2
votes
2answers
84 views

“Partitioning” an uncountable set “equally”

I just observed a simple fact about the real numbers, that if U is an uncountable subset of the set of real numbers, then there exists a real number r, such that both the set V of all elements of U ...
0
votes
0answers
35 views

Flaw in the proof that a set is countable [duplicate]

Q: Let S be the set containing all sequences of 0's and 1's. i.e $S = \{(a_1,a_2,a_3,a_4,\ldots) : a_i = 0 \text{ or } 1\}$ Show that S is countable. Proof(Flawed) : Let $A_i$ be the ...
18
votes
2answers
292 views

Covering $\mathbb R^2$ with function graphs

Suppose we have a countable family of function graphs (each function is $\mathbb R\to\mathbb R$, not necessary continuous). Obviously, they cannot cover the whole plane $\mathbb R^2$, because they ...
1
vote
1answer
78 views

Theorem of Galvin, Mycielski and Solovay

I don't know is this the right place to ask this question, but can someone tell me where I can find the proof of the theorem of Galvin, Mycielski and Solovay. Theorem that says that a subset $X$ of ...
2
votes
3answers
152 views

An example of an ordered, uncountable set in $\Bbb R$?

Is there an example of an ordered, uncountable set in $\Bbb R$? My Calculus professor, who likes to keep things simple, defined a sequence in $\Bbb R$ as an "ordered and infinite list of real ...
1
vote
0answers
73 views

When the continuum hypothesis settles the uniqueness (upto isomorphism) of the Hyper-reals doesn't it mean the hypothesis should be an axiom?

One useful consequence of $ZFC$ is that the real numbers can be shown to be unique upto isomorphism. According to wikipedia: The use of the definite article the in the phrase the hyperreal ...
22
votes
4answers
1k views

Is there a simple, constructive, 1-1 mapping between the reals and the irrationals?

Is there a simple, constructive, 1-1 mapping between the reals and the irrationals? I know that the Cantor–Bernstein–Schroeder theorem implies the existence of a 1-1 mapping between the reals and the ...
3
votes
1answer
250 views

Cardinality of set of discontinuities of cadlag functions

I know that non-decreasing cadlag functions (functions that are right continuous with left limits) on $[0,\infty)$ have at most a countable number of discontinuities. Does the same result hold for ...
8
votes
3answers
243 views

Is Zorn's lemma necessary to show discontinuous $f\colon {\mathbb R} \to {\mathbb R}$ satisfying $f(x+y) = f(x) + f(y)$?

A UC Berkeley prelim exam problem asked whether an additive function $f\colon {\mathbb R} \to {\mathbb R}$, i.e. satisfying $f(x + y) = f(x) + f(y)$ must be continuous. The counterexample involved ...
1
vote
2answers
49 views

Can we find a family of sets over the whole space $X$ that is “larger” than the power set of $X$

Don't know if the statement in the title is true or not. Can anyone help me out? Thanks!
1
vote
2answers
196 views

How is the extended real number line modeled?

In set theory, numbers are often constructed, e.g. from nestings of sets which eventually contain the empty set. The operations are defined in term of taking unions etc.etc. The extended real number ...
4
votes
2answers
101 views

Bounded non-convergent sequence with respect to an ultrafilter

Let $\mathfrak{U}$ be an ultrafilter on $\mathbb{N}$ and $(x_n)$ be a real sequence. We say that $(x_n)$ is: bounded with respect to $\mathfrak{U}$, if there exists $M>0$ such that $\{n \mid ...
3
votes
2answers
106 views

Cardinality and Measurability

We can show that $\mathbb{R}$ and $\mathbb{R}^2$ or ($\mathbb{R}^n$) have same cardinality using the following one-to-one and onto mapping: Say x = (0.123456789....) Then f(x) = ...
4
votes
0answers
79 views

Partition of $\mathbb{R}$ into nullset and 1st category set

Let $\{a_i\}_{i=1}^{\infty}$ be an enumeration of the rationals, $\mathbb{Q}$. Let $I_{ij}$ be the open interval centered at $a_i$ and having length $1/2^{i+j}$, and define $G_j = \cup_{i=1}^\infty ...
5
votes
3answers
168 views

Alternate proofs (other than diagonalization and topological nested sets) for uncountability of the reals?

