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15 May 2022
@_discord_103550118136909824:t2bot.ioBabySnake because its axioms lead to the principle of explosion 21:12:07
@_discord_870115701279584326:t2bot.ioDiscoDoug You’d want to be able to show you satisfy the axiom with something like bijection and not simply say “they’re equal” 21:12:28
@_discord_870115701279584326:t2bot.ioDiscoDoug Garkelein showed why this axiom is necessary. 21:13:02
@_discord_103550118136909824:t2bot.ioBabySnake if you add a theorem which is provably not true given the other axioms as an axiom your new axioms are not consistent 21:13:02
@_discord_103550118136909824:t2bot.ioBabySnake the assertion that all E which satisfy the first 4 axioms are all isomorphic is provably untrue 21:13:37
@_discord_103550118136909824:t2bot.ioBabySnake so if we assert it /is/ true, then we have a statement which is both true and false 21:13:50
@_discord_103550118136909824:t2bot.ioBabySnake which is the principle of explosion 21:13:55
@_discord_103550118136909824:t2bot.ioBabySnake which makes the theory inconsistent 21:13:59
@_discord_870115701279584326:t2bot.ioDiscoDoug Proving that you can’t invent something is clearly harder 21:14:06
@_discord_103550118136909824:t2bot.ioBabySnake all models of N being isomorphic is a consequence of the second order inductive axioms 21:14:29
@_discord_870115701279584326:t2bot.ioDiscoDoug I’m not sure why I’m not getting my point across but I’m running out of ways to say the same thing. 21:14:56
@_discord_870115701279584326:t2bot.ioDiscoDoug It isn’t a consequence. 21:15:05
@_discord_870115701279584326:t2bot.ioDiscoDoug That’s why there even is a fifth axiom. 21:15:24
@_discord_870115701279584326:t2bot.ioDiscoDoug You don’t toss in an axiom for giggles. 21:15:42
@_discord_103550118136909824:t2bot.ioBabySnake but its not an axiom 21:15:49
@_discord_103550118136909824:t2bot.ioBabySnake I dont know where he got it from but thats not an axiom 21:15:56
@_discord_870115701279584326:t2bot.ioDiscoDoug I can only wish you good luck at this point. 21:16:22
@_discord_103550118136909824:t2bot.ioBabySnake Its a theorem of PA when you have the inductive axioms 21:16:28
@_discord_103550118136909824:t2bot.ioBabySnake Garklein where did you get that original image from? 21:16:34
@_discord_103550118136909824:t2bot.ioBabySnake Unless were implicitly defining E to be from some predicate P, then its the inductive axiom 21:17:35
@_discord_103550118136909824:t2bot.ioBabySnake but thats what I said the other day is maybe its a weird way of writing the inductive axioms 21:17:47
@_discord_870115701279584326:t2bot.ioDiscoDoug I’m out of ways to make this point. Sorry. 21:18:07
@_discord_870115701279584326:t2bot.ioDiscoDoug From the wiki on Peano axioms: “The intuitive notion that each natural number can be obtained by applying successor sufficiently often to zero requires an additional axiom, which is sometimes called the axiom of induction.” 22:38:04
16 May 2022
@_discord_351172817519116301:t2bot.ionikyq#8590 changed their display name from nikyq to nikyq#8590.06:18:34
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@kamila:coven.palaiologos.rockskamilai love spotting tiny gems like these in source code18:06:24
@_discord_671689100331319316:t2bot.ioBrian E#0926 lol 18:13:12
@_discord_613847106993782835:t2bot.ioGarklein#9297 it's from oskar31415's youtube channel banner 18:59:06

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