[theWheez]

This, alongside the second incompleteness theorem, showed me that any formal system cannot prove itself, and any system deemed "consistent" can only be considered as such from outside that system.

A great way that I visualize it (thanks to Hofstadter) is like Escher's never ending staircase painting. If you look at each step, it looks just fine. In fact, you can say that each step on its own is "consistent". But it is only when you take a step back, outside the system of each isolated step, that you see the paradox.

Similarly, the theorem applies this to mathematics, and by extent, any formal system. Just as we can see each "step" going from one to the next, in mathematics we must hold an axiom as truth without proof, so that subsequent theorems may use it as the next "step". It is, in a sense, a paradox which we agree to use.

In practical terms, I may extrapolate that further and more philosophically than was intended (or that I have commonly seen), but the subsequent Halting Problem combined with the Curry-Howard isomorphism shows me that Godel's Theorem has major real world effects. The idea that there can be something that is true but not provable rings true to me in many regards. I, like many, spend nearly every moment of the day unconsciously putting faith into the "truth" that I am not alone in the universe, thereby fending off solipsism.

I cannot prove "I" exist. But it is true. What are the other truths that cannot be proven? That question guides my thinking.

[psychometry]

The FIT is a statement about finitary number theory. Even Godel, who was an ardent platonist, wouldn't try to extrapolate it to philosophy.

[theWheez]

Yes, I understand that, and am well aware of Godel's position on its application. Hence my statement that I extrapolate it further than intended.

My extrapolation is influenced by the Curry-Howard isomorphism, equating computer programs and mathematical proofs. Extrapolation to "reality" and philosophy, to me, comes from my belief of computing as universal and that consciousness is not so separate from computing.

[coldtea]

Thankfully we're not constrained to only consider it the way Godel did, but are free to extrapolate it to philosophy...