• TonesInDeepFreeze
    2.3k
    I don't know anyone who has said that all others are ignorant. You are ignorant on the subject. That doesn't entail that others are ignorant on it. Indeed, there are people who critique classical set theory who are extremely knowledgeable about it. Critiques of set theory are quite fair game and bring profound insights into the subject. But those are knowledgeable, responsible and thoughtful critiques. And better yet, they are critiques that are followed up with actual mathematical alternatives to classical set theory.
    — TonesInDeepFreeze

    I didn't say all others are ignorant. I just said there are people who are like this. I did not specify who.
    Philosopher19

    You are very confused. Yes, you didn't say all others are ignorant. And I didn't say that you said that all others are ignorant. Rather, as now you mentions again, you said that some people have regarded all others as ignorant.

    You didn't specify anyone in particular. Good. Because there is no one who has even hinted at a suggestion that all others are ignorant. You take the sneaky road of impugning but leaving it open-ended who you are impugning though it is obvious who you mean. And my point stands: You are ignorant on the subject. That doesn't entail that others are ignorant on it.
  • Lionino
    849
    Did you read anything from the link I gave you?Philosopher19

    I skimmed through two pages. It seems to be a collection of semantic games. I am more concerned with what issues you solve with your beliefs. That your beliefs are not contradictory (big claim) is not a selling point for others to adopt it.
  • TonesInDeepFreeze
    2.3k
    It is blatantly contradictory for x to be both x and not x.Philosopher19

    It's even contradictory just to say that x is not x.

    And set theory does not say there is an x that is not x, nor that there is an x that is x and not x.

    You ma
    It is blatantly contradictory for a set to be both a member of itself and not a member of itself.Philosopher19

    Correct! Indeed that is a crucial point that is used in an important proof I gave you.

    Yet you want to persist by saying things like the above.Philosopher19

    That's a lie. Stop lying. I never said anything like that.
    Once again:

    It is blatantly contradictory for x to be both x and not x. It is blatantly contradictory for a set to be both a member of itself and not a member of itself.
    Philosopher19

    That is not "once again". Previously you said that "a set cannot be both a member of itself and a member of other than itself". That is different from "a set cannot be both a member of itself and not a member of itself".

    I wondered a while ago whether you did not actually mean "a set cannot be both a member of itself and a member of other than itself" but actually meant " "a set cannot be both a member of itself and not a member of itself". But in my reply I addressed the former in such a way that if you hadn't meant it, then you could revise to what you did mean.

    It is blatantly contradictory for x to be both x and not x. It is blatantly contradictory for a set to be both a member of itself and not a member of itself.

    Who would reject this but the contradictory/unreasonable/irrational/absurd/insincere?
    Philosopher19

    Indeed. (Well, except for dialetheists and paraconsistent-ists.)
  • Lionino
    849
    I will say that for diagonal paradoxes, this thread is much better.

    Relevant: Lawvere's fixed point theorem.
  • Vaskane
    643
    Says the guy who tried arguing Cardinalities don't have size yet they do, as per the theorem I produced to prove you wrong. Since some Cardinalities are greater than others, we can say that some infinities are larger or even smaller than others. That you got your ass handed to you by someone suffering from "dunning-kruger" just goes to show you've got a lot to learn, but I'm happy to correct you any time pal.
  • jgill
    3.5k
    is there real math behind the north pole of the riemann sphere?Mark Nyquist

    Point at Infinity

    :cool:
  • TonesInDeepFreeze
    2.3k
    Says the guy who tried arguing Cardinalities don't have size yet they do, as per the theorem I produced to prove you wrong. Since some Cardinalities are greater than others, we can say that some infinities are larger or even smaller than others. That you got your ass handed to you by someone suffering from "dunning-kruger"Vaskane

    You are egregiously and flagrantly putting words in my mouth.

    I never said cardinalities don't have size.

    But I'll say now that cardinalities are sizes.

    Two sets are equinumerous iff there is a bijection between them.

    The cardinality of a set is the cardinal number with which the set is equinumerous.

    'the size of the set' and 'the cardinality of the set' are synonymous.

