>you can make two sphere from one sphereWhy doesn't this convince logicels that zfc is unsound? Having no contradiction doesn't stop your formal system from producing slops.
>>17059885Why are brainlets assblasted about this? You have a whole uncountable infinity of points to make imaginary objects with. Why should you stop at 1?
>>17059885The consequences of no AoC are arguably worse.
>>17059885So? The outer measure isn't countably additive on all of R. Their result simply implies (in a grandiose way) the Lebesgue measure isn't countably additive on all of R^3. Why is this surprising?
>>17059890Agreed.
>>17059890You will it le bug.
>>17059893Good point. It should not be surprising at all in 3D. The shear amount of space in R^3 as compared to R^2 always surprises me.Poor drunken hobos in R^3 never come back. Where do they all go?
>>17059885>>17059885>Why doesn't this convince logicels that zfc is unsound?Yep thats the only conclusion. Straight forward. Just as the thread with the Schrödingers cat is proof copenhagen interpretation is unsound.Just as einsteins theory is unsound from the start.But modern science lifes with sancrosanct "facts", and oen of math is that the jewish cantor was a genius and set theory cant be wrong.
>>17059893its surprising because mass is intuitively thought of as a measure of quantity.and mass is preserved, not duplicable.
>>17059998>massMaths, son. Not physics.
>>17060027>assume a perfectly spherical hotdog
>>17059992>someone like this allegedly studied mathflunked out but still
>>17059885No what this result does is show that the real world is unsound. Or more accurately that it is unintuitive and complex in it's workings. It doesn't obey the AoC because it is finite and indivisible (plank length, finite speed of light ect). Whereas every natural mathematical framework does, but the real world is not natural as contradictory as that sounds. It's why people think our universe is a simulation.
>>17060219>Mathematics is real; reality is notWhat a degree in maths does to a mf
>>17059885You are aware that ZFC only changes the fact that you can tractably label the points in these operations, right? If you remove choice the problem at its core does not go away. A sphere of radius 1 and a sphere of radius 2 have the exact same number of points in a set theoretical sense, choice or not. Yes, BT is flashy because you decompose the sphere into point clouds and don't have to do a scaling operation to bijectively map one onto the other, but the fact that the map you construct for BT is made of translations and rotations is a parlor trick. At its core the real numbers just are like that.
>>17059890indeedMeanwhile, it would be a form of confirmation bias to discuss only counterintuitive consequences of the axiom of choice, without also discussing the counterintuitive situations that can occur when the axiom of choice fails. Although mathematicians often point to what are perceived as strange consequences of the axiom of choice, a fuller picture is revealed by also mentioning that many of the situations that can arise when one drops the axiom of choice are perhaps even more bizarre.For example, it is relatively consistent with the axioms of set theory without the axiom of choice that there can be a nonempty tree T, with no leaves, but which has no infinite path. That is, every finite path in the tree can be extended to further steps, but there is no path that goes forever. This situation can arise even when countable choice holds (so countable families of nonempty sets have choice functions), and this highlights the difference between the countable choice principle and the principle of dependent choice, where one makes countably many choices in succession. Finding a branch through a tree is an instance of dependent choice, since the later choices depend on which choices were made earlier.Without the axiom of choice, a real number can be in the closure of a set of real numbers X ⊂ R, but not the limit of any sequence from X. Without the axiom of choice, a function f : R R can be continuous in the sense that every convergent sequence xₙ x has a convergent image f(xₙ) f(x), but not continuous in the ε, δ sense. Without the axiom of choice, a set can be infinite, but have no countably infinite subset. Indeed, without the axiom of choice, there can be an infinite set, with all subsets either finite or the complement of a finite set. Thus, it can be incorrect to say that ℵ0 is the smallest infinite cardinality, since these sets would have an infinite size that is incomparable with ℵ0.
>>17059890 >>17060266Without the axiom of choice, there can be an equivalence relation on R, such that the number of equivalence classes is strictly greater than the size of R. That is, you can partition R into disjoint sets, such that the number of these sets is greater than the number of real numbers. Bizarre! This situation is a consequence of the axiom of determinacy and is relatively consistent with the principle of dependent choice and the countable axiom of choice.Without the axiom of choice, there can be a field with no algebraic closure. Without the axiom of choice, the rational field Q can have different nonisomorphic algebraic closures. Indeed, Q can have an uncountable algebraic closure as well as a countable one. Without the axiom of choice, there can be a vector space with no basis, and there can be a vector space with bases of different cardinalities. Without the axiom of choice, the real numbers can be a countable union of countable sets, yet still uncountable. In such a case, the theory of Lebesgue measure is a complete failure.To my way of thinking, these examples support a call for balance in the usual conversation about the axiom of choice regarding counterintuitive or surprising mathematical facts. Namely, the typical way of having this conversation is to point out the Banach-Tarski result and other counterintuitive consequences of the axiom of choice, heaping doubt on the axiom of choice; but a more satisfactory conversation would also mention that the axiom of choice rules out some downright bizarre phenomena - in many cases, more bizarre than the Banach-Tarski-type results.--- Joel David Hamkins, Lectures on the Philosophy of Mathematics
>>17060231it is not that reality ain't real, it clearly is, it is just that it is a finite subset of mathematics
>>17060266>>17060268some mathematicians hate AoC because it is actually useful, any physicist would wipe their ass with vector spaces with no bases and similar bullshit
>>17059998>mass is preservedAs far as we know.
