I have just discovered (during homeschooling) that my children are being taught at school that infinitely many numbers exist. Where do I have to send my petitions against this devastating creationist pre-Hilbertian falsehood?
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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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Replying to @Plinz
I don’t follow why the existence of a computable algorithm is needed for numbers to exist. Are you opposed to the axiom of choice on the grounds of this argument, or do you consider functions it describes to be “constructed”?
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Replying to @lacker
As far as I understand, the axiom of choice is necessary to deal with infinite sets (which we will never encounter). I am suspicious of a function that promises to do something without also stating how it does it.
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For enumerable sets it's not a problem, and I understand it in the same way.
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