    And we say that two sets have the same cardinality iff they are equinumerous.

    And you have it backwards:

    The original poster claims that it is contradictory to say that there are different infinite sizes. I have been saying that it is not contradictory to say that there are different infinite sizes. And I have been saying that in set theory it is easy to prove that there are different infinite sizes and indeed that some infinite sets are larger than other infinite sets.

    It is amazing that you reversed it completely to characterize me as saying the opposite of what I have been saying.

    /

    There was no "ass handing" though you like the tough talk sound of that.

    /

    just goes to show you've got a lot to learn, but I'm happy to correct you any time pal.Vaskane

    I am continually overwhelmed by how much I don't know and could learn. But with you what I have learned is not about mathematics or philosophy.

    I'm happy to correct you any time pal.Vaskane

    I'm happy to be corrected any time I am incorrect.
  • jgill
    3.5k
    Relevant: Lawvere's fixed point theorem.Lionino

    Good for you. I flamed out at "epimorphism". (i.e., the beginning). And I have actually worked with fixed points in Banach spaces and specifically the complex plane.
  • Bob Ross
    1k


    Nothing about what you said demonstrated my argument was circular. How was I begging the question?
  • TonesInDeepFreeze
    2.3k
    It could not be more clear.

    You wrote:

    "if we are considering the set of all natural numbers, then we thereby know that this set is infinite because there is an infinite amount of them."

    But:

    "there is an infinite [number of] natural numbers" is just another way of saying "[the] set [of natural numbers] is infinite".

    So your argument is just that we prove the set of natural numbers is infinite because it is infinite (has infinitely many members).

    Proving that a set is infinite is the same as proving that it has infinitely many members.

    So it is question begging to assume the set of natural numbers has infinitely many members when that assumption is just another way of saying what you want to prove.

    But you can look up actual proofs that the set of natural numbers is infinite.
  • Metaphysician Undercover
    12.3k
    Here's actually some advice to all non-mathematicians (from a non-mathematician):

    If you really can ask an interesting foundational question that isn't illogical or doesn't lacks basic understanding, you actually won't get an answer... because it really is an interesting foundational question!

    Yet if the answer is, please start from reading "Elementary Set Theory" or something similar then yes, you do have faulty reasoning.
    ssu

    I disagree. The "math boys" here at the forum tend to respond with 'go read some math texts' to anyone who disagrees with them on fundamental principles. In that case, the issue is not a matter of better learning the mathematical representation of the fundamental principles, and how to apply them mathematically, as a math text will demonstrate, it is a matter of disagreement with those mathematical representations. Therefore the reply of "please start from reading 'Elementary Set Theory' or something similar", is usually just a copout, a refusal to engage with the philosophical matter at hand as if further reading of the mathematics will change a person's mind, who already disagrees with it. That's like telling an atheist to go read some theology, as if this is the way to turn the person around.
  • TonesInDeepFreeze
    2.3k
    The "math boys" here at the forum tend to respond with 'go read some math texts' to anyone who disagrees with them on fundamental principles.Metaphysician Undercover

    Again, the intellectual dishonesty of the crank in action. In this case, blatant strawman by misrepresentation of what his interlocutors have said. And even more egregiously by dint of the fact that this strawman has been pointed out to him many many times.

    It's not a matter of disagreement on principles, but rather ignorant and confused misrepresentation of the mathematics that is supposedly being discredited. It is fine and even essential that there be different points of view about foundations including critiques of classical mathematics. But it is pernicious against knowledge and understanding when the attacks on mathematics claim things about the mathematics that are crucially false, when the attacks are premised in an ignorant prejudice that the mathematics works in certain ways that it definitely does not. After the crank's error about this have been explained to him over and over and over and he still persists to spread the disinformation, then the best thing is to recommend that he get a basic textbook to inform himself in the subject that he has spent so much time already cultivating his self-imposed terrible misunderstandings.

    That's like telling an atheist to go read some theology, as if this is the way to turn the person around.Metaphysician Undercover

    It's nothing like that. It's the reverse. It's like telling the zealot denouncing scientific theories to get a textbook in biology.

