>Zorn's Lemma makes sense>Well-Ordering theorem makes no sense>Axiom of Choice is impossible to decideBut they're all equivalent. How is that possible?
>>16474407All of them are impossible to decide, and intuition is subjective.
Niggers in your anus
>>16474407Well-ordering theorem refutes the AOC. How the FUCK can [math]\mathbb{R} [/math] be well-ordered?
>>16475045This. Well-ordering theorem is the strongest argument against ZFC I know.
>>16475045By assigning an ordinal number to each real number. There are more ordinals than real numbers so that is not a problem.
>>16474407these things are equivalent and all of them hold in the Godel's constructible universe (which in turns do exist in every model of vanilla ZF).
>>16475045>>16475106The Axiom of Choice is literally the tool you use to well-order the Reals.
>>16475267Actually it can only prove that such a well-ordering exists in the abstract, and different models satisfy that statement in different ways.
>>16475983You literally use the Axiom of Choice to well-order the real numbers. You cannot complete the process, and you never reduce the size of the remaining unordered set, but the process is clear and obvious.
>>16474407It's because we are using our intuition of a physical world when thinking about mathematical objects. Take AoC, for example. Most people think they understand it because they look at your picture which portrays several sacks with balls in them and they think 'Well, it's reasonable to suppose that we could pick a random ball from each sack'. You could, for example, order the balls by weight and choose the lightest. Or you could physically reach inside a sack and pick a ball. But this isn't what AoC says, because it doesn't deal with physical objects. It says that you can choose one ball from a set of balls which are completely indistinguishable from each other by every characteristic including not only weight, but even their position in space (because if balls occupied different positions in space you could just pick a ball that's closest to your hand reaching into a sack). But fine, let's say that you can pick a random ball from a sack of indistinguishable balls. Or you could throw a die and get a completely random number between 1 and 6 which nobody could predict even if they had perfect information about the initial conditions of the die and the surrounding environment. Or you could, in Russell's analogy, pick a random sock from a pair of indistinguishable socks. This throws away all our intuition about everyday physical objects we interact with, but it's still a somewhat reasonable proposition. 1/2
Well, you don't actually need AoC for that. The examples I've listed above follow from axiom of finite choice, which isn't an axiom at all, it's just a theorem provable in ZF. What you actually need to invoke a new axiom for is saying that you could pick a random ball from infinitely many sacks of indistinguishable balls. Or, alternatively, you could pick a random real number. Which sounds easy, surely you can choose a number. But keep in mind that you would have to choose a completely random, meaningless, infinite Kolmogorov complexity number. It would have to be uncomputable (since the set of computable numbers is countable and there exists a trivial choice function). Well, fine, let's say that it still sounds somewhat reasonable of a proposition. You can't compute a number and you need infinite time and space to choose it, but let's say that you could choose it. Well, you still don't need to invoke AoC for that. Everything I've described above follows from a weaker Axiom of countable choice. What do we need an actual AoC for? Honestly, I'm not sure. Of course, we can add it to ZF, derive and prove theorems from it. ZFC, as far as we know, is a system that doesn't contradict itself. What it contradicts is common sense and our understanding of physical world. 2/2
>>16476613>What do we need an actual AoC for?Well-ordering the real numbers.
>>16476625You're a funny guy.
>>16476610Does it even matter when all of them are the same? In the end is just decrementing the cardinality of each set by one. Also is not correct to compare it to throw a dice because that would require the elements to be labeled and there is no such requirement.
>>16474407>>16475045Choice can be restated as "All cardinals are comparable." From there, it's obvious that the continuum must be in bijection with some ordinal, which would be its well-ordering. There are very few restrictions on what the cardinality of the continuum could be.I object on the grounds that I don't see why the powerset of the naturals should have any relation to the set of all countable well-orderings at all, so I find Determinacy preferable.
>>16474407What's the second least number on the interval [0,1]
>>16476688The second one you choose using the Axiom of Choice.You think "<" is the only possible ordering. It is not.
>>16476707But if I choose "<" for ordering, what's the next number after 0
>>16476709"<" is not a well-ordering of the reals.They exist, because of AoC, but that is clearly not one of them.
>>16476715May I see it?
>>16476715How can there be a well ordering of [0,1] , let alone R, if the ordering expires before you get out of [0,1^-epsilon]?
>>16476723Almost certainly not (>>16476168). Doesn't mean they don't exist.Are you starting to see or do you need more hints?
>>16476734I understand >>16476168 and the anon you replied to isn't me. How can there be a well ordering of [0,1], let alone R, if the ordering expires before you get out of [0,1^e]?
>>16476730>>16476746By using the Axiom of Choice.Tautologies are tautological, which is the point. The reals are well-orderable because the Axiom of Choice let's you create a well-ordering of the reals.That's it.Without AoC, you can't do it.It's not magic. It just is. If you want well-ordered reals, you need the AoC to do it. If you want the AoC, you will need to accept that the reals can be well-ordered.
>>16476753But how can the reals be ordered if any subset of the reals can't be ordered with respect to any other subset.
>>16474407Set theory is a joke. Reminder ZFC can't prove that S>T -> P(S)>P(T)
>>16476759Oresme's harmonic divergence and Cantor's powerset inequivalence are incompatible.
