uncool

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  1. Spark vs Rank of a matrix

    The statement "the spark of a matrix is zero" expands to mean "There is a set of columns of size zero that is linearly dependent." Which isn't true. Spark of a full rank matrix is something of a convention. Spark increases as linear dependence decreases - and a full rank matrix is maximally linearly independent, so you want the spark to be large, not small. Choosing it to be infinity is likely the best convention.
  2. Continuity and uncountability

    1) There is a school/philosophy of mathematics that TheSim is referencing, called constructivism. It is a relatively rare view. 2) To be honest, you don't have the experience, knowledge base, or understanding to deal with the differences between constructivism and the standard view. I would highly recommend that you study the basics for quite a while longer.
  3. Continuity and uncountability

    And that's a nice demonstration of what I said. The set of real numbers is not countable, pengkuan.
  4. Continuity and uncountability

    It also doesn't really get rid of the philosophical confusion - I'd even argue it adds to it. Someone should have a firm grasp of the basics of countability and set theory before attempting to get into computability theory.
  5. Sets vs Omniscience

    The powerset of the empty set is the set of the empty set, P({}) = {{}}. It has one element: the empty set.
  6. Continuity and uncountability

    That's what cardinality means. And no, that doesn't explain why it should be a natural number. There are infinitely many even numbers. There is no natural number n such that the set of even numbers is in bijection with {0, 1, ..., n - 1}.
  7. Continuity and uncountability

    I'm not sure what you mean by that. i can be an arbitrarily large finite number. We can say that the rationals are countable because we can construct a bijection between the natural numbers and the rational numbers. That's all there is to it.
  8. Continuity and uncountability

    I'm not sure what you mean by "at the nth step, we stop the count"; if you mean "for each i/j, we can stop the count at some n and be at i/j", yes. The other two parts are both correct. That is a way to look at it; that your attempt to match N to its powerset doesn't work for the set of all even numbers because there is no finite number that matches it. Not really. Countability is about the existence of some matching - some bijection; the fact that some map doesn't work as a bijection doesn't mean there can't be another. For example: the set {1, 1/2, 1/3, 1/4, 1/5, ...} with an added 0 (i.e. union with {0}) is countable, even though the map n -> 1/n clearly "misses" 0. It is countable because of the bijection 1 -> 0, n -> 1/(n - 1) for n > 2.
  9. Continuity and uncountability

    No, that definition of finitude is not the definition used to prove that a set is countable, because countability and infinitude are different concepts. However, I want to draw your attention to something. Note that every fraction appears after a finite number of steps. That is, if you gave me a fraction, I could tell you exactly at what step you reach it - and I could represent the number by writing it as "1 + 1 + 1 + 1" and eventually stop. For example, 2/4 is reached in the 12th step, that is, the 1+1+1+1+1+1+1+1+1+1+1+1th step.
  10. Continuity and uncountability

    Infinite means not finite. A set A is infinite if it is not finite, that is, if for any n, there is no bijection f: A -> {0, 1, ..., n-1}. It's not hard to prove that the set of even numbers is infinite.
  11. Continuity and uncountability

    I can give several definitions that are equivalent under the usual axioms of ZFC. The one that seems useful here is: A set A is finite if there exists a natural number n such that A is in bijection with the set {0, 1, 2, ..., n-1}.
  12. Continuity and uncountability

    That's not a precise definition. However, even using that imprecise definition, it is clear that the set of even numbers is not finite.
  13. Continuity and uncountability

    How do you define a finite set?
  14. Continuity and uncountability

    Among other things: 1) The definition you are attempting to use would make every set finite. 2) The specific method you are attempting would have to use the definition that a set is finite when there is a natural number corresponding to its size. That does not in any way match the definition you attempt to use to justify your claim there.
  15. Continuity and uncountability

    "K e is a natural numbers for all n, thus K e has finitely many digits when n increases endlessly." That's not how any of this works. You are still making the same error, after all these years.