    I don't know anything about microbiology, so I don't spout a bunch of nonsense about. If I did, I should expect someone to kindly tell me to shut up about it and get an introductory text.
  • jgill
    3.5k
    The "math boys" here at the forum tend to respond with 'go read some math texts' to anyone who disagrees with them on fundamental principlesMetaphysician Undercover

    You have mentioned, for example, that the limit concept is flawed, although it works well most of the time. But I don't recall your argument beyond that point. A more complete knowledge of space and time and points and continuity? Oh yes, something about the Fourier transform and the Uncertainty principle. What are your suggestions to fix that up? Intuitive mathematics? Remind me where doing something specific makes it better.

    Are you working on a change in the fundamentals of math that might calm your concerns? I hope so, no one said math as it stands is perfect.
  • Vaskane
    643
    Then you're a dumbass for arguing with me when I was correct that Infinities do indeed have different sizes if that's your stance too. Either way, you're an "egregious" dipshit. "Oh, this guy is arguing the same thing as I am, and he's being comical, I should "correct," him about infinities not having sizes even though that's MY STANCE! Oh, wait, it's my stance, noone else can have it!" Eitherway, the fact is, you're a dumbass when it comes to communication.
  • Philosopher19
    273


    . I am more concerned with what issues you solve with your beliefs.Lionino

    I believe the solution to Russell's paradox is in here:
    http://godisallthatmatters.com/2021/05/22/the-solution-to-russells-paradox-and-the-absurdity-of-more-than-one-infinity/

    There are also other things on the website. I think they are appropriately titled with regards to what they try to do or highlight or solve or discuss.

    It seems to be a collection of semantic gamesLionino

    A triangle is triangular (or the angles in a triangle add up to 180 degrees) is not a semantic game. It is use of semantics in a non-contradictory manner. Counting to infinity, or there being no set of all sets is use of semantics in a contradictory manner.
  • Metaphysician Undercover
    12.3k
    You have mentioned, for example, that the limit concept is flawed, although it works well most of the time. But I don't recall your argument beyond that point. A more complete knowledge of space and time and points and continuity? Oh yes, something about the Fourier transform and the Uncertainty principle. What are your suggestions to fix that up? Intuitive mathematics? Remind me where doing something specific makes it better.jgill

    I think the obvious point to start with is divisibility. Generally, mathematics provides that a quantity, any quantity, can be divided in any way. We can call that "infinite divisibility". In reality, there is very clearly many division proposals which simply cannot be done. Because of this fact, that there are real restrictions on divisibility, there is a very big difference between dividing a group of things, and dividing a single object. Each of these two types of division projects has a different type of restrictions or limitations on it.

    For example, to divide a group of seven human beings into two equal groups is a project that cannot be done, even though common math would say seven divided by two is three and a half. So we'd have to chop a person in half. But then we'd have eight objects instead of seven, because we'd have have two halves, which are two objects, but unequal to the other six objects. So we have to conclude that the way we divide a group, or quantity of things is seriously restricted.

    Further, the way that we quantify something dictates the way that the quantity can be quantized. So if we use weight for example, to measure the volume of a group of grains of sand, we do not count the grains and divide the number of grains evenly, we look at the sand as one thing, with one weight, and divide that weight however we will. But there will still be a issue with precise division, when we get to the point of needing to divide individual grains of sand.

    This leads into the problem of dividing single objects. An object is a unit, and this is fundamentally a unity of parts. If there is an object which is not composed of parts, like the ancient atomists proposed for the "atom", this object would be indivisible, and provide the basis for the rules of all division projects. However, such an object has not been found, so the guidelines for dividing a unit must follow the natural restrictions provided by the divisibility of the type of object. Different types require different rules, so mathematics provides for all possibilities (infinite divisibility). What physicists have found, is that the true restrictions to divisibility of all things, are based in mass and wave action, rather than composite "parts".

