't think we want to stop talking about the material world as if we knew what it is.

I'm not sure without time anything would exist. But maybe it's the other way round, time maybe is just the counting of what things do.

On substance, it seems possible to reach infinity because, apparently, that's something we do all the time. Again, just because we can conceive of infinity as something we can reach doesn't entail the existence of this kind of infinity but this in turn doesn't entail that this kind of infinity doesn't exist. All we can do is try and ensure we remain logical in our ideas. Logical ideas seem to tell us what we can try in practice without fear of getting hurt.

And of course, that we're unable to count an infinite number of anything like we can count our five fingers doesn't entail there isn't an infinite number of anything.

Personally, I can conceive of an infinity which has a limit, or even several limits, or even an infinity of limits. I fail to see where there would be a logical contradiction in that.
EB

 

From Speak Pigeon Post 32

You need some formal training in logic & mathematics.

Speak Pigeon: Most folks do not have a precise definition of an infinite set & their intuitive notions are incorreect. You need to read Cantor's essay on infinite numbers. He gives a succinct definition of an infinite set, known to few people.

An infinite set is a set which can be put into a one-to-one correspondence with a subset of itself.​

 

Why would our intuitive notions about the infinite be incorrect?! You'd need to argue this extraordinary claim.

On the contrary, I think we seem to all have perfect notions. And it also seems we tend to ignore what other people mean when they talk of the infinite or anything else. We just so love to air our views and feel we are right.
Cantor's definition is not what infinite sets are. It's a property infinite sets have.
And it's actually not true. I'm not going to go there. I'm just going to make a mental note that it's something I'll need to look at.
Still, it's true that some infinite sets have that property.
And it seems Cantor is used to making mistakes.
EB

 

From SpeakPigeon post 34

Please note the following.

(1 & 2)
(2 & 4)
......
(n & 2n)​

Above is a on-to-one correspondence between all the members of a set & a subset of those members.

This is not possible for finite sets & is considered the defining property of infinite sets.
​

What do you mean by the following pairs of statements

​

If you disagree, please supply your definition of an infinite set. A formal definition, not some intuitive notion of yours.

Perhaps you should find a book containing Cantor's essay & read it.

 

Also from From SpeakPigeon post 34

I claimed that infinite sets can be put into a one-one correspondence with a subset & that is the defining property of infinite sets. Note the following for the infinite set of integers

The above is not possible for finite sets & is considered the defining property of infinite sets.

You & many others do not seem to be aware that the above property is the defining property of infinite sets (read Cantor's essay). Because you & many others are not aware that the above property is the defining property of infinite sets, I claim that that you & many others have incorrect intuitive notions about infinite sets.

 

All sets are subsets of themselves. Every set is in 1-1 correspondence with itself.

 

uhClem: I suppose that the definition of subset might include a set itself as a subset.

I believe there is a term which I should have used (perhaps it is Proper Subset) which excludes the set itself as a subset.

Note that the infinite set of all integers allows for the following one-to-one correspondence with a subset which excludes the odd integers. Such a pairing is not possible with a finite set.

(1 & 2)
(2 & 4)
. . . .
(n & 2n)
​

Such a correspondence is not possible with a finite set (There are not enough even integers).​

Excuse me for not posting Cantor's complete essay relating to the above, which avoids possible objections to my remarks.

BTW: Cantor's essay deals with infinite sets by pairing of members to decide which of two sets is larger. The smaller set runs out of members first.

For example, it is easy to prove that the set of all subsets is always larger than the set itself (except for a set consisting of one member) . This seems to be intuitively obvious, but relying on one's intuition is likely to result in erroneous conclusions when dealing with infinite sets.

The pairing of members to decide which of two sets is larger avoids errors due to faulty intuition & allows for deciding when two infinite sets have the same numbers of members & deciding which (if either) of two infinite sets has more members.

Note that the set of all even integers can be matched one-for-one with the set of all integers, leading to the counter-intuitive notion that both sets have the same number of members.
​

 

OK, I'm good with that.

I agree it doesn't seem possible for finite sets.
Is considered the defining property of infinite sets by whom?
Not me.
Please try to be a bit more explicit whenever you're making a claim. It's important to specify whether it's mathematicians, all of them, some of them, etc. who have adopted this definition. I suspect it's "not all of them". But I don't actually know that, so thank you eventually to tell me.

I should have said, "It's a property I understand some mathematicians say infinite sets have and I disagree".

Please, don't bold your comments. It's rude and not justified here. I can read what you posted all good without that. Thank you.
So, yes, I disagree with Cantor, if it's what he said, and no I won't supply any definition, formal or not. This thread is about the infinite, not about whether mathematicians get their definition wrong.

Thanks for the advice but I'm already busy up to the gill and I definitely won't do that now. But as I said, I took note of the issue and will certainly look into it if I live long enough. That's only the second issue I have with Cantor so I'll to read whatever he said or something more current perhaps.
EB

 

All you would need to do is post the definition Cantor gave. Perhaps you need to read Cantor's essay to do that properly.

I'm not sure about that. Any erroneous conclusions may have to do more with some weird definition mathematicians use.

Pairing doesn't seem conclusive to me when dealing with infinite sets. Which suggests an interesting question: can we always prove that a proof is conclusive? I doubt that but maybe you can tell.

That's not counter-intuitive. It's just that the notion of counting an infinite set is a creation of mathematicians and therefore unfamiliar to the non-specialist.
EB