# uncountable

1. ### Enumerating An Uncountable Ordinal

Disclaimer: I have a sense of humor and a purpose, so Iâ€™m going to try and enumerate $\omega_1$ itself. Clearly that is a crankish proposition, so this disclaimer is meant to clarify what I mean to accomplish and acknowledge that there should obviously be an error if I conclude that I have...
2. ### Well Ordering of an Uncountable Set Question

Let $I = \{ x : x \text{ is an infinite binary string} \}$. Let $F = \{ x : x \text{ is a finite binary string} \}$. For each $a \in I$, $a = a_{1}a_{2}a_{3}\dots$ where each $a_i$ equals $0$ or $1$. For each $a \in I$, let $f(a) = \{ a_{1}, a_{1}a_{2}, a_{1}a_{2}a_{3}, \dots \}$. Let...
3. ### Uncountable Subset

Which of the following is an uncountable subset of $R^2$ ? A) {(x,y) \epsilon R^2 : {x \epsilon Q} { or} {(x+y ) \epsilon Q} } B) {(x,y) \epsilon R^2 : {x \epsilon Q} and {y \epsilon Q} } C) {(x,y) \epsilon R^2 : {x \epsilon Q} { or} {y \epsilon Q} } D) {(x,y) \epsilon R^2 : {x...
4. ### Countable and uncountable sets

Does the number of all irrational elements divided by the number of all rational elements equal the number of all transcendental elements divided by the number of all algebraic elements? Are these ratios otherwise comparable?
5. ### Reals are uncountable - without Cantor diagonal argument

Gist of proof: length of unit interval = 1, if the reals are countable, the length = 0, contradiction. Proof: Arrange reals into countable list. Let x be an arbitrarily number > 0. Cover first point on list by interval of length x/2, second point by interval of length x/4, third point by...
6. ### Ruler Postulate and bijection between uncountable sets

why the Ruler Postulate itself is an another postulate ??? i mean there are postulates to define "Lines" "Points" and "Real Numbers" so why one cannot prove the existence of the ruler postulate using them ??? :confused: if it has to be an another postulate then there must be some building...
7. ### Ruler Postulate and bijection between uncountable sets

why the Ruler Postulate itself is an another postulate ??? i mean there are postulates to define "Lines" "Points" and "Real Numbers" so why one cannot prove the existence of the ruler postulate using them ??? :confused: if it has to be an another postulate then there must be some building...
8. ### Cantor: The reals are uncountable

The following proof is incorrect From: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument -------------------------------------------------------------------------------------- In his 1891 article, Cantor considered the set T of all infinite sequences of binary digits (i.e...
9. ### Countable vs uncountable vs logic

I cannot seem to understand in what sense real numbers are uncountable. No matter how many real numbers (rational or irrational) you give me I can use naturals to count them. Î•, Ï€, and Ï† are three irrationals that I just counted and found to be three. What is my faulty logic?
10. ### How to prove P(N) is uncountable ?

How do i prove that P(N) is uncountable, I can't use the diagonalization method for sure here. Can anyone explain how do I go about solving this ? One hint is that I have to equate P(N) to another uncountable set such as set of Real numbers. or a infinite number of sets which contain infinite...
11. ### The minimal uncountable well-ordered set

I had given up on finding this, but since we're having so much fun talking about infinite sets in another thread I thought I might give it another go. My Topology text makes a lot of use of S_{\Omega}, the "minimal well-ordered set, such that every section is countable." Two questions...
12. ### show that M is uncountable

If M is connected and has at least two points, show that M is uncountable. (Hint: Find a nonconstant, continuous, real-value function on M) I have no idea how to start this proof, need help on it.
13. ### Prove that f: N -> {0,1} is uncountable

I need to prove that f: N -> {0,1} is uncountable. Can anyone help me with this proof? Thanks!
14. ### Uncountable

Can anyone explain why this function ins't countable? Real #'s with decimal representation of all 1's or all 9's.
15. ### countable, uncountable

so we are told that if E1, E2, ... are finite sets, and E:=E1 x E2 x ... :={(x1, x2, ...) : xj in Ej for all j in N} then E is countable. We are to prove this or show a counterexample to disprove it. So in the back of the book it tells us that this is false. I am totally confused about this...
16. ### Set Theory: Prove the set of complex numbers is uncountable

How to prove the set of complex numbers is uncountable? Let C be the set of all complex numbers, So C={a+bi: a,b belongs to N; i=sqrt(-1)} -------------------------------------------------- set of all real numbers is uncountable open intervals are uncountable...
17. ### Surjection, Uncountable Set

Prove that there exists a surjection from \mathcal{P}(\mathbb{N}) onto \omega_1. Notation: \omega_1 denotes the first uncountable ordinal. The solution may have some aspects in common with the proof of Hartog’s theroem. [we may not use the Axiom of Choice]
18. ### Function uncountable problem

Hello, Please help me with this question: Complete the definition of the function f in the argument below designed to prove that the set S of total functions from N to { 11, 17 } is uncountable. Suppose f0, f1, f2, ... is any list of elements of S. The we construct a function f : N -> { 11, 17...
19. ### Sets, Countable and Uncountable

Is there a therom that says a countable number of items can be removed from an uncountable set and it will remain uncountable? I believe this to be true, but I'm looking for a proof or a way to prove. My logic is that an uncountable set is akin to an infinate number. An infinate number minus...