>>17060270>Maths contains the set of reality and moreLmfao
>>17060268>both options are nonsenseNot very convincing. If non-aox covers aox then it is the most likely to follow through on because it can still manage to cover a possible axiom which addresses all issues internal or external of aox.
>>17059890>The consequences of no AoC are arguably worse.Also simply no AC does not guarantee measurability on all subsets.>>17059992>But modern science lifes with sancrosanct "facts", and oen of math is that the jewish cantor was a genius and set theory cant be wrong.To create a function that Lebesgue measures every subset of R you need far more Jewish nonsense than the innocent and intuitive Axiom of Choice (which is Ernst Zermelo's btw, a German); see https://en.wikipedia.org/wiki/Solovay_model. And as a result, you can, for example, "partition R into disjoint sets, such that the number of these sets is greater than the number of real numbers", see >>17060268. Also strange.>>17059998>its surprising because mass is intuitively thought of as a measure of quantity.>and mass is preserved, not duplicable."Mass" on the real line can be very strange sets. Dropping AC would imply equivalently or even weirder stuff (see my comment above).
>>17059998>its surprising because massSo... uh... what's the mass of a purely abstract sphere that's actually an infinitely thin shell made up from uncountably infinite points with no volume?
>>17059998>>17060463... because if you say "zero", mass is indeed preserved.
>>17060457I don't think it contains it. I think math can reproduce the perfect (Platonic) representation of our world. And it's a good framework. You can never produce a physical object with a given exact length for example (may this length be in Q or R/Q). On the real line you can draw it, manipulate it, etc.
>>17060465Every object with a length is exactly its given length, thoughbeit.
>>17060241>A sphere of radius 1 and a sphere of radius 2 have the exact same number of points in a set theoretical sense, choice or not.>At its core the real numbers just are like that.This.>>17060466Yes, and Euclidean space is a great framework in which to represent that object.
>>17060266>>17060268Thanks, bro. Whole new reading list for me to rabbit-hole into!
>>17060241Perfection.
>>17060469>Yes, and Euclidean space is a great framework in which to represent that object.Kek. I would like one (1) unit length, please.>any particular directionDealer’s choice.>ah, a roots of unity man, take your spinThe 13th of the 17 roots, my favorite.
>>17060241>A sphere of radius 1 and a sphere of radius 2 have the exact same number of points in a set theoretical senseI'm interested in understanding this, can you elaborate?In a physics-like approach ie it's a real, well-defined 3D space that has a certain max resolution ie max number of points per unit volume, this isn't the case. But in the case of a loosely defined and purely mathematical space, all bets are off because points can be considered to have a size of zero (or approaching zero) so infinitely many points can fit into an infinitely small space and at this point, my understanding breaks down how such a statement can possibly be correct then.Can you explain?
>>17060463Mass isn't a math concept. You can talk about volume and length, width, surface area occasionally, but not mass, that's not a thing. If you insist though, then just define a value for density and then mass = volume * density. But then, the idea of mass is just a derived number from density and isn't anything unique nor necessary at all so might as well not even think about it.
>>17059992YOu do not know what a Borel set or Lebesgue measure is; you are not armed with the knowledge to discuss these topics.>Oh no the world at microscopic scales doesn't fit neatly into my intuitions which I gained from living in a Newtonian scale. Must mean the world is wrong.
>>17060715>the world isn't like my made up math gobbledygookYes, we know
>>17060711Thanks for letting me know, anon. I was actually hoping for that guy to tell me the mass of a sphere but I guess he literally can't.
