1. B

    Gödel's theorem is invalid his G statement is banned by axiom of the system he uses

    Godels theorem is invalid as his G statement is banned by an axiom of the system he uses to prove his theorem a flaw in theorem Godels sentence G is outlawed by the very axiom he uses to prove his theorem ie the axiom of...
  2. Z

    Axiom of Regularity is a Theorem

    Zylo's Axiom (ZA): A = B → A ∉ B can be an axiom, or Theorem:: Assume A = B and A ∈ B, B={A} = {B} = {{A}} = {{B}} = {{{A}}} = ...... indefinitely → A ∉ B Theorem (Extended Axiom of Regularity} Every member of A which doesn't contain another member of A is disjoint from A. Proof: Lst...
  3. Z

    ZFC Axiom of regularity

    "In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set A contains an element that is disjoint from A. In first-order logic, the axiom reads: {\displaystyle \forall x\,(x\neq \varnothing...
  4. A

    Axiom of Computable Power Set

    I'm guessing something like this already exists or has been proposed and I'm simply not aware, but I am wondering what the consequences would be and am inclined to spark general discussion. Most are familiar with the power set axiom, as it is one of the basic axioms of ZFC set theory...
  5. M

    The Completeness Axiom Of the Reals

    Hi, I've been taught the completeness axiom for the real numbers in my calculus lectures, but whilst reading an analysis book I discovered an axiom with the same name, but different in its presentation: (lecture definition) every convex subset of the real numbers is an interval. (book) every...
  6. R

    What effect does including the axiom of choice have on ZF?

    I imagine the axiom of choice does something for Zermelo-Freankel set theory, as in it modifies it in some non-negligible way, otherwise there would be no point in having a ZFC in addition to ZF. What is its effect? What theorems can be proved with the axiom of choice that can't be proved...
  7. Z

    Zermelo Frankel Axiom of Regularity

    Axiom of Regularity: For any set x\neq0 there is a member y of x such that y\capx=0. x={y,{x}}, y\capx=0 But x is circular.
  8. Z

    Axiom of Regularity is wrong

    AR: ZF Axiom of Regularity*: x has a member y st x\capy=0. a\notina. Proof 1) Let x={a} 2) x\capa=0, AR requirement 3) members of x: a 4) members of a: b (a has to have a member) 5) x\capa=0 implies b\neqa 6) \therefore a can not be a member of itself. Problem with proof: Step 4)...
  9. T

    Archimedes axiom

    I am having trouble understanding what this short proof is about. I will type out what it says in the book then explain my confusion. book: Archimedes axiom states that, given any real number x, so N\leq x and x-N\leq 1 this is the part tripping me up. How can x-N\leq 1 be true beacuse if...
  10. L

    Theorem axiom of separation and yours implications in methaphisics

    In philosophy we begin with a "being" concept, what generate your contrary, or, "don't being". but I propose beginning with concept, "possibilities of existence", because is correct talk the possible, became ancient a concept of being. Well, let's use the axiom of title above to create a set...
  11. B


    Probability of an event made of two or more sample events is the sum of their probabilities. Cold some one give me example for this. Thank you in advance
  12. R

    The axiom of choice and the well ordering theorem

    I wanted to see if I could prove on my own the equivalence of the axiom of choice and the well-ordering theorem. I came up with a proof, which I will now state somewhat informally. I would like to know if my reasoning in this proof is sound. Axiom of choice: For any collection of nonempty...
  13. R

    Is the completeness of the real number line equivalent to Dedekind's axiom?

    Dedekind's axiom of continuity states that if the points of a line be divided into two classes such that every point in the first class is to the left of every point in the second class, then there exists one and only one point $P$ such that every point in the first class is to the left of $P$...
  14. V

    Hello everyone

    Hello everyone! Not sure where to begin, I love math, and science. But I believe the way that I learned math isn't the best way to have learned. Unfortunately i never paid to much attention in geometry class, which now i realize was kinda important. I have been wanting to go back and start...
  15. R

    Propositional logic question

    I've encountered something I don't understand while reading Bertrand Russell's Principles of Mathematics. In chapter 2 he lists ten axioms for propositional logic, which are supposed to be the rules of deduction for all of mathematics. I'm guessing these are equivalent to the ten rules of...
  16. U

    H&B's 2nd axiom for successor function contradicts 1st?

    Hi all, I'm reading Charles Petzold's "The Annotated Turing" for fun, and I'm no mathematician, so this might be a silly question. I have a question about page 226, where Hilbert and Bernay's axioms for the successor function are defined. Specifically, the second axiom: (ÆŽx)(y)-S(y,x) I...
  17. Theta

    Axiom or theorem?

    Let $a,m,k \in \mathbb{Z^+}$ Is it an axiom or a theorem that $[a(m)]k=a[m(k)]$ ? Since it is proved that $ab=ba$ I suppose that the aforesaid proposition needs to be proved too, or maybe it is to be conceived as an axiom? BTW Don't be scared to answer :)
  18. P

    False axiom

    Hi guys. Haven´t been here for a long time, but glad to come back. As usual i have a problem. let me say: "from any given point in the space we can map any other point in that space". Is this true? Let´s assume Earth (hardly a mathematical point), can we reach any...
  19. S

    Axiom of Choice

    I do some exercises in set theory. Among them, there is one that says: Prove, using the Axiom of Choice, the following: a) \forall x \exists y : \phi(x, y) \equiv \exists f \forall x : \phi(x, f(x)) b) \cup_y \cap_x F_{x,y} \subset \cap_x \cup_y F_{x,y} It is clear to me that point a)...
  20. S

    axiom of choice

    How can I use the axiom of choice to prove the fact that every vector space has basis?