Argument From Infinity

DragonGod · 5 Jun 2017 21:33 UTC

NOTE: This post contains LaTeX; it is recommended that you install “TeX the World” (for chromium users), “TeX All the Things” or other TeX/LaTeX extensions to view the post properly.

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?

Argument From Infinity

The Argument From Infinity

Counter Argument.

Alternative Formulation/​Charitable Hypothesis.

Comment

Alternative Formulation/Charitable Hypothesis.