Does True Randomness Actually Exist? ( ^&*#^%$&#% )

Where we differ..... you assert that it is possible to flip an infinite number of heads (or tails). I say that's mathematically impossible. 
A billion, or a trillion......yes. Infinite....no.

I've had a quick look at Wiki and noted a couple of "Laws of large numbers" where Borel's seems to be somewhat counter to Bernoulli's. However, there's no doubt that Elroch has made a mistake in equating large numbers with infinity. These ideas have nothing to do with infinity and apply to large numbers only. It does make a big difference and I'm completely satisfied that in an infinite sequence, all we can say is that the numbers of heads and tails are equal. Not approximately equal, since that would be nonsense, as infinity can't be counted.

Bernoulli and Borel drew somewhat opposing conclusions, regarding the tendency of results to display trends as numbers of tests increase. There can be no question at all that Elroch is mistaken. It is impossible for an infinite sequence to contain all heads and no tails. There are bound to be even quantities of each. Probability isn't applicable regarding the overall results. It only works for measurable sequences and an infinite, measurable sequence is impossible.

@1

Let's take a look at a traditional definition which is along the line of predictability. basically if an action outcome is unpredictable, then it must be random. right? kinda, because if no one can predict an outcome, it’s absolutely random for practical purposes like engineering,

For the record, there is no definition here. You have not defined randomness....or anything else. A true definition of something places that something in a class, and then differentiates it from all other members of that class. 
Example:  An aircraft is a vehicle that can fly. 

Not a definition:  A dog is an animal with four legs and a tail. 

Also not a definition:  Anything that uses the words "basically" or "kinda".

From #1
<<Moving on. now let’s look at a modern definition along the line of “the lack of information” for example. if you write down a string of random numbers, and ask me to guess them, they are 100% random for me and at the same time 100% determined for you. Wait, what? that doesn’t sound “truly random” at all! exactly. And more important for this topic, it tells us nothing about determinism nor “true randomness”. (it’s actually a good definition that I like for some purposes, but not for a thread about determinism)>>

There's a difference between lack of information and unpredictability although, obviously, information is that which allows predictability. Yet the apparently modern "no-information" definition doesn't appear to be a correct definition of randomness. It appears to fail in the same way that the example "a dog is an animal with four legs and a tail" fails as a definition. They are partial descriptions .... in one case of an animal and in the other, of a situation. To become a definition, it would have to specify that the (missing) information would be enough to predict at least sufficient to allow non-random numbers to be placed in a sub-set of candidates, either by range or characteristic.

So randomness is defined by lack of predicability and not lack of information, unless the information is described as that which would enable predicability. We could have "true predictability" as opposed to "true randomness". True randomness is, of course, an old concept, differentiating it from pseudo-randomness.

I think it's fine to ignore what others think if we simply aim to be correct and accurate, since we already have a situation regarding the (claimed) consensus of mathematicians, regarding the idea that an infinite series of coin tosses, with a fair coin, can contain, for example, zero heads.  On focussed and considered reflection, that is undoubtedly impossible, because it is counter to the very idea of infinity. This shows that mathematicians may have no need to actually understand the concepts they work with. This, then, becomes a case of "in for a penny, in for a pound". Basic concepts, assumed by some to be correct and givens, can be challenged if they are incorrect or weak.

In my simple world, random just means that something has the exact same likelihood as anything else. Rolling a die or tossing a coin therefore would generate a random number. Nothing influences the die or the coin, other than chance. Any other influence would make the outcome not random. 

Yes, fair enough to me at least.

I thought a great deal about random numbers, 30 years ago when I was a keen programmer and had to use programmed lists of random numbers for some applications. It was possible to tweak them to make them less non-random but I arrived at the conclusion that to generate a truly random number, you had to access a random, physical source and count it. It was the only possible way. These days they claim to be able to generate random numbers but I'm sceptical.

Yes....mathematicians have a hard time proving random numbers are indeed random, and that they are influenced by nothing but chance....especially when they are generated by computers programmed by humans. 

If you wanted to generate a handful of random numbers, it would be easy using a 10 sided die (the physical source you reference)....if such a thing can be constructed. But generating large quantities of random numbers is a completely different challenge.

Every specific sequence has a probability of zero -- that does not mean it's impossible.

You understand that a billion heads is possible because you understand that the odds are always 50/50 and do not depend on the flips that came before. This does not magically change at infinity, so it is not impossible, it's just that probability is zero (like every specific sequence).

An infinite series of random flips all landing heads is impossible. Not just improbable. That's due to the nature of infinity, which Elroch clearly doesn't understand. You believe what he says. You believe that you're right because you believe Elroch must be right but neiher of you are capable of understanding what infinity means. I've heard it said before that many mathematicians don't understand the fundamental concepts they use. I shelved that idea because of course my son's a mathematician, more qualified than Elroch. He's a clever guy ... he would think about things rather than repeat verbatim what he's been told by others or told that others believe.

<<You understand that a billion heads is possible because you understand that the odds are always 50/50 and do not depend on the flips that came before. This does not magically change at infinity, so it is not impossible, it's just that probability is zero (like every specific sequence).>>

This is just wrong. You're talking down to Mike just like you were talking down to me, which is why you and I fell out. It wasn't anything I did ... just you failing to understand and teaching me stuff that's wrong, like you might teach kids.

You say the nature of probability doesn't change at infinity. What does that even mean? There's no such point as "at infinity". It proves you've placed your faith in the wrong person here and you got all angry with me, because I know you're wrong and wouldn't give in to what you were talking down at me. And so you started a quarrel in another thread, quite deliberately, and also deliberately brought others into it and incited them to troll. That is not good. No good at all. You should be ashamed of yourself.