>>17060710You pretty much nailed it. Gotta define my terminology a bit before I can answer. The set theoretical sense of "number of points" (cardinality) [math] |A| [/math] is more of a grouping operation that is ordered. Specifically, [math] |A| \leq |B| [/math] if there is an injective function from A to B, because that means you can uniquely label every element of A with a unique element in B. [math] |A| = |B| \iff (|A|\leq|B| \wedge |B|\leq|A|) [/math] works just like with normal inequalities which is equivalent to saying there is a bijective (invertible) function.So far so good? Now, why do the two spheres have the same number of points.Let me start with a segment of the real number line because that is easier to imagine. There is a bijective function between [0,1] and [0,2] (f(x)=2x with inverse f^-1(x)=x/2), so line segments of different lengths have the same cardinality. For a sphere you can do the same trick by taking the length 1 line segment connecting an arbitrary point on the surface of the sphere with the center and doubling that while preserving the direction. In Cartesian coordinates with the sphere centered at the origin that would be like taking any point (x,y,z) and multiplying each entry with sqrt(2) (why that is equivalent would take more math to explain but it works out).The rational numbers are "just as bad" in this regard because [math] 2 \mathbb{Q} = \{2x: x\in\mathbb{Q}\} = \mathbb{Q} [/math].
>>17060791Holy shit... this is wild.I actualy get it. Thank you for taking the time to write this.
>>17060824My pleasure.
>>17060791You are a good person, Anon.Even if you pretend not to be sometimes.
>>17060241Imho the problem is the whole concept of set theory.
>>17059893Thats nice math speak for "our basic model is completely unrealistic"
>>17060715>YOu do not know what a Borel set or Lebesgue measure is; you are not armed with the knowledge to discuss these topics.I have done countless calculations with them. But this is 100% irrelevant to the discussion.You dont even grasp the problem which is way deeper, it should disproof set theory all together since it gives an obvious contradiction.
>>17060461>https://en.wikipedia.org/wiki/Solovay_modelWow yeah this is some next level halluzination "math". The face of this solovay guy tells everything.
>>17059885This world is ruled by quantum mechanics. This world doesn't have what is necessary to make two spheres from one sphere. Simple as.
>>17059885ZF, ZFC, ZF+ Global choice, ZF+Global choice + generalized continuum hypothesis and even ZF + AC negated: all these theories prove exatly the same arithmetical sentences (since these sentences are absolute between the ambient universe and its constructible ubuniverse where V=L holds, hence global HC and GHC; for the negation of AC, pick the Cohen model which violates AC in e.g. the boolean valued approah to forcing: he shares the same L as the generic extension, which shares the same L as what is de facto an elementary extension of the ground model).A mathematical claim which is not an arithmetical sentence (all claims about what a computer program does are of this form) has no physically testable content (the existence of a real number such that this or that happens has a metaphysical content only). So there is no real issue with choice, it is only a flavor of the infinite vs another flavor, a.k.a. a byzantine concern.
>>17061005>I have done countless calculations with them. and clearly learned nothing since your understanding of contradiction ("not what my gut feelings tell me") is that of a smarmy undergrad. Were you seething throughout all your studies having to use mathmatics that according to you are "wrong"? Did you ever get to debate your professors?
>>17059885How is it slop? It's the leading candidate theory for dark energy.
>>17061047The Universe is The Ultimate Free BOGO Lunch.
>>17060457Of course!
>>17060469>Euclidean space is a great framework It isn't but it should be, which is the whole point of this thread
>>17061012>A mathematical claim which is not an arithmetical sentence (all claims about what a computer program does are of this form) has no physically testable contentProof?
>>17061006He clearly yearns for the inaccessible cardinal.
>>17061065Once he finds the largest prime and finishes factoring the natural numbers and reversing their Gödel numbers back into their original mathematical sentences, he will.
>>17060931Thanks, Anon. I can be hard on people here because I am very disillusioned with the state of the board but I am happy to share what knowledge I have.
>>17060266>>17060268Aren't all of these just issues with "interpetation" of what it means for an existence statement to be true (that an example is constructible(ish) in some sense)? E.g. you If you phrase it in the way of "there is no explicit method/algorithm writeable in finite symbols to construct an infinite path for any such a tree" rather than "the path does not exist"? Then this does not seem anywhere as pathological at all. Same for the others (there are different algebraic Q closures because without choice we have no canonical way to construct an isomorphism to the standard one for each?)(Ofc. we still got LEM so it's not completely constructive yet but it's constructiver than with AoC.)
>>17061126This is exactly why we have these conversations in the first place.If we can break the concept of “choice” into three more fundamental concepts then math has advanced.What is the minimal amount of machinery we need to accomplish the goal?That’s the point.
>>17061065>>17061067You can take as a definition of "arithmetical sentence" any sentence where every quantfier is bounded by V_omega (the set of hereditarily finite sets) and encoding everything here; integers, finite lists and so on.
>>17061148Do we still get to explore the inaccessible cardinals?If so, I’m in.
