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whoa here i am just clicking around and this is the strongest argument against believing the principle of mathematical induction i've ever seen
golem.ph.utexas.edu/category/2011/
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discussion of ontological nebulosity in the foundations of arithmetic
trying to decide whether i'm changing my mind about whether the natural numbers exist 🤔
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had a conversation with a friend once about how many numbers we had personally verified existed by counting up to them and he said once as a teenager he counted up to several thousand or something just to double-check they were all there
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Hmm, is the claim approx the following?
Nat Nums n are defined w/ a starting point 0. For each n, it takes some amount of time to "compile" its representation as a function of just 0. For some cleverly defined k and all N>k, this compile time cannot complete in a finite universe
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read the actual leivant paper imo it's super good
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also, bellantoni & cook's predicative recursion is totally a modal type theory
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you know what i've linked this a couple times in the past few days so here's a version w/ typos fixed, extra note, & not-technically-necessary but conceptually helpful use of metavariables
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I'd like to use this as an opportunity to plug coinduction as an everyday bread-and-butter epistemic primitive for resolving boring finitist confusions
twitter.com/dailectic/stat
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"Infinite" objects with no well-founded closed form, naturally represented via yoneda-flavored domain, can still have well-defined finite properties
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