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The Argument From Infinity

If you live forever then you will definitely encounter a completely terrible scenario like being trapped in a black hole or something.

 
I have noticed a tendency, for people to conclude that an infinite set implies that the set contains some potential element $Y$.
 
Say for example, that you live forever, this means that your existence is an infinite set. Let’s denote your existence as $E$.
 
$E = {x_1, x_2, x_3, …}$
Where each $x_i$ is some event that can potentially happen to you.
  The fallacy of infinity is positing that because $E$ is infinite, $E$ contains $x_j$.
 
However, this is simply wrong. Before I prove that the infinity fallacy is in fact a logical fallacy, I will posit a hypothesis as to the underlying cause of the fallacy of infinity.
 
I suspect it is because people have a poor understanding of the nature of infinity. They assume, that because $E$ is infinite, $E$ contains all potential $x_i$. If $E$ did not contain any potential $x_i$, then $E$ would not be infinite, and since the premise is that $E$ is infinite, then $E$ contains $x_j$.
&nsbp;

Counter Argument.

I shall offer an algorithm that would demonstrate how to generate an infinite number of infinite subsets from an infinite set.
 
Pick an element $i$ in $N$. Exclude $i$ from $N$. You have generated an infinite subset of $N$.
There are $\aleph_0$ possible such infinite subsets.
Pick any two elements from $n$ and exclude them. You have generated another infinite subset of $N$. There are $\aleph_0$ \choose $2$ possible infinite subsets.
In general, we can generate an infinite subset by excluding $k$ elements from $N$. The number of such infinite subsets generated is $\aleph_0$ \choose $k$.
 
To find out the total number of infinite subsets that can be generated, take
$$\sum_{k=1}{\aleph_0} {\aleph_0 \choose k}$$

However, these are only the infinite subsets of finite complements. To get infinite subsets of infinite complements, we can pick any (finite) subset of $\aleph_1$, and find the product of that set. Take only all multiples of that set, or exclude all multiples of that set. That gives you $2$ infinite subsets for each finite subset of $N$.
I can generate more infinite sets, by taking any infinite sets, and adding any $k$ excluded elements to it—or similarly subtracting $k$ elements from it.
However, this algorithm doesn’t generate all possible infinite subsets of $N$ (e.g the prime numbers, the Fibonacci numbers, coprime numbers, or any infinite subset that satisfies property $P$ e.g solutions to equations with more unknowns than conditions etc). The total number of possible infinite subsets (including those not generated by my algorithm) is $>= \aleph_1$ (around the same cardinality as the real numbers).
 
To explain the counter argument in simple terms:

There are an infinite number of even numbers, but none of them are odd.
There are an infinite number of prime numbers but none of them are $6$.
There are an infinite number of multiples of $7$, but none of them are prime save $7$ itself.

The number of possible infinite subsets is far more than the number of elements in the parent set. In fact, for any event $x$ (or finite set of $x$), the number of infinite sets that do not include any $x_i$ is infinite. To posit that simply because $E$ is infinite, that $E$ contains $X_i$, is then wrong.  

Alternative Formulation/Charitable Hypothesis.

This states a weaker form of the infinity fallacy, and a better argument.

If you leave forever, the probability is arbitrarily close to 1 that you would end up in a completely terrible scenario.

Let the set of events anathema to you be denoted $F: F = {y_1, y_2, y_3, …, y_m}$.
 
We shall now attempt to construct $E$.
For each element $x_i$ in a set $A$, the probability that $x_i$ is not in $F = \frac{# A - # F}{# A}.
 
$${# A - # F}{# A}{# A} \to 0 \,\,\, as \,\,\,\, #A \to \infty$$.
Thus, when $# A = \infty$
$Pr($\neg bad event$) = 0$ $Pr($bad event$) = 1 – Pr(\neg$ bad event$)$.
$1 – 0 = 1$.
$\therefore$ the probability that you would encounter a bad event is infinitely close to $1$.
 

Comment

I cannot comprehend how probability works in the face of infinity, so I can’t respond to the above formulation (which if valid, I’ll label the “infinity heuristic”).
 
