Can an Infinite Sequence of Coinflips Come Up All Heads?

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Intro

If there is an infinite sequence of coinflips (for simplicity, assume that heads or tails are the only possible outcomes for each coinflip, and that this is a fair coin), is it possible for the coinflip to come up heads? I think so, but how do we prove it?

Background

This issue bears relevance to the Eternal Society Argument against an infinite past. Roughly (in the paper the Eternal Society Paradox was published), an Eternal Society is a society that has existed for a beginningless, infinite duration of time and has the abilities of ordinary human beings in each year of its existence; e.g. in each year people in the society can flip coins, write books, sing songs, and pass on information possessed in the current year to the next year. Because of the society’s extremely modest abilities, it seems like an Eternal Society would be possible if an infinite past were possible (note that by “possible” in this article I’ll be referring to metaphysical possibility, as opposed to e.g. physical possibility).

Now imagine the Eternal Society has the following Annual Coin-Flipping Tradition: each year they flip a coin. If the coin comes up heads and they never did a particular chant before, then they do the chant; otherwise they do not do the chant for that year.

The coinflips are probabilistically independent events, so any particular infinite permutation of coinflips is equally unlikely but also equally possible. Consider scenario S1 in which the coin came up heads for the first time last year for the Eternal Society practicing the aforementioned Annual Coin-Flipping Tradition. The Eternal Society gets together to do the chant for the first time. This seems like it would be possible if an infinite past were possible (an eternal society with the ability of ordinary humans, by which I mean the society has the ability of ordinary humans in each year of its existence, could surely do something like this), but this scenario is provably not possible.

Again, the coinflips are probabilistically independent events, so if scenario S1 were possible, then another scenario, that we can call scenario S2, would be possible: the coin came up heads each year of the infinite past for the Eternal Society engaging in the Annual Coin-Flipping Tradition. If the coin came up heads each year, did the Eternal Society ever do the chant? They would have had to have done the chant some year, because they would have done the chant last year if they hadn’t done it yet (since the coin came up heads last year). And yet any year you point to, there is a prior year in which they would have done the chant if they had not done the chant before. So they had to have done the chant (since the coin came up heads last year), yet they could not have done the chant (there is no year they could have done it), and so this scenario creates a logical contradiction.

Although scenario S1 is not directly self-contradictory, scenario S1 is impossible because it implies the possibility of a logical contradiction. The Eternal Society argument against an infinite past goes like this:

1. If an infinite past were possible, an Eternal Society would be possible.
2. If an Eternal Society were possible, then scenario S1 would be possible.
3. If S1 would be possible, then S2 would be possible.
4. S2 is not possible.
5. Therefore, an infinite past is not possible.

The Eternal Society Paradox Argument Against an Infinite Past is a deductively valid argument—the conclusion (line 5) follows logically and inescapably from the premises (lines 1-4). A sound argument is a valid argument with all true premises, so the only way the argument can fail to be sound is with a false premise.

One could deny premise (1) particularly since that seems to be the most vulnerable premise, but as the Eternal Society Paradox paper says, “Surely there is something metaphysically suspicious about an infinite past if an eternal society with the abilities of ordinary humans can actualize a logical contradiction.” The idea that an infinite past is possible but an Eternal Society is not possible strikes me as overly ad hoc due to the Eternal Society’s extremely modest abilities (the abilities of ordinary humans in each year of its existence).

An Objection

One objection I’ve seen is that it’s just not possible for a coin to come up heads infinitely many times, such that S1 is possible (since both heads and tails coming up infinitely many times is possible) but S2 is not (it’s impossible for an infinite sequence of heads to happen). The reasoning behind this may stem from a confusion of the law of large numbers or thinking that if the probability of heads is 50% for each trial, an infinite sequence of such trials leads to a probability of 0, and a probability of 0 means that it’s impossible (this is not true; any infinite particular permutation of coinflips where both heads or tails has a “probability of 0” as the number of trials goes to infinity; so this “probability of 0” reasoning would imply that no outcome is possible for an infinite sequence of fair coinflips). Regardless, can we mathematically prove that an infinite sequence of heads is possible under the conditions of the Eternal Society Paradox? We can.

The Proof

To define some terms (with the caveat that different sources may define these terms slightly differently): in probability a random experiment like flipping a fair coin is called a trial. In the case of an individual coinflip for the scenario used here, the outcomes are binary meaning that there are only two possible outcomes (heads or tails). Mutually independent trials are where the probability of given trial’s outcome is unaffected by the outcomes of other trials, including any combination of the outcomes of the other trials. A sample space is the set of all possible outcomes of a random experiment.