>>16476755You need to let go of "<".You can, because AoC says you can. It is nothing more than that.Look, you are dealing with transfinites. Thing are going to get weird and non-intuitive, but that was sort of the point Cantor was making. Transfinites have weird properties, but we can formalize those ideas and start to get a handle on them.
>>16476765But if (0.0, 0.1) can't be fully ordered, how can you prove that every number in (0.1, 0.2) is greater than any number in (0.0, 0.1)
>>16476779eyeroll.emoji, keep working on it, Kid.Good luck.
>>16476670Your post is actually incomprehensible because of your poor English. I actually don't know what you were trying to say.>Does it even matter when all of them are the same?Well, random choice comes into play when you choose from a set where no element is different in at least one aspect from all others. A set with two blue and three red balls which are otherwise identical is another example.>compare it to throw a diceI brought dice up because the existence of true randomness or ability to generate a random number is a famous point of contention in philosophy. You can debate it either way, but you should certainly be aware that you're taking side on a very complex issue when you say that you can choose something randomly.
>>16476789It's a serious question. How can any subset of the real not have a completed order but all subsets can?
>>16476798And you have been given nothing but serious answers. Seriously, good luck.Pro-tip: Use the Axiom of Choice, if you want to.
>>16476807How can any subset of the reals not have a completed order but all subsets can? Why can't you simply answer the question?Protip. Use the axiom of choice on (0,1) and on (1,2)
If there were a simple argument proving that the reals are somehow immune from their subsets, I would prefer to agree with that argument. Say something to convince me.
>>16476759>Reminder ZFC can't prove that S>T -> P(S)>P(T)And why should that be true? You're faulting ZFC for being too weak to prove things you think are intuitively obvious, when by many standards it's already an unnecessarily strong theory for what it's trying to do. The only reason we even take choice (and by extension the linear hierarchy of cardinals) for granted is that it made life easier for people working in other branches of math.
>>16476807Why are you pretending to know what you're talking about? The axiom of choice is a statement, not a method. The well-ordering theorem is non-constructive.
>>16474407>But they're all equivalent. How is that possible?theyre all bullshit
>>16476886Feel free to point out the post which contains the explicit choice function which offended you.
>>16476168>Make this idealized cake please>Ok what do I need to do?>Just add more eggs first>Then more flour>Then more eggs>Then...>When do I get to bake the cake?>Oh no keep adding ingredients, you'll bake it once we run out of real numbers.
axiom of choice is bullshit and i'm tired of pretending it's notit can be safely ignored for a lot of great math, but pretending that there is always an algorithm that exists to pick elements from any set is hiding your head in the sand if you know anything about (legit honest to jesus) computer science
>>16477557Let n -> u_n be a sequence containing all partially defined and algorithmic maps from N to itself (their set is necessarily at most countable). Then the map v:= n -> {0 if u_n(n) is undefined, max(0,1-u_n(n)) if u_n(n) is defined}. From a simple diagonal argument, we can see that v is not algorithmic (if v = u_m then u_m(m) is defined and u_m(m) = max(0,1-u_m(m)) which cannot happen).What kind of formal theory do we need to formalize the result above and its proof? IMHO very little (a fragment of Peano would suffice, not to mention any mainstream set theory). The argument "we must only talk about algorithmic functions" was bogus from the start, math cannot afford to limit itself in that fashion.
>>16477557anyways, again, AC is not BS because, using only vanilla ZF (without AC nor even foundation) you can build a class L called "the constructible universe", in which a stronger form of AC holds (the class L has an explicit global well ordering) and has exactly the same hereditarily finite sets as the ground universe which means that all the arithmetical statements proven in L (hence relying on global AC and also HGC) hold in plain ZF.There is a reinforcement of this called the Shoenfield absoluteness theorem btw.
>>16475045pedo alert
>>16477655Now do L(R).
>>16474594I concur with this particular thesis
>>16477557Classical mathematics is not algorithmic in any way even if you don't use AC.
>>16477654>What kind of formal theory do we need to formalize the result above and its proof?It's not really a result. You relied on a diagonal argument, which is a process that is never going to halt, which means you haven't achieved any result. >math cannot afford to limit itself in that fashion.Why? You really need theoretical circlejerk which explicitly relies on your ability to do the thing you can't actually do? That's like saying that biology cannot limit itself to real animals and has to include ghosts and demons. Like, sure, you can do this as a fun pastime, but maybe don't put that in the center of the entire field as a foundational principle?
>>16477557There is no algorithm that can prove the totality of multiplication without suitable axioms, even though it's obviously true, and also sufficient to imply incompleteness. Algorithms can, however, enumerate every implication of ZFC using the rules of inference. So it doesn't really matter what algorithms can do.
>>16476746If you are finished with [0,1), you can repeat the same process countably often with the other intervals [k,k+1) and you get your well-ordering of R.
>>16477557You're confusing an algorithm with a function
>>16474407I think Zorn's lemma is the one that actually makes the least sense of the three when you think about infinite sets. It's true for finite sets, but why would this property generalize to infinite case? Doesn't really matter ofc since all three are equivalent to each other.
>>16478284If he could, Zeno would be laughing from his grave rn
>>16478284But he said you can't finish [0,1)
>>16474407Do any of these concepts exist in reality, or do they only exist to verify the validity of your symbol game? If it's the latter, why bother?
