>>11309926i mean, probably a thousand people have already told you about the things that don't make sense in your screenshot, so whatever i say will ring on deaf ears, but just for one moment reconsider whether you might actually be the most special person in the world or whether you're just insanely arrogant:
it's fine to add $\pm \hat{\infty}$ to the real numbers and just say "alright, these are the real numbers with two new points added". however, there are a bunch of unaddressed questions:
- what is the norm/absolute value $\| \hat{\infty}\|$? how is it defined?
- how do you define objects like $\hat{\infty} - a$ for positive reals $a$? just saying "$\hat{\infty}$ is like $\infty$ but without additive absorption" does not actually give any meaning to the numbers $\hat{\infty} - a$. you prove in theorem 1.10 that, retrospectively, these objects were real numbers all along, but that runs into the next problem:
- theorem 1.10 relies on the ordering you defined on objects of the form $\hat{\infty} - a$ to fulfil the property $x < y \implies x \neq y$. however, you have not proven anything about the order that you define, but you have only written that you "suppose this order exists". and sure, you're free to define a relation between your objects $\hat{\infty} - a$ for different positive reals completely as you like, but without proof, who is to say that this relation actually defines an order?
- why does $\|\hat{\infty}\| = \|\infty\|$ imply that $\overline{\mathbb{R}} = [-\hat{\infty},\hat{\infty}]$? we do not even know what $\|\hat{\infty}\|$ is supposed to be, yet it somehow tells us something about the shape of the extended real numbers? and what is $[-\hat{\infty},\hat{\infty}]$ even supposed to be? the real numbers with $\pm \hat{\infty}$? the real numbers with $\pm \hat{\infty}$ and $\pm \infty$? the real numbers with $\pm \hat{\infty}$, $\pm \infty$ and all objects $\hat{\infty} - a$ for all positive a?