    This means that in order to provide the proper rules or guidelines for the division of units, unities, we need to understand the real nature of space and time. Mass is a feature of temporal extension at a point in space, and waves are a feature of spatial extension at a point in time. Where the common principles of mathematics mislead us is the assumption of "continuity", and this is closely related to the simplistic notion of "infinite divisibility".

    Now we have two closely related, but faulty principles of mathematics, infinite divisibility and continuity. They are applied by physicists, and people believe they provide a true representation of reality, when physicists know that the evidence indicates the presence of discrete quanta rather than an infinitely divisible continuity. Therefore our representations of spatial and temporal features need to be completely reworked. To begin with, as I've argued in other threads, representing space with distinct continuous dimensions (Euclidian geometry) is fundamentally flawed. The separations within space indicated by quantum physics, must indicate distinct incommensurable parts. These distinct parts are the parts which may be represented dimensionally, and the parts which cannot be represented that way. However, they must be incorporated together in a way which adequately represents what's real. At the current time, we have a dimensional, continuous line (numberline), with non-dimensional points (real numbers) which may divide the line infinitely, but this is just an unprincipled imaginary concept which in no way represents the real divisibility of space, and it becomes completely inapplicable when physicists approach the real divisibility of space.
  • Philosopher19
    273
    Correct! Indeed that is a crucial point that is used in an important proof I gave you.TonesInDeepFreeze

    'There exists a z such that for all y, y is a member of z' contradicts this instance of the axiom schema of separation: For all z, there is a x such that for all y, y is a member of x iff (y is a member of z and yis not a member of y).TonesInDeepFreeze

    No, I believe it is a crucial point that is used in an important proof I gave you, which to put it in as short a manner as possible is: An item in a subset cannot be both a member of the subset and the set. If it is a member of the subset, it is a member of the subset. If it is a member of the set, it is a member of the set.

    That is not "once again". Previously you said that "a set cannot be both a member of itself and a member of other than itself". That is different from "a set cannot be both a member of itself and not a member of itself".TonesInDeepFreeze

    If it's a member of other than itself, this means that it's not a member of itself. So my question to you is, what is the difference?
  • Philosopher19
    273
    So that this point is not misunderstood:

    An item in a subset cannot be both a member of the subset and the set. If it is a member of the subset, it is a member of the subset. If it is a member of the set, it is a member of the set.Philosopher19

    The following must be considered properly:

    Something cannot be both a member of itself and a member of other than itself at the same time. For example, take z to be any set that is not the set of all sets, and take v to be any set. The z of all zs is a member of itself as a z (as in in the z of all zs it is a member of itself). But it is not a member of itself in the v of all vs, precisely because in the v of all vs it is a member of the v of all vs as opposed to a member of itself. If we view the z of all zs as a z, it is a member of itself. If we view the z of all zs as a v, it is a member of the set of all sets. You can't view it as both a member of the z of all zs and a member of the v of all vs at the same time. That will lead to contradictions. In other words, we can't treat two different references as one (as in are we focused on the context of vs or the context of zs?)Philosopher19
  • TonesInDeepFreeze
    2.3k
    Then you're a dumbass for arguing with me when I was correct that Infinities do indeed have different sizes if that's your stance too. Either way, you're an "egregious" dipshit. "Oh, this guy is arguing the same thing as I am, and he's being comical, I should "correct," him about infinities not having sizes even though that's MY STANCE! Oh, wait, it's my stance, noone else can have it!" Eitherway, the fact is, you're a dumbass when it comes to communication.Vaskane

    I have never argued against the point that there are different sizes (cardinalities). And I have never said that other people may not also point out that there are different sizes (cardinalities).
  • Vaskane
    643
    Put it this way, you're the epitome of my initial mockery, and too dumb to realize. It's caused by what Nietzsche calls "the masculine stupidity." Im engaging in it too, wasting my time with a bullheaded objective thinker.
  • TonesInDeepFreeze
    2.3k
    An item in a subset cannot be both a member of the subset and the set.Philosopher19

    Do you mean: If S is a subset of some set T and x is member of S, then x cannot be a member of T ?

    That's incorrect. By the definition of 'subset', if S is a subset of T, and x is a member of S, then x is a member T.

