Congratulations, you've committed yourself to Descarteian solipsism. Unless you doubt your mind's existence, however, that's a contradiction.
The completeness theorems don't talk about axioms. It's about the logical consequences of using a set of axioms.
When we argue about the existence of external reality, we're arguing about the validity of self evident (axiomatic) claims.
However, solipsism is pretty silly to everyone.
Medicine isn't really medicine if it permanently harms people. Not that everything harmful is outlawed, but alcohol, cigarettes, and vice companies don't really attempt to fool you on selling you a lie of a healthy product.
My implied point is that transgender surgeries could be considered dishonest medicine. Might need to call it a snake oil or vice industry and keep it as a separate entity from real medicine.
Anyways, your views are very libertarian with bodily automony, so I guess you would have to take a strong stance on legalizing all drugs including hard ones like heroin to be consistent. It's not hard to see how curtailing heroin might save lives and the same logic can be applied to people who want transgender surgeries.
In other words, not only that you can define things into existence, but proving existence without definitions is impossible, because senses are flawed.
I think there's a better way of putting this. The brain makes assumptions or internal models of world and uses our senses to test the accuracy of its predictions. I wouldn't go as far as to say our senses are necessarily flawed.
We also see the world through a set of perspectives dependent on our beliefs, concepts, and knowledge, filtering our perception of the world.
Also, Godel's theorems don't necessarily apply to consciousness because consciousness might not be computational. I'm personally agnostic on the idea.
If consciousness does operate on a formal system, then I think it means there are truths or facts about things (especially math) which we can not know. As it applies to observable existence, I think it kinda weak.
I used to believe objective truth, but now I think Nietzsche is right that "there are no facts only interpretations." I haven't read Nietzsche, but I studied snippets of his philosophy. I now subscribe to the theory of perspectivism. I came to this conclusion because the mind isn't a mirror of of objective reality. The mind participates in the active construction/interpretation of reality. This interpretation of reality doesn't have to match objective reality; it merely needs to be useful enough for us to navigate the world.
I'm not a relativist because I think some perspectives are better than others; instead, I'm a pragmatist and believe knowledge must be useful to action and survival.
I think perspectives can inform us about an underlying objective reality, but they represent incomplete approximations of what reality "is." Therefore, I think multiple differing perspectives can be counted as knowledge/truth because our human ability to understand the world is imperfect.
I draw a lot of my current philosophical inspirations from Kant, Nietzsche, and Jordan Peterson, but I study philosophy as an amateur.
