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Set-Theory, a paradox from Zeno and Russell/Cantor confusion.

  1. Aug 20, 2011 #1
    I can't come to a comfortable conclusion which doesn't make positive, negative, real, imaginary, complex, irrational or transcendental numbers seem much different fundamentally.

    Zeno (to the best of my interpretation) illustrates a large part of my problem with:

    "If there are many, they must be as many as they are and neither more nor less than that. But if they are as many as they are, they would be limited. If there are many, things that are are unlimited. For there are always others between the things that are, and again others between those, and so the things that are are unlimited. (Simplicius(a) On Aristotle's Physics, 140.29)"

    My question started after I read 'Logicomix', which involved Russell's paradox asking about sets containing themselves. After long, confusing and generally baffled discussions with the coaches in the MLC (math learing center) I spent a couple hours a week arguing theories with a specific coach who was interested in philosophy and numbers.

    Eventually the topic died at how a set can never really contain itself on account of constant change in all things... and how mathematics can and often is structured practically -- where the infinitesimal change in something representing "one thing" isn't ever really a permanent (1), and how all that isn't really worth considering if you just want to figure out how many 1.5ml wine bottles you have.

    To me it seems like the sets, " A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] and of our thought - which are called elements of the set." - Georg Cantor... are all constantly changing in every sense, containing and being contained, and we try to capture them to be used by treating them numerically like unchanging snapshots which stand perfectly still until we can mimic the motion and behavior of the reality.

    I'm certainly no math major... but it feels like there has to be more on this sort of thing.
     
  2. jcsd
  3. Aug 22, 2011 #2
    "If there are many, they must be as many as they are and neither more nor less than that."

    This is where Zeno was wrong. In fact, this statement (which Zeno takes to be universal) is considered by modern mathematicians to actually be the definition of "finite."

    We say that two sets S and T are "the same size" if if there is a function that maps every element of S onto an element of T, so that no two different elements of S are mapped to the same element of T. Put another way, this means that two sets are the same size if we can take their elements and "line them up in pairs, so that neither set has any leftover elements."

    Most books on modern set theory will give the following definition of an infinite set (in laymen's terms):

    If a set S is the same size as one of its subsets, then S is an infinite set.

    If you're interested in this kind of topic, I strongly recommend the late David Foster Wallace's Everything and More. It's a brilliant exposition on the history of mathematics with an emphasis on mathematicians' attitudes towards and understandings of the concept of infinity, written very lucidly for the non-mathematician.
     
  4. Aug 22, 2011 #3
    A couple other things. What the coach at the math learning center told you (specifically, that a set cannot contain itself) is absolutely false.

    Also, a note on the quote " A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] and of our thought - which are called elements of the set." - Georg Cantor

    Cantor is considered the father of modern set theory, and what he says here is absolutely true, however, the converse is not, which is what leads to the development of axiomatic set theories.

    You see, what he says in that quote is that if something is a "set" then it is a "gathering together of ... distinct objects." However it is not the case that "every gathering together of ... distinct object" is a set.

    For instance, Russel's paradox arises from the fact that "the collection of all sets that are not members of themselves," despite being a gathering together of distinct objects, is in fact not a set. It doesn't exist. Actually, Russel's paradox isn't a paradox at all! It's just a proof of the fact that that particular "set" doesn't exist!

    So basically, all of these paradoxes arise because we "assume" that certain sets exist when in fact they do not, so enter axiomatic set theory. Axiomatic set theories give a list of rules (axioms) that dictate which sets exist. They are things like:

    • The empty set exists.
    • If to sets exist, then their union exists.
    • If two sets exist, then their intersection exists.

    These are very simple, obvious things, intuitively (an "axiom" always should be), and if we follow these strict rules, then it's not possible to construct sets like the one that caused Russel's paradox, so we're safe from those sorts of failures.
     
