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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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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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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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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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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