In the case of an infinite sequence of coinflips, each trial has exactly two possible outcomes, each trial is mutually independent (in the sense that the probability of the outcome is unaffected by the outcomes of other trials), and the probability is the same in each trial. Some additional assumptions to make the reasoning a bit easier to follow if nothing else: each candidate outcome for a trial (heads or tails) has a nonzero probability, and any outcome with a nonzero probability is possible. Let H represent “heads” and T represent “tails.”

Recall that for each coinflip, both H and T have nonzero probabilities and thus both are possible. So the fact that the probability of given trial’s outcome is unaffected by the outcomes of other trials entails that which outcomes are possible for each trial is also unaffected by the outcomes of other trials.

So for each trial (and thus each sequence element in the infinite sequence of coinflips):

(a)	both outcomes (H or T) are possible; and
(b)	which outcome is possible is unaffected by the outcomes of other trials.

Given the aforementioned conditions, since every element in the sequence has the property of heads or tails both being possible values regardless of the values of the other sequence elements, this permits any infinite sequence of the binary values.

To illustrate, suppose we have a fair coin and the coinflips are mutually independent in the sense I described earlier, and there are an infinite number of coinflips. Can the first coinflip be heads? Yes, since for each sequence element, H or T is a possible outcome, which would include the first coinflip (confer (a)). Given that the first coinflip is heads, can the second coinflip also be heads? Yes, since the possible outcomes of the second coinflip is unaffected by the outcomes of other coinflips (confer (b)); hence both H and T can be used here. If the first two coinflips are heads, can the third one be also? Yes, because the possible outcomes of the third coinflip is unaffected by the outcomes of other coinflips (confer (b)), hence both H and T can be used here, and so on ad infinitum for all of the remaining sequence elements. And since the outcomes of heads is arbitrary here (e.g., we could just as well have used T, H, T for the first three sequence elements, and then choose whatever binary sequence we wish for the remaining sequence elements) this can be generalized so that the sample space consists of all infinite binary sequences.

But since the set of all possible outcomes consists of all infinite binary sequences, this would include a sequence of coinflips coming up H for each trial.

Eben Alexander’s Questionable NDE

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Intro

Eben Alexander was on Capturing Christianity’s channel recently (as of this writing) on NDEs, titled Eben Alexander Discusses His Wild NDE with Gary Habermas. I mentioned some things in the comment but couldn’t post links (the YouTube algorithm erases my comments when I do for some reason) so I’m writing this short article here.

Fraud?

At around 1:18:08 to 1:26:00 Alexander responds to a question of mine, “What is Eben Alexander’s response to reported evidence that his NDE experience was fraudulent (e.g., testimony from one of the doctors who treated him)?”

The genesis of this, as a casual internet might turn up, is the August 2013 issue of Esquire which published an exposé in an article called “The Prophet” which can be read for free online. Alexander’s says his FAQ responds to the lawsuits, which it does, noting that they were settled (lawsuits usually are, particularly when it's not a close call) but doesn't respond to the e.g., the doctor's testimony against him. The NDE article he refers to is apparently Eben Alexander's Near-Death Experience: How an Esquire article Distorted the Facts by Robert Mays (a guest editorial). That article does not interview Laura Potter, the key physician witness of the Esquire article (apparently she was unresponsive to telephone calls by this person). The article does claim a quote by her nonetheless supposedly issued to the Associated Press which if veridical would seem to cast doubt on the Esquire article, but I've been unable to verify the veracity of the quote (a google search turns up nothing solid).

One of the reasons for suspicion is that at the time Eben Alexander had a far higher than normal lawsuit load against him, something he left out in this interview when he talked about his lawsuits (settlements are the norm, not the exception, and are usually done when it's not a close call about who would win). The Esquire article reports:

By the time all his pending cases are resolved, Alexander will have settled five malpractice cases in the last ten years. Only one other Virginia-licensed neurosurgeon has settled as many cases in that time period, and none have settled more.

Alexander had a powerful motive to acquire cash (settlements aren't necessarily cheap, especially if you undergo them far more than usual as Alexander did). Alexander had his alleged NDE when there was a $3 million lawsuit pending. He made a lot of money with his NDE, selling webinars and even co-founding an organization called “Eternea” where (at the time) if you paid $1,200 a year or more, you could qualify for a membership status called “archangel.” There was also a “Governor’s Guild” in which membership dues began at $10,000 per year. Lifetime membership was offered to anyone who made a major lump sum gift of $25,000. To insinuate that there was no financial gain here (1:25:36 to 1:25:59) strikes me as somewhat misleading.

