# Mathematical Logic and the Philosophy of Life

# Table of Contents

# Peano Arithmetic

# Exploring Extraterrestrial Life

Or, how to prove that there are other forms of life in the universe, without - or before - finding other life forms.

Perhaps you imagine that discussing mathematical logic and the Fermi Paradox would not fit into a LinkedIn post.

I accept the challenge, with the possibility of making a fool of myself.

Given that our time is short, I present all the first 9 axiomatic sentences from the image, leaving the last line for the reader to ponder.

I will focus on dissecting the last proposition, and only it. For there lies a not-so-obvious line, which has implications for the question: Are we alone in the universe?

\begin{equation} \begin{aligned} \left( F(0) \land \forall x (F(x) \implies F(x+1)) \right) &\implies \forall x (F(x)) \end{aligned} \end{equation}

Let me explain...

# Dissecting the Proposition

Let's look at the left side of the expression.

We need to try to contextualize the meaning of the predicates.

Predicate ( F(0) ): There is life (on Earth), and this is case zero; empirical and tautological.

Now, we have:

$$ \forall x (F(x) \implies F(x+1)) $$

What would it mean for this to be true?

In words: if I know that a condition ( F ) has the following property: if it is true for ( x ), it must necessarily be true for ( x+1 ).

In our example, what would be the contextual condition?

"If the physics of the universe implies life, then necessarily". In other words, "given that we find a planetary system that can have life, it will necessarily be a case of life". This would be the positive meaning of this proposition in this hypothetical context.

**Note**: This may or may not be factually true.

Suppose such a theory has been formulated and proven in a "laboratory" and replicated! And it is as well-established a scientific fact as the Pythagorean theorem.

Logically, if there is life on our planet, and every planetary system that can have life will have it. Then, in a universe with infinite planetary systems in conditions similar to ours, there will be multiple occurrences of life.

Thus, we conclude

$$ \forall x (F(x)) $$

Throughout our universe, which has infinite possibilities of life, there are (infinite) forms of life inhabiting it.

\begin{equation} \begin{aligned} \left( F(0) \land \forall x (F(x) \implies F(x+1)) \right) &\implies \forall x (F(x)) \end{aligned} \end{equation}

Q.E.D.

# Conclusion

**Disclaimer**: This text is a reflection on how we can reach conclusions about a topic of space exploration without leaving Earth. Using logic. This text does not aim to prove that there is other life. Nor that the theory that would allow such a conclusion has already been formulated and proven. Nor that any such life will necessarily exist.