    If it's a member of other than itself, this means that it's not a member of itself.Philosopher19

    Every set is a member of certain other sets. For example, every x is a member of {x}.

    With the axiom of regularity, no set is a member of itself. So with the conjunction:

    x is a member of x and x is a member of other sets

    the first conjunct is false, therefore the conjunction is false even though the second conjunct is true.

    Without the axiom of regularity, it is not precluded that there are sets that are members of themselves. Therefore, without the axiom regularity, we cannot infer that there is no set that is both a member of itself and a member of other sets too.

    /

    Or maybe your phrasing is not what you mean. When I take your phrasing literally, as best I can, I take you to be saying that a set cannot be a member of another set and also a member of itself.

    But if all you mean is that a set cannot be both a member of itself and not a member of itself, then, of course, we have no disagreement.
  • Philosopher19
    273


    I take you to be saying that a set cannot be a member of another set and also a member of itself.TonesInDeepFreeze

    Yes. The list of lists that list themselves is a member of itself in that list alone. Even though it is also a member of the list of all lists, it is not a member of itself in the list of all lists precisely because it is a member of the list of all lists and not the list of all lists that list themselves.

    Note that the above shows the need to distinguish between "member of self" and "not member of self". To say no such distinction exists or is possible is to have an incomplete/contradictory theory in my opinion (contradictory because it argues the semantics of "member of self" and "not member of self" do not exist in Existence).

    Do you mean: If S is a subset of some set T and x is member of S, then x cannot be a member of T ?TonesInDeepFreeze
    That's incorrect. By the definition of 'subset', if S is a subset of T, and x is a member of S, then x is a member T.TonesInDeepFreeze

    I get where you're coming from and I believe I completely get what you're saying. I don't deny the set of all natural numbers encompasses the set of all even numbers. Confusion occurs when one views sets that are not members of themselves (precisely because they are members of the set of all sets and not themselves) as being members of themselves.

    To meaningfully talk about "member of self" and "not member of self", a set/context/reference is needed and adherence to it is necessary for the sake of consistency. The best way for me to convey to you what I'm saying in response to what you're saying, is the following:

    Take z to be any set that is not the set of all sets, and take v to be any set. The z of all zs is a member of itself as a z (as in in the z of all zs it is a member of itself). But it is not a member of itself in the v of all vs, precisely because in the v of all vs it is a member of the v of all vs as opposed to a member of itself. If we view the z of all zs as a z, it is a member of itself. If we view the z of all zs as a v, it is a member of the set of all sets. You can't view it as both a member of the z of all zs and a member of the v of all vs at the same time. That will lead to contradictions. In other words, we can't treat two different references as one (as in are we focused on the context of vs or the context of zs?)Philosopher19
  • Mark Nyquist
    729

    I'm thinking of starting a DIPSHIT PRIZE that we can nominate for and pass around here.

    TDF really isn't that bad. He tries hard. You can be in charge.

    Edit: To management...Not serious.
  • jgill
    3.5k


    Thanks for your thoughtful and intelligent reply.
  • Count Timothy von Icarus
    1.8k


    That's an interesting post. I've seen a few arguments that the success of eternalism in physics, to the extent that many popular physics texts openly suggest "eternalism is what physics says is true," largely flows from similar assumptions in mathematics. That is, it's a similar case of "this is how we think of mathematics, so this is how the world must be."

    I am not sure if this is so much a problem with mathematics though as it is with how it gets applied to the sciences and philosophy. It seems to me that infinite divisibility might be worth investigating even if it doesn't accurately reflect "how things are."
  • TonesInDeepFreeze
    2.3k
    distinguish between "member of self" and "not member of self"Philosopher19

    That is one thing you say that makes sense and is correct.
  • Metaphysician Undercover
    12.3k
    I am not sure if this is so much a problem with mathematics though as it is with how it gets applied to the sciences and philosophy. It seems to me that infinite divisibility might be worth investigating even if it doesn't accurately reflect "how things are."Count Timothy von Icarus

    I think there is a very close relationship between "mathematics" as the principles, rules etc., and the application of those principles. As Plato said, the people who use the tools ought to have a say in the design of the tool. And in reality they do, because the ones using the tools choose and buy the ones they like, therefore design and production is tailored for the market of application.

