Btw I wish I had learned at school that numbers have to be constructed to exist. To notice that, and to understand the implications took me decades, and it's crucial for understanding the nature and models and how our minds represent reality.
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Replying to @Plinz
As long as nonconstructable mathematical entities are consistent and useful, why not consider them to exist?
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Replying to @lacker
Because this assumption leads to contradictions. If you understand mathematics as a universal Platonic code library, it means that you introduce critical and unfixable bugs into the kernel.
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Replying to @Plinz
I don’t think it leads to contradictions to allow the existence of an unconstructable entity. The axiom of choice doesn’t lead to contradictions, right?
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Replying to @lacker
Gödels proof concerning the Entscheidungsproblem lays out the core of the problem, Turing's Halting problem nails it. If you don't prove the existence of a computable algorithm that can compress your statements to the axioms, you cannot claim their semantic equivalence.
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But Gödel also tells us we may make statements that are true, which we cannot prove. Also, dealing with the infinity of numbers, and larger infinities seems quite standard practice?
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Replying to @BaileInis @lacker
Gödel had proven that the classical semantics of truth was ill-defined. It came as a shock to him, because he found it hard to change his convictions in the nature of truth. If you define truth as the output of a computable function you are fine.
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The more standard take on this (at least from the authors I read) is that Gödel proved that almost all mathematical truths have no proof. Similarly Turing showed that almost all mathematical functions have no program.
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Replying to @BaileInis @lacker
Yes, that's backward. The parts of mathematics that worked have always been only the computable ones. Gödel and Turing have shown that all the other parts are woo
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Certainly not denying the computable functions 'work', but it seems we have discovered a much larger class of functions that we may reason about.
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Yes, there is an abundance of things we can say about things that cannot exist.
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