Comments

  • Infinites outside of math?
    ↪AgentTangarine


    Given any function f from N to [0 1], the diagonal proof constructs a member of [0 1] that is not in the range of f.

    I feel pretty safe in thinking that you don't know the diagonal proof.

    Moreover, even if the diagonal proof were found to be incorrect (it won't be) then that would not constitute a proof of its negation.

    I've told you a couple of times already: To prove your claim, you must prove that there is a function whose domain is N and whose range is R and that it is 1-1. And I won't even ask that your proof be constructive by showing a particular such function, only that one exists, even though Cantor's proof is constructive: showing, for any given function from N to R, a particular real number not in the range of that function.
  • Infinites outside of math?
    Cannot N be mapped onto 0.1-1? — AgentTangarine

    Do you mean to suggest that there is a 1-1 function from N onto 0?

    By the way, you claimed that I have no sense of humor. Well, you haven't said anything funny. Neither have I very much, since I don't find this context with you to motivate me to make jokes. That is not a lack of sense of humor. You don't know me.
  • Infinites outside of math?
    think it's the continuum that confuses me, — AgentTangarine

    Start by adopting a specific definition of 'the continuum'. The term is often used flexibly, but I would settle on the continuum understood to be the pair of the set of reals with the standard ordering on the reals:

    c = <R less_than_on_R>

    Or, if you prefer to think of it as merely the x-axis:

    {<x 0> | x e R}

    Or when we refer to "the cardinality of the continuum" we are thinking of the continuum as merely the set of real numbers.
  • Infinites outside of math?
    where am I wrong if I say that N^3 can be mapped on R? — AgentTangarine

    I told you. You don't have proof of it. You only think you do.

    Also, you have a serious problem when it's pointed out to you that you are in contradiction with yourself and your response is "So?".
  • Infinites outside of math?
    Math, for some reason, refuses to let me unravel its secrets. — Agent Smith

    I cherish the mysteries of mathematics. That is not impaired by formalization. On the contrary, formalization leads to even deeper mysteries for me.
  • Infinites outside of math?
    making a definition precise (usually) means losing some/all of the feelings that go with the intuition. — Agent Smith

    (1) How do you know that unless you've interviewed mathematicians about it?

    (2) I highly doubt that mathematicians very much regret whatever such loss of feelings there might be, as I would think mathematicians are primarily eager to communicate their notions clearly and objectively to other mathematicians and to prove their results.
  • Infinites outside of math?
    infinitesimals. — Agent Smith

    That's almost a good example. But it's better described as the centuries-ago formulations being more than vague intuitions yet not adequately formalized. Then, centuries later, it was discovered with mathematical logic and model theory how to vindicate the notion rigorously.

    Of course, one can look back centuries, even to the ancients, to see that their mathematics has seen been formalized.

    What I thought you had in mind though are cases where mathematicians had vague notions and then they or their contemporaries formalized those notions themselves.
  • Infinites outside of math?
    ↪AgentTangarine


    Yet you reject other theorems from the same axioms.
  • Infinites outside of math?
    You can map N to all reals between 0.1 and 0.999999... — AgentTangarine

    You keep saying that. It's dogmatism.

    Where am I wrong? — AgentTangarine

    In thinking that the fact that in your own mind you imagine that it must be so implies a mathematical proof. And in thinking that your disconnected and mathematically unsyntactical dribblings are too mathematical proof.
  • Infinites outside of math?
    ↪AgentTangarine


    Then I bet you really would not like Banach-Tarski.

    Anyway, stepping back, do you understand the proof of the equinumerosity of N and NxN?
  • Infinites outside of math?
    I never told you to consider other posts. — AgentTangarine

    By "the rest of what you posted" I meant the rest of what you posted in that post, just as I was responding exactly to your complaint that I hadn't quoted more of your post.
  • Infinites outside of math?
    You don't need the diagonal proof to realize that. — AgentTangarine

    True, there are other proofs of the uncountability of [0 1]. Cantor gave one of those other proofs.

    Every real number can be mapped from N^3 — AgentTangarine

    There is no map from N^3 onto R. And even if there were, it would prove the countability of R not the uncountability. You are again completely backwards and confused.
  • Infinites outside of math?
    You are exclusionary and dogmatic. — AgentTangarine

    You have not shown any dogmatism by me. Nor any exclusion other than of ignorant confusion and misinformation.

    That's not what the proof is about. It just shows that [0-1]is uncountable. — AgentTangarine

    It shows that [0 1] is uncountable by showing that any map from N to [0 1] is not onto [0 1]..
  • Infinites outside of math?
    Where did I do that? — AgentTangarine

    Just now, and in the other thread that was deleted yesterday.
  • Infinites outside of math?
    The diagonal proof of Cantor says you leave numbers out. — AgentTangarine

    The diagonal proof shows that any map from N to R is not onto R. That is, there are real numbers not mapped to.
  • Infinites outside of math?
    a number (x,y) say (0.678567, 0,98678) is contained in a single number 0.65456456. — AgentTangarine

    Whatever you mean by an ordered pair being "contained" in a number, what we have is each number mapped to an ordered pair. The claim of the prover is that the whole mapping is 1-1 and onto RxR. All it takes then is to see that no ordered pair is mapped to by two different numbers, and that each ordered pair is mapped to by a number.
  • Infinites outside of math?
    ↪AgentTangarine


    Yes, a pair of numbers. Not a number as you wrote.

