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

The limit of certain expressions would approach 50%, yes.
The limit of others would approach zero, which is interesting. For example (x choose x/2) / (2^x) is the probability of there being an equal number of H and T after x flips (if x is an even number) and this gets smaller for a larger number of flips. Intuitively this makes sense as the more flips you have, the more chance there is for something to "go wrong" so to speak.

The point about all heads is if the experiment is flipping a coin infinitely many times, but also, you repeat that experiment an uncountably infinitely many times. In that case, you get every possible infinite sequence, which will include all heads.

^^ You can't repeat an infinite sequence twice, thrice or an infinite number of times, consecutively. Only concurrently and after all, it would still be an infinite number. Wouldn't make any difference there so that's glib and incorrect.

That's clever and yes.

This is not true. If you toss a billion times, and you repeat that an infinite number of times, there will eventually be a case where you get a billion heads. That is a certainty. However, that is not the case with an infinite number of tosses. In that case, there is only one outcome.... 50-50.  There is never a case where you toss an infinite number of times and get an infinite number of heads. 

There is an infinite number of infinite sequences, but they all have one thing in common...an equal number of heads and tails. That's the nature of infinity. It is not the nature of any finite number, however large.

A more extreme example, perhaps, of my own thinking on the subject. It accepts infinity as a perfect ideal.

To be fair I had to edit that part. You have to repeat the experiment an uncountably infinite number of times since there are uncountably infinite sequences (as Elroch pointed our earlier, and as Cantor's diagonal argument shows).

As for there being only one outcome which is 50/50, first of all, infinity divided by 2 is not defined, so it doesn't make sense to talk about half of infinity. It's better to write expressions and then take limits because that's explicit. In the expression I gave earlier, the probability of having exactly 50% is zero. In a law of large numbers formulation, the limit would be the expected value, which would be 50%.

I'm going to stand on what I said. In an infinite number of infinite series, every one of those infinite series will break 50-50 heads / tails. It's a mathematical impossibility to have an infinite number of tosses and have them all be heads......or any distribution other than 50-50. 

And I don't have to divide infinity by two to claim that the distribution of heads and tails will be 50-50. It just means that the distribution gets closer to 50-50 as you approach infinity, and 50-50 is the limit as n approaches infinity.....as I learned in first semester freshman calculus. 

Every specific infinite sequence has probability zero.

The probability of getting one specific sequence after x number of flips is 2^-x

An infinite number of infinities is ridiculous. As if infinity isn't sufficient.

As I pointed out earlier, Elroch is often wrong. Very interested in the account of possible differences in how the maths works, dependent on the mathematician's beliefs regarding the nature of infinity. I rather think I've won that argument. Be very surprised if there IS a difference!

@4110

I think you're confusing sequences with distributions.  Again....in an infinite number of tosses, there can only be one distribution.....50-50.

Yes, that's right, the proportion approaches 50% even though the linear difference grows (I assume to infinity)

A very large number divided by infinity is zero which is why that works I suppose... funny things happen with infinity, so that's why you need specific expressions.

Not to mention that half of infinity .... seems like infinity to me.
Half of zero is zero too.

Yes, half of infinity is indeed infinity. In an infinite number of coin tosses, the number of heads and the number of tails will each be infinite.....but it will be a one to one correspondence....for every head there will be one and only one tail, consequently a 50-50 distribution. It is a certainty.  If you believe in mathematics, you ultimately have to believe this.

There's something to be said for simply one person to bounce ideas around with. Thanks @Mike.

There's also a problem with the somewhat positivist interpretation. It really stifles creativity. Where would we be if we all had to think like dead mathematical theorems which don't even contain an innate truth but more like an artificially super-imposed rationality?

Again, we need specific expressions. The probability of each result is zero, but that doesn't mean each result is impossible.

I agree that as the number of flips goes higher, the proportion approaches 50%.

I also say that flipping all heads is as likely as flipping any other infinite sequence (that is, the probability is zero) and that if you perform the experiment uncountably infinite times, all combinations will happen.

And I say this is 180 degrees wrong.  It would be right for any finite number of tosses, but it's not right for infinity. 

I don't believe the mathematics shows that at all.

The proportion is 50%, but you're guaranteed to NOT have a one to one correspondence.

An unintuitive result for sure

You're welcome....I think. Most of these debates go over my head, but my patience has paid off and finally there is one I can at least participate in.

Not necessarily. It's an infinite sequence, so in every sequence it will even out to an even number of heads and tails.

Elroch seems to be misapplying the Bernoulli Sequence. He doesn't understand what infinity is: it's as though it can be a constant which can be manipulated. That's what it can be in maths but not when it's translated to a need to imagine the reality of that reality. It's a different logical set. All this time I've had to imagine the ideas after not really having thought about them much before, at least for many years and it was difficult. Sometimes, new ideas that should have been instantaneous took two days to emerge. Finding someone else who thinks alike is empowering.