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

I do not know what you mean.

The sort of thing I mean is this. Someone is observing a spaceship moving past at tremendous speed and sees two events occurring on the spaceship that are (of necessity) separated by more than a Planck time in their frame. But in the time-dilated frame of the spaceship these two events occur closer together. Maybe less than a Planck time apart. If so, their times cannot be distinguished.

I feel there is some understanding to be gleaned by resolving this.

I’m certainly meaning from a not of this worldly  ‘sense of perspective’ . Only from the true spiritual reality. Something I’m absolutely certain of, given ONE is All in All, One that is not even allowed to be discussed. …..sadly to say =/ 

…..The  ONE  That IS Life.

…. lol I could give a flying flip fiddle to any worldly perspectives 😂

The thing is that the Planck time is a fixed time, about 10^-43 seconds.
In some cases - the Planck mass and Planck energy - the quantity is not a minimum in any sense, since the mass is much larger than those of known particles and atoms (and the energy enormously greater than that of most photons etc.)  It has a meaning as the smallest possible black hole mass, I believe, for technical reasons.

sleep tite opti......

Can you elaborate on the highlighted portion? By definition this is impossible yes?

An infinite number of flips of a fair coin will contain all possible sequences of finite length. So while some groups of sequences will have the proportion of heads be 1% higher, there will be an equal number of groups with tails 1% higher due to symmetry.

If it does not contain all finite sequences, then it wasn't an infinite number of flips (or wasn't a fair coin).

Yes the "ground fanishes" in the sense that we need axioms, but I'm not sure about the highlighted statement.

Although, this is kind of gross... I've never thought about the cardinality of infinite coin flips heh...

An infinite number of flips could contain an infinite subset of infinite flips the same way the counting numbers have an infinite subset of even numbers, an infinite subset of numbers divisible by 5, etc.

So, for example, if you flip a coin an uncountably infinite number of times, even if it's a fair coin, it could come up heads an infinite number times? This is unintuitive and werid.

Nah, quantum stuff breaks casuality all the time, e.g. Bell's theorem.

I guess I don't understand what you mean by "this is true with probability 1"

I'm trying to think of a better way to say what I was thinking.

Let's say we try to categorize infinitely long sequences of a coin flip. Some will contain all H, some H T H T repeating, etc while others (due to having no pattern) will contain every possible sequence. I'm guessing there must be some theorem about there being essentially infinitely many more sequences that have maximum entropy.

So could we say that the probability of producing a low entropy sequence after infinite flips is actually zero?

Well, as an example, it is certainly possible for an infinite sequence of coin flips to all be heads. (It is just as possible as any other specific sequence of results).

It also has probability 0

To prove this, observe that, for any N, the probability P of this is no more than the probability of getting N heads, which is 2^-N.

Thus P < 2^-N for all N

and the only positive P that satisfies this is P=0.

The flipside of this is that NOT getting all heads has probability 1.

Sure, just like each event in a CDF had probability zero, but infinitely many of these events sum to 1.

I guess it goes back to what you're saying about a more complex event.

Sure, each flip is independent, and every sequence is equally likely.

But I suppose the set of infinite sequences that have maximum entropy is infinitely larger than sets with less entropy... maybe not true for coin flips, but I imagine in most cases this is true. Maybe this is why in reality we never flip an infinite sequence of H... or maybe that's just a convenient way of thinking about it.

The events are independent, so it's possible to flip heads indefinitely.

But sure, the limit of 1/2^n as n -> infinity is zero.

And as a practical matter you can't repeat an event infinitely many times.

First, it's worth headlining that I have been glibly writing as if there is a measure on the set of all sequences of flips of a fair coin. There is a powerful general result about the existence of a measure on the cartesian product of an infinite number of measure spaces but this does not mean that every set of sequences is measurable. It's worth trying to do this and see where it leads.

With a set of sequences, for each N, consider the set of truncated sequences with only N flips from the starts of the sequences in S. This defines a number

P(S, N) = K / 2^N

where K is the number of distinct initial sequences of length N in S.

P(S, N) is clearly (easily checked) a decreasing sequence, so its limit is a number in the interval [0, 1]. I would like this limit to be the probability of S, but it would need to satisfy the axioms.

To be a probability measure for all sets of sequences (seems too much to hope for) the complement of a set S (denoted ¬S) would have to have probability that is 1 - P(S).  To prove this would require proving:

P(¬S) >= 1 - P(S)

and

P(¬S) <= 1 - P(S)

The first follows from the fact that for any N,  it is quite easy to see that P(S, N) + P(¬S, N) >=1

This is clear because every sequence of N flips has to appear in either S or ¬S (it could be in both). So K + K' >= 2^N

(K' being the number of initial sequences in ¬S)

Is it possible to prove the inequality the other way? I think not.

It's easy to define a set S so that both S and ¬S contain every possible finite sequence at the start.  All you need to do is included all sequences all but finitely many of whose values are "H" in S and include all sequences, all but finitely many of whose values are "T" in ¬S.

[Note there are a lot of other sequences whose location is not specified]

Both S and ¬S contain sequences with every possible finite start. So that demolishes my naive attempt at defining the probability of an arbitrary set of sequences.  It's a safe bet this can't be fixed.

However, the attempted definition of probability is in some sense a meaningful bound on the size of a set of sequences. It works fine for probability zero, when we find that the probability of the starts of the sequences goes to zero so it is clear that it makes sense to say that the probability of the set of sequences (a smaller subset of the set of sequences than all sequences with those starts) is zero.

Likewise where I referred to "probability 1" this can make sense if we can bound the complement to have probability 0.

For example, consider the set of sequences that does not contain a single specific subsequence SS, let's say it's of length N. Then it doesn't contain SS at index 1:N, and it doesn't contain SS at index (N+1):2N and so on.  Each of these conditions reduces the bound on the probability by a factor of (1 - 2^-N). Thus the upper bound on the probability of this set of sequences is smaller than any positive number, so it is zero.

So that glib claim was meaningful.

Since a countable union of sets of probability zero has probability zero, we can see that it is also meaningful to say that the probability of a sequence containing all finite subsequences is 1. (The probability of a finite subsequence lacking at least one of them is zero, from that infinite union).

im not really sure why ppl keep using small-time coin-flipping as the answer to all things random. plz tell me u u/s that it isnt. and if u sadly do ?...plz tell me gravity is NOT affecting the toss yet gravity is not static anywhere ? so plz stop w/ the arithmetic and go to summa our U-laws ?...like space & temperature & light & stuff like that.

help my frustration - pretty plz ?

This is not a logically consistent position.

To see this, observe that EVERY specific infinite sequence of results has probability zero. Do you really want to say that every infinite sequence of results is impossible? This is not useful as it contradicts the existence of such sequences, hence the entire analysis.

The lack of identification between "possible" and "non-zero probability" is characteristic of measures on uncountably infinite sets (probability distributions on countably infinite sets are not much different to those on finite sets).  For example, quantum mechanics considers the probability distribution of the location of a particle. For all but extreme distributions, the probability of a set containing a single location is zero. But it would not be useful to say all locations are impossible.