    You keep resorting to saying that I must consider the rest of what you posted. But each time it turns out that the rest of what you posted doesn't actually qualify into correctness the initially incorrect statements you make.

    But funny, <x y> actually is a number. It's a complex number.
  • Infinites outside of math?
    start life more as vague intuitions — Agent Smith

    Example?
  • Infinites outside of math?
    the definitions in math give me the impression that true understanding is being sacrificed for logical rigor. — Agent Smith

    I have never had any such impression. Very much to the contrary.
  • Infinites outside of math?
    Rigor comes much, much later if I'm not mistaken. — Agent Smith

    To know, we would have to have access to the mental states of mathematicians. We would have to know how long was the time between their first pre-formal musings and then putting them down in concrete formulations. There is no reason to believe that for many mathematicians that time might be very brief.

    Anyway, if cranks said, "Here are my pre-formal musings, maybe something could come of them", then that would be one thing, but instead cranks insist that their view and only their view is correct; that ordinary mathematics (and even the alternative systems that the crank is ignorant of) are wrong. It is the crank, not the mathematician, who is dogmatic and exclusionary.
  • Infinites outside of math?
    .
    a number (x,y) — AgentTangarine

    That is not a real number, you understand, right?
  • Infinites outside of math?
    conflating facts with opinions — Agent Smith

    No way. One can offer alternative systems; I enjoy reading about them if they are rigorous. And one can even stipulate one's own terminology, and if it is rigorous, then we can accept it for purpose of discussion. But whether a proof is correct from given axioms is not a matter of opinion. Indeed, in principle, it is machine checkable. And the matter of what, in fact, mathematicians mean by the terminology is empirical fact, not opinion.

    Saying in a case like this "Oh, it's all opinion anyway" is intellectual dereliction.
  • Infinites outside of math?
    It's making use of decimal expansions also but overlooks the majority of them. — AgentTangarine

    Name one.
  • Infinites outside of math?
    I don't take it too seriously — AgentTangarine

    The problem is not so much that you don't take it seriously, but that you take it seriously enough to stubbornly persist in claims that are false or just ersatz gibberish from your own mind uninformed about anything other than itself.

    I can't see how R and RxR can have the same cardinality. — AgentTangarine

    You could ask for more details about the proof mentioned by jgill and about the proof in the Quora thread.

    For a proof in greater generality for any infinite S, as I said, it requires learning set theory.
  • Infinites outside of math?
    Thanks for the resistence! :smile: — AgentTangarine

    I resist misinformation.
  • Infinites outside of math?
    There are a lot of good books indeed. — AgentTangarine

    And you desperately need one if you are not to remain mired in your terrible confusions.
  • Infinites outside of math?
    ↪AgentTangarine


    You still demonstrate that you don't understand these basic concepts.
  • Infinites outside of math?
    ↪AgentTangarine


    There you go what? I am the first to say that one has to use great caution trying to pick up math on the Internet. There are some excellent Internet sources, but usually the best approach is in books. I recommended the Internet to you only because I know you wouldn't bother to read a proper book on this subject.
  • Infinites outside of math?
    ↪AgentTangarine


    I redid the post.
  • Infinites outside of math?
    I got it from the net... — AgentTangarine

    So? Not everything on the Internet is the sharpest formulation.
  • Infinites outside of math?
    EDIT:

    ↪AgentTangarine


    That's not quite right since it has free variables on the right that aren't on the left.

    f is onto Y if and only if (f is a function & range(f)= Y)

    Note that that precludes Y from being a proper subset of range(f), which is a situation not usually mentioned. I suppose one could reformulate the definition to allow Y being a proper subset of the range(f) for a broader definition.

    Also, some mathematicians consider a function to be not just the graph but also that the domain and the range are specified, and sometimes the domain and a co-domain (there may be different co-domains, since a co-domain can be any superset of the range). That could be made rigorous by saying a function is a triple <domain graph range> or alternatively <domain graph co-domain>. However, in set theory, the function is the graph, period, though of course that graph determines the domain and range.
  • Infinites outside of math?
    I can send every cardinal on top of R. — AgentTangarine

    As I said, you don't know the meaning of 'onto' in mathematics. Or you're insane.
  • Infinites outside of math?
    ↪AgentTangarine


    So which is it now? You claim that card(R) = aleph_1, thus asserting the continuum hypothesis? Or you deny that card(R) = aleph_1, thus denying the continuum hypothesis?

    Hardly matters though, since it is sheer crazy talk to say that there is a mapping of a singleton onto R.
  • Infinites outside of math?
    So you are in flat out contradiction with yourself.
    — TonesInDeepFreeze

    So?     — AgentTangarine
  • Infinites outside of math?
    I can map a singleton on R too. No problem. — AgentTangarine

    Either you are insane or you don't know the meaning of 'map onto'.
  • Infinites outside of math?
    ↪AgentTangarine


    So you are in flat out contradiction with yourself.
  • Infinites outside of math?
    I can even map a single element of N to R. — AgentTangarine

    Of course one can map a singleton set INTO R.

    But there is no map of N ONTO R.
  • Infinites outside of math?
    ↪AgentTangarine


    Yet you say you disagree with the continuum hypothesis now, while you have been claiming it in dozens and dozens of posts.
  • Infinites outside of math?
    you have answered with a riddle. — AgentTangarine

    The number 0 is not a riddle.
  • Infinites outside of math?
    I can map N infinite times infinite times on R. — AgentTangarine

    Great. And I can make the sun disappear by wiggling my ears.
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