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The finite axiom of choice doesn't suffer from paradoxes of infinity. It's mostly intuitive to computer scientists that infinities don't exist ;) And we are a
by ukj 6y ago
The finite axiom of choice doesn't suffer from paradoxes of infinity.
It's mostly intuitive to computer scientists that infinities don't exist ;)
And we are also in the habit of proving/realizing our choice-functions rather then assuming them axiomatically.
https://en.wikipedia.org/wiki/Nondeterministic_programming https://en.wikipedia.org/wiki/Nondeterministic_programming
- cjfd 6y agoWell, if one assumes finiteness all of these paradoxes go away and this discussion is not about anything anymore. I studied physics and am quite attached to infinities. That one constrains oneself to a finite state system mostly does not mean that one is not allowed to look out of the window towards the stars and marvel at infinity anymore.
- ukj 6y agoFrom your comment above I am making the (admittedly presumptuous) inference that you didn't study Quantum physics. https://en.wikipedia.org/wiki/Renormalization https://en.wikipedia.org/wiki/Renormalization If giving up infinities allows me to understand The Universe better, I don't see how that robs me of marveling at the stars - it only makes it more exciting! I can't comprehend an infinite universe. Nobody with finite memory/time can.
- cjfd 6y agoWell, I did a PhD in quantum field theory and as part of that published a paper on renormalization..... Of course the infinite sets involved can be approximated using finite sets if one does not mind losing important symmetries. Still, these finite sets would still be frightingly and incomprehensibly large.... If one wants to comprehend anything one may be better off just ignoring quantum mechanics..... And I am not talking about the measurement problem. I am talking about the incredible vastness of the state space. The wave function of the universe is an incredible and completely untractable thing even if one assumes it is finite. It contains all possible histories of the universe for one thing and lets all of these possible histories interact with one another.
- ukj 6y agoI hear you (and I obviously made a bad inference re: your background - apologies), but there is a distinction to be made between large-but-finite (computable? halting?) and infinite (non-computable? non-halting?) sets. We can't comprehend the universe using numerical methods - it's too complex for our brains/computers. But we can understand complexity using symbolic methods. Symbolism/representationalism (religion) is rather inevitable part of the human condition. ∞
- cuspycode 6y agoInfinite sets are a useful abstraction, but uncountable infinities can probably be avoided since the Löwenheim-Skolem[0] theorem says that any countable first-order theory has a countable model. [0] https://en.wikipedia.org/wiki/Löwenheim-Skolem_theorem https://en.wikipedia.org/wiki/Löwenheim-Skolem_theorem
- ukj 6y agoLöwenheim-Skolem doesn't hold for higher order logics such as type theory/Lambda calculus. Basically, it doesn't hold for any constructive logic.