It is generally understood, although not necessarily accurately, that the number "2" means "2.000...", that is the number two with an infinite set of zeroes after the decimal.
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(As an aside this is also exactly the same as a number one with an infinite sequence of nines after the decimal.)
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Not exactly.
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Replying to @JustCanadian7 @phyphor and
Yes, exactly. 1.99999999999999... means the limit of the series 1+0.9+0.09+0.009+0.0009+...., which is 2.
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Replying to @JustCanadian7 @IMJackRudd and
Well this is ironic I mean, sure, no one can force you to accept this as true if you don't want to -- that's the whole point of this conversation! Everyone has their own idiosyncratic language about numbers they use on their head to understand the world and that's fine!
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Replying to @arthur_affect @JustCanadian7 and
But from the point of view of standard, accepted, classroom mathematics you're just wrong 0.9999... = 1 is, according to the normal rules, true You can get into an extremely formal proof of it via real analysis, but you can demonstrate it informally with simple algebra
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Replying to @arthur_affect @JustCanadian7 and
Ever since this debate became an Internet meme there's been mathematicians pointing out that *you don't have to* accept this if you don't want to You don't have to do anything, in math, math is a construct you choose to adopt for your own purposes
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Replying to @arthur_affect @JustCanadian7 and
A friend of mine had a maths professor once who didn't believe in the Axiom of Choice. Religious objection. He still taught it, but he absolutely personally believed it to be false.
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Replying to @iridienne @JustCanadian7 and
One of the ironies of the discourse over Gödel's incompleteness theorems is Gödel, himself, was a Platonist who saw himself as illuminating the limits of constructivism
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Like, speaking informally, he thought it was obvious that the Gödel sentence, "This sentence cannot be proven to be true", IS TRUE, and your awareness that it's true but inability to prove it shows intuition's primacy over logic
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Replying to @arthur_affect @iridienne and
Could Gödel create a sentence even he couldn't prove was true?
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