    So in the case of "infinite" for example, the principles of calculus allow for the representation of an operation which is carried out without a limit. The limit is infinite, which essentially means there is no limit, and the operation proceeds endlessly. This representation proved to be very useful in application.

    The issue we can look at, as philosophers, is what exactly is the effect of such an untruthful representation. First, we need to accept the fact that it is untruthful. To allow into any logical "conclusion", that an operation has been carried out without end is a false premise. In reality, the need to carry out the operation endlessly would deny the possibility of a conclusion.

    The next step I believe, is to apprehend the level of ignorance which this untruthful representation propagates. There are some very specific problems produced from our conceptions of the continuity of space and time, which were demonstrated by Zeno. The mathematical representation (or more properly misrepresentation) as a premise in calculus, creates the illusion that these problems have been resolved, and so there is denial and ignorance concerning the reality of the problem amongst many people.

    Finally, we can see how allowing this untruthful representation actually magnifies the problem rather than resolving it. When the usefulness of the misrepresentation is apprehended and recognized, it, and similar forms are allowed to pervade throughout the logical system (we can call this the propagation of self-deception). This creates the issue pointed to by the op, the need for different types of infinities, infinities of infinities, and the transfinite in general. The issue with the transfinite being, that some applications require the truth, a finite number, while others require the impossible, or false representation of a conclusion drawn from an endless operation, so some applications require a relationship between the two, hence the "transfinite". We can look at it as a bridge between the untruthful, and the truthful, a bridge which enables the self-deception.
  • Count Timothy von Icarus
    1.8k


    I mean that all makes sense, although my understanding was that the question of whether or not space-time is infinitely divisible was an open one. I know there are a lot of physicists who claim that the universe must be computable, in which case it cannot truly require the reals to represent it, and space cannot be truly infinitely divisible. But from what I understand, experiments to support this idea have been wholly inconclusive. There are also folks like Gisin who argue that intuitionist mathematics is the better structure for representing physics, but they are a small minority (albeit seemingly a growing one). But there certainly do seem to be contrary voices who argue that mathematics based on true continuities, infinite points between any two points, works best for predicting empirical results precisely because it does represent the world. That is, perhaps not all elements of the world are infinitely divisible, but some, like space-time, would be.

    I know Paul Davies claims to have an experiment that might settle this issue but it's currently impossible to actually perform, involving an insane number of beam splitters and accuracy. Other experiments looking at how light travels from distant stars have made predictions about how it must travel if space conforms to a finitist model, but the data ended up supporting a continuity, although from what I understand, these are no way definitive experiments. And then we can consider that certain discrete limits, like the Landauer Limit, appear in experimentation to turn out not to exist.

    So, part of the problem might be that there seems to be informed disagreement on what represents truth here. Although, I do agree that it is problematic when a position becomes the "default" through inertia, despite not having strong evidence for it being the case over the contrary position. I would say reductionism is a strong example of this, where actual empirical support would seem to leave the question undecided, but it remains the default anyhow.

    I find the intuitionist view fascinating because it would seem to allow for potential infinites of division, but not actual ones, a sort of near reintroduction of Aristotle.
  • Lionino
    849

    Honestly, I am having trouble dissecting the arguments used here.
    Thoughts, @jgill ?
  • Metaphysician Undercover
    12.3k
    I mean that all makes sense, although my understanding was that the question of whether or not space-time is infinitely divisible was an open one.Count Timothy von Icarus

    It appears to me, like the quantum nature of energy demonstrates quite conclusively that the reality of space and time cannot be infinitely divisible. You see, the wave-function represents a continuity, but what it represents is not an observable aspect of reality. Observations indicate discrete occurrences of so-called particles (quanta), with not necessary continuity between the occurrences. If there is a true continuity, it is not represented by the wave-function, which represents possibilities. And, it is not the continuity of space-time, which fails at the quantum level. So it hasn't yet been determined.
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