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201 Beacon of Hope

About uncool

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  1. I assume this is the post? The problem is that you seem to have misunderstood the well ordering principle by accidentally rearranging quantifiers. It doesn't say that there is a minimum that works for all sets; it says each set has its own minimum; that is, while multiple sets can have the same minimum, they are not all required to. z may not be a minimum of T, but another thing can be. Your proof uses nothing about the real numbers. If it worked, it would apply equally well to the natural numbers, which are well-ordered. Try it out, and see if you can understand why it doesn't work there. Alternatively, try writing your proof formally - you should be able to see where you have wrongly exchanged quantifiers.
  2. Continuity and uncountability

    Precisely - which means you have no natural number corresponding to the set of even numbers, which means you have no bijection.
  3. Turn my right arm into a whip

    There is a genetic component, however, that says "When you are getting these signals, do this." I think the relevant part may be Hox genes; an example is when scientists were able to cause flies to grow antennae in the place of legs. https://en.m.wikipedia.org/wiki/Antennapedia
  4. Cardinality and Bijection of finite sets

    The axioms are designed to work together. Just saying "limited by ω" doesn't really work. For the rest of your post, the idea of formal sums covers what you are trying to do. You are trying to deal with the concept of formal sums over an ordered set.
  5. Cardinality and Bijection of finite sets

    This is not in any way how the usual natural numbers work. I'd recommend using a different name. N' or EN or something like that. So you've defined these new "pseudo-natural numbers". OK. Now what do you plan to do with them? Note that anything you prove with these "pseudo-natural numbers" does not affect statements about actual natural numbers.
  6. Cardinality and Bijection of finite sets

    I can't figure out what that means, no. You're clearly using "+" to mean something different from how it is usually used. To answer a later question in your post, yes, since you are using "+" in a nonstandard way, I do expect you to define it.
  7. Continuity and uncountability

    Any finite number? So the string "101010101010" by itself represents the set of all even numbers? Can you type out the specific string you think represents the set of all even numbers? No ellipsis - the exact string.
  8. Cardinality and Bijection of finite sets

    If you're adding an axiom, you will have to demonstrate that your new system is consistent (or at least, equiconsistent).
  9. Cardinality and Bijection of finite sets

    You would have to prove there is such an ω. When considering the integers, there is no such ω; when working with extensions, you will have to define your operations (addition, subtraction, etc.) If you are truly usinng cardinalities, then there is no subtraction. ω-1 does not make sense. I'm not "creating them". B does not contain elements that aren't in A. For any integer x, 2x is also an integer.
  10. Cardinality and Bijection of finite sets

    Again: a bijection doesn't exist "step by step". It exists all at once. For every number, it is possible to double that number. This process is injective, and by definition of even numbers surjective to the even numbers. Therefore, it is a bijection.
  11. Continuity and uncountability

    N has infinitely many members, each of which has finitely many digits. So there is no natural number "n" that denotes the number of elements of N.
  12. Cardinality and Bijection of finite sets

    A bijection isn't (generally) a step-by-step thing. It can be built step-by-step, but the function simply is. For example: the identity map on the natural numbers, that takes each number to itself. This can be built step-by-step, sending n to n, but it simply exists: given any n, it returns n. It is clearly a bijection, just by checking definitions.
  13. Continuity and uncountability

    As strange has said, that is not a natural number. If you think otherwise, then please write that many '1's in a row. For example: I can write 2 1s: 11 I can write 15 1s: 111111111111111
  14. Continuity and uncountability

    As with all your other attempts, pengkuan, this fails because you exhaust the finite subsets, then claim to have exhausted all subsets. What natural number corresponds to the set of all even natural numbers?
  15. definition of derivative

    How did you get those series in the first place? They usually would come from calculus, but you are trying to define calculus in the first place. It would look a lot like (be isomorphic to) the field of rational functions on R, with order given by "eventual" behavior/leading coefficient.