  5. Aug 22, 2011 #4

    pwsnafu

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    There is an axiomatic system called http://en.wikipedia.org/wiki/New_foundations" [Broken], which contains a universal set. And yes, the universal set contains itself.
    But the universal set does not exist in http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory" [Broken] though.
    That is not a mathematical argument. That is philosophy of mathematics.
    Concerning the highlighted section. Mathematics has nothing to do with reality. Applying mathematics to model reality is the realm of physical sciences. Mathematics itself
    is unconcerned with reality.
     
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  6. Aug 23, 2011 #5
    Thank you alexfloo. That information was very helpful.
     
  7. Aug 23, 2011 #6
    Could you clarify that?
     
  8. Aug 24, 2011 #7

    pwsnafu

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    I can give you an example: the http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox" [Broken] states there exists a way of partitioning a sphere into finite number (usually five) disjoint parts, such that, by only using translations and rotations, it is possible to reassemble the partitions to obtain two spheres of the original radius.
    This is a mathematical truth if you accept the http://en.wikipedia.org/wiki/Axiom_of_choice" [Broken]. But it is not replicable in the real world.
     
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  9. Aug 24, 2011 #8
    The thing is that I believe in math and the axiom of choice is logical. It isn't difficult to deconstruct or point out logical inconsistencies with mathematics and real world interactions just like it isn't difficult to point out that my pocket knife can't cut a mountain in half. This is where a strong curiosity of the philosophy of numbers and logic becomes a slippery slope.

    When applied by a reasonable person the tools of mathematics benefit and evolve... though the people who produce these things are rare (Zeno, Kurt Godel, etc) and wildly outnumbered by people whose philosophy and logic are completely uninformed/uneducated, communicated over zealously with too many flaws or just plain crazy.

    That said, mathematics is initially dependent on reality, unless you consider it a separate form of consciousness independent from humans...or something (which hopefully doesn't happen... I've seen the movies :eek:) . However, after that, it does not have to continue to be so. In these cases, using one of our most complex tools (mathematics) we can accomplish something like Banach-Tarski's sphere. Though in fifty years we may have a outbreak of logicians and philosophers (who are credible) which alter the foundations of how we use these tools by creating new perspectives for scientists, making the sphere mathematically illogical, but also producing far better more profound mathematics.

    Personally... I'd like some mathematics for 3d fluid mechanics.

    In my opinion it's the people who ask themselves questions like Godel, Aristotle, Zeno - etc did who produce the start of what becomes change in the sciences. Which is then taken on and massively improved by what I would call specialists (mathematicians, physicists, chemists). And from the specialists you get better technology for biochemists, civil engineers, programmers and so on. Though this has to occur at critical thinking and questioning in a reasonable sense from a knowledge of things.

    P.S. Sorry if that was a bit of a random lecturish post. Had a bit of wine tonight.

    Edit: Also, I'm not implying any sort of restrictions when referring to philosophers, mathematicians, logicians, physicists - etc.
     
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  10. Aug 24, 2011 #9

    Hurkyl

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    I've always found this silly. The correspondence between "subset of the sphere" and "an object you can cut out of a real-life spherical object" has broken down long before you get anywhere close to non-measurable sets appearing in the proof of the Banach-Tarski theorem.
     
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  11. Aug 24, 2011 #10
    I think he was just illustrating his point on mathematics in itself not being concerned with reality. Like 2+2=4 ... doesn't require reality to prove given that it is, in its own system, a true statement and works independently from reality as a result.
     
  12. Aug 24, 2011 #11

    Stephen Tashi

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    For a set A, we have to consider the distinction between "A is a subset of A" and "A is a member of A", before we pass judgement on the coach. "Contain" is an ambiguous verb in the context of sets.
     
  13. Aug 24, 2011 #12
    I'm aware of the distinction, and I certainly wasn't passing judgement, I was simply stating that what he said was not necessarily correct. However, I admit to call it "absolutely" false was to misspeak. It is something which depends on the axioms chosen, and so varies from theory to theory.