At 1:23:51 to 1:51:55 he says the Esquire article author (Luke Dittrich) was “obviously blackballed” by the industry, but the evidence he cites for this (not having published an article in the press for several years) seems insufficient, since it's unclear that writing articles for newspapers and magazines is his intended primary source of income. However, Alexander's claim that some 200 scientists questioned Dittrich's book appears to be factual (one can find it reported in Scientific American) to at least some degree; there was dispute about Dittrich's characterizations of an MIT neuroscientist.

I'm not saying that I know Eben Alexander is a grifter, but I think some amount of skepticism is warranted.

An Objection to the Eternal Society Paradox

Home  >  Philosophy  >  Atheism/Theism

Intro

The Eternal Society Paradox is one in which an eternal society with the abilities of ordinary human beings in each year of its existence uses its modest abilities to create a logical contradiction, thereby casting doubt on the metaphysical possibility of an infinite past. If the past is finite, this in turn can be used as part of an argument for the universe having a transcendent personal cause. Jimmy Akin wrote an article that among other things critiqued the Eternal Society Paradox argument against an infinite past. The original link is here but in case something disastrous happens, the archived link is here.

Background

Roughly (in the paper the Eternal Society Paradox was published), an Eternal Society is a society that has existed for a beginningless, infinite duration of time and has the abilities of ordinary human beings in each year of its existence; e.g. in each year people in the society can flip coins, write books, sing songs, and pass on information possessed in the current year to the next year. Because of the society’s extremely modest abilities, it seems like an Eternal Society would be possible if an infinite past were possible (note that by “possible” in this article I’ll be referring to metaphysical possibility, as opposed to e.g. physical possibility).

Now imagine the Eternal Society has the following Annual Coin Flipping Tradition (ACFT):

For each year y in the beginningless, infinite past: the society flips a coin, and they do the chant if and only if these two conditions are met: (a) the coin comes up heads; (b) they never did the chant in a year prior to y.

Less technically but somewhat more roughly: ACFT is where every year they flip a coin and they do a chant in that year if and only if (a) the coin comes up heads; and (b) they never did the chant in a prior year. Even more crudely put: each year they flip a coin. If the coin comes up heads and they never did a particular chant before, then they do the chant; otherwise they do not do the chant for that year.

The coin flips are probabilistically independent events, so any particular infinite permutation of coin flips is equally unlikely but also equally possible. Consider scenario S1 in which the coin came up heads for the first time last year for the Eternal Society practicing the aforementioned Annual Coin Flipping Tradition. The Eternal Society gets together to do the chant for the first time. This seems like it would be possible if an infinite past were possible (an eternal society with the ability of ordinary humans, by which I mean the society has the ability of ordinary humans in each year of its existence, could surely do something like this), but this scenario is provably not possible.

Again, the coin flips are probabilistically independent events, so if scenario S1 were possible, then another scenario, that we can call scenario S2, would be possible: the coin came up heads each year of the infinite past for the Eternal Society engaging in the Annual Coin Flipping Tradition. If the coin came up heads each year, did the Eternal Society ever do the chant? They would have had to have done the chant some year, because they would have done the chant last year if they hadn’t done it yet (since the coin came up heads last year). And yet any year you point to, there is a prior year in which they would have done the chant if they had not done the chant before. So they had to have done the chant (since the coin came up heads last year), yet they could not have done the chant (there is no year they could have done it), and so this scenario creates a logical contradiction.

Although scenario S1 is not directly self-contradictory, scenario S1 is impossible because it implies the possibility of a logical contradiction. The Eternal Society argument against an infinite past goes like this:

1. If an infinite past were possible, an Eternal Society would be possible.
2. If an Eternal Society were possible, then scenario S1 would be possible.
3. If S1 would be possible, then S2 would be possible.
4. S2 is not possible.

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5. Therefore, an infinite past is not possible.

The Eternal Society Paradox Argument Against an Infinite Past is a deductively valid argument—the conclusion (line 5) follows logically and inescapably from the premises (lines 1-4). A sound argument is a valid argument with all true premises, so the only way the argument can fail to be sound is with a false premise.

One could deny premise (1) particularly since that seems to be the most vulnerable premise, but as the Eternal Society Paradox paper says, “Surely there is something metaphysically suspicious about an infinite past if an eternal society with the abilities of ordinary humans can actualize a logical contradiction.” The idea that an infinite past is possible but an Eternal Society is not possible strikes me as overly ad hoc due to the Eternal Society’s extremely modest abilities (the abilities of ordinary humans in each year of its existence).

The Rebuttal

When using the phrase “Eternal Society Paradox” the author seems to have in mind specifically scenario S2. From the article:

…the solution [to the paradox] is straightforward: The Eternal Society Paradox is presupposing a logical contradiction.

How is this a solution? The fact that the Eternal Society Paradox (in scenario S2) entails a logical contradiction is part of the point; it’s not a solution to simply to concede part of the claim.