But generating large quantities of random numbers is a completely different challenge.

I can gen up to abt a 1T on my clunky laptop. Luv2hava antminer !...but im having a out-of-money experience.

just tell me what u want & i can probably code one up...if u trust rng's. that way u dont hafta sample off a 21-sided magic stone til u needa operation on ur elbone. 

An in finite series of random flips...

...so are u saying that infinity does NOT physically exist ?

random just means that something has the exact same likelihood as anything else

really ?...random to me means kinda like arbitrary or maybe better for da he!! of it. y-day april (who rains random stuff) came over and we took a harborwalk just for the he!! of it.

I'll go on record here.....In an infinite series of coin tosses, it is a statistical certainty that the results will be exactly 50% heads and 50% tails.  I'm talking an INFINITE number of tosses....not just some large number. 

if infinity is a physical number then i guess ∞/2 (50%) would be one too. now all u gotta do is convince a math genius that ∞ is a #. good luck luv !

I've always avoided this thread, based on how it began, which was with not much discussion of the actual topic.  I see my name is being tossed around lately, though.  Maybe I need to come in here and hold court for several months?  I mean my presence must be wished for, obviously...

Or maybe just stop talking about me in threads I am not participating in...you know, like civilized people who aren't gossips.

P.S. Elroch and Llama are on the right side of this discussion, and Axel and Silver needed to start an art thread, long, long ago.

Set theory is the foundation of all mathematics.

From the basic axioms and logic, it is possible to prove that there is an infinite sequence of increasingly large infinite sets, in the sense that for each set there is no surjective function from that set to the following one (a function that has every element of the second set in its image).

A simple example is to take the set of natural numbers then to apply the power set operator iteratively (the power set of a set is the set of all subsets of that set) to get larger and larger infinite sets.

But it doesn't stop there. The union of that sequence of sets is trivially larger than any of the individual sets.

Then you can apply the power set operator iteratively to that set to get another infinite sequence of much larger infinite sets...

Viewing the whole procedure we have just described twice as a single operator, you can do the same thing with this operator as we just did with the power set operator and get much larger sets. 

Then repeat that with even more powerful operators over and over again.

But that is merely scratching the surface. To learn about much larger infinite cardinals, see:

Large Cardinals

Note on a technical point, in mathematics we always have to be careful to express things in a way that is meaningful. To say that half the coin tosses in an infinite sequence are heads is not meaningful as it stands. Hilbert's Hotel shows you need to be careful when you do arithmetic with infinite numbers.

Like in calculus, you can instead use limits. I would define a bound on the fraction of heads thus:

There is an upper bound F on the fraction of heads in a sequence if there is a number N such that the fraction of heads in the first M flips is less than or equal to F if M > N.  The set of upper bounds has an infimum (it may not itself be an upper bound).

Similarly you can define a lower bound on the fraction of heads in a sequence, and there is a supremum of the set of lower buounds 

Then you can define a sequence as having a specific fraction  F of heads if all numbers greater than F are upper bounds and all numbers less than F are lower bounds, so F is both the infimum of upper bounds and the supremum of the lower bounds. The fraction of heads gets closer and closer to F as you get to larger heads of the sequence.

There is no reason a sequence of flips has to have such a fraction F of heads at all. For example, consider the sequence that starts with 1 head, then 10 tails, then 100 heads, then 1000 tails, then 10000 heads and so on. You can see that the head of this sequence is alternately very largely heads for a while, then very largely tails for a while, and so on.  There is no meaningful fraction of heads.

There are, in general, well defined infimum of upper bounds and supremum of lower bounds for the fraction (as defined above). They just don't have to be equal.

But what do you mean by saying that these sets exist? They've been invented. They've been imagined into place. Of course someone can define infinity to be capable of producing a set of coin tosses that is all heads but it isn't something that one would necessarily agree with.

The question really is "is anything gained by it which can't be achieved by a more natural and, dare I say, more honest analysis of the problem of infinity?"

I've looked at this again and agree with my own instincive comment. In fact, I disagree with the general thrust, that the set of powers of the set of natural numbers to infinity is any different. Logically it is not different but simply a quicker way of getting nowhere. It gets nowhere because in any case, infinity is unbounded so it must be incorrect to claim that there are different infinities.

To explain it using a trivial example, it's more like the concept of infinity can be applied to different concepts, such as the set of natural numbers or that set multiplied by two, to give an illusion that it's a different set.  What you or they are doing is saying that, hypothetically, there can be infinitely different types of snail and infinitely different types of hedgehog, therefore there are different infinities. It's equivalent to saying that this jacket is blue and that hat is blue and so there are different types of blue. Types of quantity are no different. This is definitely a mistake that has been made by mathematicians. They're wrong. Mike and I are correct. I believe I could win this argument against anyone who was capable of following it, at any rate.

They are part of the body of mathematical truth. They happen to be useful in the foundations of mathematics. Gödel was interested in how they related to completeness. The existence of large cardinals causes more true statements to be provable using transfinite induction.

Large cardinals and determinacy - Stanford encyclopedia of philosophy

This advanced topic has no consequence to the simple fact that a sequence of an infinite number of heads is one of the uncountable set of possible sequences (all of which individually have probability zero). That is a triviality.

The set of sequences of coin flips is the set of functions from the natural numbers to the set with two elements, H and T.  It is of course trivial that an element of this set is the function f defined by:

f(n) = H for all n in ℕ

ℕ is a special character, the "double-struck N", unicode +2115, which represents the natural numbers in mathematics.