    In ZFC, however, what he said is correct. In naive set theory, it is not, which is an inconsistent system, so that hardly matters. Any set theory which contains the axiom schema of specification cannot have such a set, because then we could construct Russel's set. I'm not aware of a proof, however, that a consistent theory cannot have such a set, although there is certainly a lot I'm not aware of.
     
  14. Sep 3, 2011 #13
    I'd like to point out that the this discussion was a dialectic between myself and the coach. When the topic died it was on the point that the set can never contain itself, but that such a point was primarily moot since we approximate numbers to degrees that we need (like Pi or any irrational number), and going deeper, trying to approximate to a finite number was mostly a waste in an, "Okay, that makes sense logically, but it doesn't in any way dismiss the usefulness and applications of set-theory/math/numbers in general".

    I.E. You have one rock, and you have one moon. Both are never really a finite (1) since the cardinality of that set (one rock, or one moon) is not only constantly in a state of change (rapid or slow... all relative to the size of the original set) but the original set itself is in a constant state of change. However the original sets state of change is infinitesimal, as are the identified subsets... the moon, or rock, a week from now is going to have changed far too little for it to matter. Same with the things reasonable enough to consider subsets, or elements, of the set.

    The idea of a pure, finite (1) doesn't seem logically possible, but arguing pretty much all of the former as an extreme flaw in set-theory doesn't get anyone anywhere. And trying to account for all those extremely infinitesimal changes generally seems like a bad idea (unless it isn't, but a situation where that's true is more than likely beyond me).
     
  15. Sep 4, 2011 #14
    The set containing "the moon and the rock" will not change, at all, ever. The actual moon and rock that manifests itself may be a different moon and a different rock, but that set is exactly the same set at all times and under all circumstances.

    Mathematical objects are infinitely more absolute than anything observable in the physical world. "The moon" that is in that set is not necessarily the one in the sky, but a particular one that was/will be in the sky at a particular time. (Or in fact, it may never have been in the sky at all.)

    "To which physical object does the term 'the moon' refer? If every cell in my body dies and is replaced, then how can I be uniquely identified by the same name that identified me years ago? It I remove and replace each board on this boat one by one, is it the same boat? At which point does it become a new boat? Certainly not after I replace one board? After the fifth? The sixth?"

    These are reformulations of your question in different terms. The question you're considering is a question of natural language, not of mathematics. Ludwig Wittgenstein and other linguistic philosophers write on these questions at great length; few have any real answers, but most have explanations that are more than sufficient for modern natural language processing applications.
     
  16. Sep 4, 2011 #15
    Basically everything you mentioned here is why the topic died. :tongue2: Though it was discussed in far more length until it was obviously a waste of time.

    It started as something interesting, then we approached it from every angle we could think of to maybe produce an idea that had practical applications, instead of just sounding like most of the things in 'Tractatus Logico-Philosophicus'. Neither of us could produce that idea.
     
  17. Sep 8, 2011 #16
    @Ntstanch

    You should check into Type Theory for a more tangible understanding of logic. (Set theory is old and full of messy philosopher sh_t).

    It starts off with the Curry-Howard isomorphism:

    http://en.wikipedia.org/wiki/Curry–Howard_correspondence

    Which states a deep compatibility between programming and logic. Propositions are types. Proofs are terms. Implications are functions. Ands are ordered pairs. Ors are disjoint unions. Foralls become Pi types. Exists become Sigma types. Induction becomes recursion. Paradoxes become infinite loops.

    That last one is especially nice. People get fascinated by the idea of a paradox. They become mystified. Does a barber who shaves every man who doesn't shave himself shave himself? If he shaves himself, then he doesn't. But if he doesn't, then he does! But expressed as a program, you end up with an infinite loop. Does he shave himself? Does he not? Does he? Does he not? Does he? Does he not? Does he? ...

    Infinite loops are mysterious. They are annoying. They haven't enlightened you about the futility of meaning in the universe. They mean you have messed up. You assumed something that wasn't safe to assume.

    So can a set contain itself? That's between you, your axioms, and your rabbi.

    But if you do accept it as an axiom, you will be able to prove any statement in your langauge (and its negation).
     
