Whenever I get around to studying modern mathematical logic I have this feeling of awe, as if I managed to get a glimpse of the potential of the "true nature of the world".
Take the Löwenheim–Skolem theorem (mentioned in the post). It essentially tries to answer "what is the potential of illusion" within the first order logic, and shows that the potential is pretty damn big. You can be mind-blowingly confused about the "true nature of your reality". As you, a first-order logic creature, walk around, explore and interact with your space you can fool yourself into believing all kinds of marvellous things (for example that real numbers are "continuous", that they're inherently more complicated than naturals), you can even prove such things like our good friend Cantor did, never knowing that you were in fact always in a boring, aleph-zero universe. A universe built out of strings, or maybe natural numbers, nothing more than that. It's just that the daemon (engineer) ruling your world is incredibly clever and is adeptly laying out the bricks as you walk around, always putting them down just in time before you manage to turn your head around. And you'll never be able to catch that daemon and unravel the whole scheme.
I am looking forward to the next parts! I have also always wanted to understand forcing.
Take the Löwenheim–Skolem theorem (mentioned in the post). It essentially tries to answer "what is the potential of illusion" within the first order logic, and shows that the potential is pretty damn big. You can be mind-blowingly confused about the "true nature of your reality". As you, a first-order logic creature, walk around, explore and interact with your space you can fool yourself into believing all kinds of marvellous things (for example that real numbers are "continuous", that they're inherently more complicated than naturals), you can even prove such things like our good friend Cantor did, never knowing that you were in fact always in a boring, aleph-zero universe. A universe built out of strings, or maybe natural numbers, nothing more than that. It's just that the daemon (engineer) ruling your world is incredibly clever and is adeptly laying out the bricks as you walk around, always putting them down just in time before you manage to turn your head around. And you'll never be able to catch that daemon and unravel the whole scheme.
I am looking forward to the next parts! I have also always wanted to understand forcing.