Another popular form of the argument from infinity:

If you put a million monkeys on a million type writers and let them type forever, the entire works of Shakespeare would eventually be produced.

There is an actual proof of this which is sound. This implies, that a random number generator on any countable set will generate every element in that set. The entire sample space would be enumerated. However, there are several possible infinite sets that do not have all the elements in the set. It bears mention though, that I am admittedly terrible at intuiting infinity.

The question remains though: is the argument from infinity a fallacy or a heuristic?  
What do you guys think? Is the argument from infinity the “infinity heuristic”, or is it just a fallacy?

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Unless the probability of the terrible scenario (taken for example as an annual probability) is constantly decreasing, it will indeed happen almost surely.

Not true. For example, if the probability of the terrible scenario at some point drops to zero, that is sufficient.

Zero? What's that?

It's that which is not a probability :-P

Yes, I was assuming that we are assigning reasonable probabilities.

What is the current probability of Nikita Khruschev ordering a nuclear strike on the United States? What was this probability 55 years ago and what happened to it?

The current probability is pretty low. It was higher before and dropped pretty dramatically in 1971. It is also currently getting lower since it is becoming more difficult to resurrect him. But it is nowhere near zero yet.

On the other hand, this may satisfy the condition of constantly getting lower, so it may never actually happen.

since it is becoming more difficult to resurrect him

Tell me more about this resurrection thing and why do you think it's becoming more difficult as time passes.

More scattering of information, presumably.

The second law of thermodynamics I see...

"summation of {i = 0} to n of (n combination i) = 2^n"

This is not a proof that "2^{aleph_0}" is the cardinality of the set of the subsets of natural numbers. You assume it works in the infinite cardinal case (without proving it), and then say that you thus proved it. You got confused by notation.

"I shall proffer a mathematical proof to show that for any infinite set of cardinality aleph_0 (the cardinality of the set of natural numbers) there are aleph_1 (2aleph_0) distinct infinite subsets."

No. 2^{aleph_0} is /by definition/ the cardinality of the set of the subsets of the natural numbers. It's named that way to allow the intuition of "summation of {i = 0} to n of (n combination i) = 2n" to work with cardinalities.

"aleph_1 (2^{aleph_0})"

aleph_1 = 2^{aleph_0} has been shown to be independent from ZFC. ie, if we haven't worked within inconsistent math for that past 60 years, what you just said is unprovable. You might have confused aleph and beth numbers.

I have edited the article to be more accurate.

When K is a cardinal, 2^K is defined as the cardinality of the set of subsets of K (or alternatively, the cardinality of the set of functions from K into {0,1}, but it is easy to show that these are equivalent). Your "proof" of this is completely wrong. For any finite k, (aleph_0 choose k) = aleph_0, so the summation (k=0 to infinity) of (aleph_0 choose k) is (aleph_0)^2, which is also aleph_0. That's because you only added together the subsets with finite complement, of which there are only countably many. Also, aleph_1 is not necessarily the same as 2^(aleph_0). I'm also not sure why you even brought this up.

I don't think anyone has actually committed the fallacy that you describe, and when people say things like that, they mean something more like your alternative formulation (described more precisely by entirelyuseless).

There is another fallacy: that an infinite path includes the infinite number of elements. However, we could imagine circular path, which has the finite number of elements. Eternal return by Nietzsche is a (wrong) example of such path.

It may help reject some false ideas about immortality and create the more positive image of it, without entrapping into black holes or need to remember infinite number of things.

It may help reject some false ideas about immortality and create the more positive image of it

I don't think that most people would think circular immortality is especially positive.

If we have linear and circular immortality, we could create morу complex constructions of pathways in the space of possible minds with merges, attractors, dead ends, two-dimensional figures etc. It completely dissolves fear of bad infinity associated with the naive understanding of immortality.

I don't see a convincing argument here. In fact, I don't see any argument at all, convincing or otherwise.

Do you think the argument from infinity is in fact a valuable heuristic?

It is not a heuristic; it is a mathematical proof, given the condition I mentioned (namely that the probability does not diminish indefinitely. A circular path which does not contain the element has a probability of zero for the element, but we have no reason to expect circular paths.)