It presupposes a first and a last element to a supposedly infinite series, so the Eternal Society Paradox commits the First-and-Last Fallacy.

Simply calling something a fallacy doesn’t make it so. The “first-and-last fallacy” is described as follows:

The First-and-Last Fallacy occurs if and only if a person envisions a supposedly infinite series as having both a first and a last element.

I didn’t envision it, and neither did scenario S2. Indeed, part of the reason there’s a contradiction is that the scenario (if anything) envisions that there is no first element.

Another problem with the objection is that the Eternal Society Paradox Argument is logically valid, so if the argument is unsound, which premise is false? At first blush at least, this objection doesn’t actually attack any premise of the argument! My conclusion was initially that this objection is a red herring (for more on red herrings, see my red herring video). But see the update below:

Update

After personal communication with the author in 2023-07-31-MO I discovered I hadn’t quite correctly understood his objection. While at the time it wasn’t clear to me which premise he was attacking, it seems he was attacking the premise that says scenario S2 is impossible. Although proof of its logical impossibility can be proven via symbolic logic, we won’t need to go into something so complex here. Here was Akin’s understanding of the Annual Coin-Flipping Tradition that I’ll label ACFT*:

The society flips a coin each year of the infinite past. If the coin comes up heads, then they do the chant if and only if it’s the first time the coin comes up heads.

The objection goes as follows: scenario S2 is the conjunction of ACFT* and the coin coming up heads each year. Since there is no first year the coin came up heads, they never do the chant. There is no self-contradiction with ACFT* and the coin coming up heads each year; the society just never does the chant in any year. Therefore, S2 is not self-contradictory and premise (4) is false. To think that in S2 they had to do the chant assumes that there was a first year the coin came up heads (as well as a “last time” the coin came up heads, since there is a present year); hence a “First and Last Fallacy.”

The problem is that ACFT* is not synonymous with ACFT. To see this I’ll use a reductio ad absurdum argument, which assumes the opposite of what we want to prove (the reductio assumption) and show that it leads to an absurdity (such as a self-contradiction). Recall that ACFT says that the society does the chant when these two conditions are met: (a) the coin comes up heads; and (b) they have not done the chant in a prior year. Assume for reductio that when following ACFT the society never did the chant in any year and the coin comes up heads each year. Now consider last year: are both conditions (a) and (b) met?

• Condition (a) is met, since the coin came up heads last year.
• Condition (b) is also met since they never did the chant in a prior year. (Since they never did the chant in any year.)

So on AFCT they would have done the chant last year since both conditions are met, which contradicts the original assumption that they never did the chant in any year. Thus it is logically impossible for them to have never done the chant if the coin came up heads each year and they followed AFCT. It also follows that ACFT and ACFT* are not equivalent; when the coin comes up heads each year, the society never doing the chant is possible on ACFT* but logically impossible on ACFT (since last year, both conditions (a) and (b) would have been met and they would have done the chant last year).

Somewhat more formally, the reductio argument goes as follows, where the premise (6) is true by definition of the ACFT:

6. If ACFT is true, then for any year y, the society does the chant in year y when (a) the coin comes up heads in year y; and (b) they did not do the chant prior to y. (Follows from the definition of ACFT)

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7. In scenario S2, ACFT is true, the coin came up heads each year, and they never did the chant in any year. (Reductio assumption; we’ll show that this leads to an absurdity.)
8. Last year, the coin came up heads (follows from 7) and thus condition (a) is met.
9. Last year, they never did the chant in a prior year (follows from 7) and thus condition (b) is met.
10. Last year, conditions (a) and (b) are met. (Follows from 8 and 9.)
11. If conditions (a) and (b) are met last year, then they did the chant last year. (Follows from 6.)
12. They did the chant last year. (Follows from 10 and 11.)
13. They did not do the chant last year. (Follows from 7.)
14. They did the chant last year and they did not do the chant last year. (Follows from 12 and 13.)
15. Therefore, (7) is false. (Follows from reductio and 14.)

We know that scenario S2 includes ACFT and that the coin came up heads each year, so the part of (7) that must be false is the idea that they never did the chant in any year.

Conclusion

This illustrates an important lesson in spotting straw men; sometimes statements that might initially appear to be synonymous are actually subtly different, and sometimes that subtle difference matters when evaluating objections. When evaluating an objection, consider how that objection fares against the position as it was originally stated. In the case of the Eternal Society never having done the chant in scenario S2, when we apply ACFT’s conditions (a) and (b) to last year we see that they would have done the chant last year, whereas on ACFT* they would not have done the chant. I can understand why one might think that ACFT and ACFT* are synonymous, but that wasn’t quite true; they were strongly similar but subtly different, and that subtle difference made all the difference in the world.