  18. Sep 17, 2011 #17
    The language seems too vague in both theories. You need to create new methods to fit the obvious infinite logical looping inherent in the programs structure... the program is built off the present comprehension of the logic and, from what I can tell, a far more prominent isomorphism between late 1800's logician thought and the same current problems concerning the whole deal.

    When given a set of rules the program, based on a foundation of the current comprehension concerning mathematics/logic and applied to the programming languages, (not the most informed in how these various programs work in great detail, but I have a general idea) you are left with a yes or no "baseball pickle". Or in other words, the program is stuck in an infinite 'thought' with a technically finite series of "yes/no/yes/no" looping set stuck, in the allowed rules. When I say "technically finite" I am referring to the fact that outside the loop, from any persons perspective, it can stop becoming that infinite loop through adapting conscious logic/knowledge/understanding of the issue.

    "People get fascinated by the idea of a paradox. They become mystified. Does a barber who shaves every man who doesn't shave himself shave himself? If he shaves himself, then he doesn't. But if he doesn't, then he does!" -- However a person can realize that this question is a set of rules which become aimless language and direction at some point.

    "If I have exhausted the justifications, I have reached bedrock and my spade is turned. Then I am inclined to say: "This is simply what I do." - Ludwig Wittgenstein -- From my understanding this is how programs approach paradox. And the person (the programmer) is now back to the late 1800's logician frame of mind concerning all of mathematics and its logic. Where creating a better spade seems like the best choice.... If you want to keep dividing by zero that is fine, but to me it's like hitting the bedrock with the spade(s), and not stepping back to improve. Or not stepping back enough.
     
  19. Sep 18, 2011 #18
    "When I say "technically finite" I am referring to the fact that outside the loop, from any persons perspective, it can stop becoming that infinite loop through adapting conscious logic/knowledge/understanding of the issue."

    You're alluding to the fact that a computer could "recognize" the problem, and just stop. This would mean proving that the current loop is infinite, which requires induction, so what you mean in a precise sense by "current comprehension" is non-inductive logic.

    Here's the thing though: when we talk about an infinite loop, we don't really care about the fact that it goes on forever, we care what it tells us about the underlying though.t That is, "the thought is not well-formed," or "the though is self-referential," etc. We don't really care if the program can "notice that it's looping and stop doing it," because if it did that (even though the program would stop running) it's still infinite in the sense above. That is, it still just doesn't tell us anything.

    Keep in mind "running time" is a physical concept. It's a property of a program (which is abstract) in conjunction with an actual physical machine (which is not) to execute that program. Therefore, in this type of mathematics, we dont (and couldn't if we wanted to) care about "how long" the program runs for. If the machine can find a shortcut, and skip the whole program because it knows what's going to happen, that's awesome if you're a programmer, but inconsequential if you're interested in programs as formal arguments.
     
  20. Sep 19, 2011 #19
    I wrote that response at around 4am, drunk... then realized salvaging it would waste peoples time. :P

    Not quite sure how to reply... I know close to nothing about complex programs. Only thing I'm curious about is if they have programs which operate on a different set of logic. Like a "Russell program" to snap a "Cantor program" out of its ill-formed thought "trap". If a program is based on highly structured rules, and a paradox pops up couldn't there be a safeguard which simply identifies this and snaps the program out of it?

    *Shrug* ... No idea how any of it works really. Though this topic (for me) has run drier than it did last spring. Think this area of thought is dead as far as being useful for me... oh, and the topic seems kind of dead as well.
     
  21. Sep 19, 2011 #20

    Stephen Tashi

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    G. Spencer-Brown (http://en.wikipedia.org/wiki/G._Spencer-Brown) developed a system of logic where (by my interpretation), a true statement is represented by the infinite sequence (1,1,...), a false statement is represented by (0,0,...), a statement of the form "A implies not-A" is represented by (1,0,1,0,...). His book, "The Laws Of Form", is a mathematical curiosity, but I don't think anything important came out of it.
     
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