I recently started studying set theory and am having quite a bit of difficulty accepting Cantor's diagonal proof for the uncountability of the reals. I also saw a topological proof via nested sets for ...
3
votes
2answers
179 views

how to create the set of hyperreal numbers using ultraproduct

As title says, can anyone explain how to create the set of hyperreal numbers using ultraproduct from the set of real umbers?
5
votes
2answers
291 views

Are there any continuous functions from the real line onto the complex plane?

Is there any measurable continuous differentiable analytic surjective function $f:\mathbb{R}\to\mathbb{C}$?
0
votes
1answer
44 views

The set S of all alignments of digits obtainable by sequence of positive integers is countable?

Is $\left\{ n_k \right\}_{k=1}^{\infty}$ a strictly increasing sequence of positive integers (written in decimal notation). Consider the alignment (infinite) set of digits {${n_1 n_2 n_3}$} format ...
1
vote
0answers
102 views

Questions about $\sigma$-algebra, algebra and topology

I know the definitions of $\sigma$-algebra, algebra and topology, but why countable/finite union, as in $\sigma$-algebra/algebra, and finite intersection, arbitrary union as in topology? What inspire ...
4
votes
1answer
105 views

Is this function constructed using AC necessarily discontinuous everywhere?

Assume AC. Let $x_\alpha$ be a well-ordering of $\mathbb{R}$. For all $\alpha < \mathfrak{c}$, let $F(x_\alpha) = x_{\alpha+1}$. Can it be proven that $F$ is discontinuous everywhere?
2
votes
1answer
257 views

Cardinality of continuous functions on subspace of R^n

I know that the set of continuous functions on $R^n$ has cardinality of R. Can this be generalized to any subspace of $R^n$? It seems intuitive, but the empty set seems to be a counterexample of this ...
0
votes
1answer
290 views

Cantor Set defined by sequence

http://www.scribd.com/mobile/doc/76236535 page 49-50 Exercise 3.19 Let $A=\{0,2\}$ and $C$ be the Cantor Set. Define $x(\alpha) = \sum_{n=1}^\infty (\alpha_n / {3^n})$ for all $\alpha \in ...
11
votes
4answers
543 views

Is there a set that is both a sigma algebra and a topology but not a powerset?

Is there a set that is both a sigma algebra, $\Sigma$, and a topology, $\tau$, but not a powerset, $\mathcal{P}$?
2
votes
4answers
156 views

What is the order type of monotone functions, $f:\mathbb{N}\rightarrow\mathbb{N}$ modulo asymptotic equivalence? What about computable functions?

I was reading the blog "who can name the bigger number" ( http://www.scottaaronson.com/writings/bignumbers.html ), and it made me curious. Let $f,g:\mathbb{N}\rightarrow\mathbb{N}$ be two monotonicly ...
-2
votes
2answers
394 views

(ZF)subsequence convergent to a limit point of a sequence

Arthur's answer; (ZF) Prove 'the set of all subsequential limits of a sequence in a metric space is closed. Let $\{p_n\}$ be a sequence in a metric space $X$. Let $B=\{p_n|n\in\mathbb{N}\}$ and ...
5
votes
1answer
137 views

Does the specification of a general sequence require the Axiom of Choice?

Many results in elementary analysis require some form of the Axiom of Choice (often weaker forms, such as countable or dependent). My question is a bit more specific, regarding sequences. For ...
5
votes
3answers
141 views

Always a value with uncountably many preimages? (for a continuous real map on the plane)

Let $f$ be a continuous map ${\mathbb R}^2 \to {\mathbb R}$. For $y\in {\mathbb R}$, denote by $P_y$ the preimage set $\lbrace (x_1,x_2) \in {\mathbb R}^2 | f(x_1,x_2)=y \rbrace$. Is it true that ...
2
votes
2answers
66 views

Elementary Question

Let be $f:X\to X$ a bijection, an $A\subset X$ a invariant subset of $X$, i.e $f(A)\subset A.$ How can see that $$f(A)=A$$ I'm trying to show that $$f(A^{c})\subset A^c$$ but I can not.
1
vote
1answer
241 views

Using Gödel Numbering to Represent Sets of Real Numbers

Recently, I was looking at Gödel numbering, and I was wondering, is it possible to represent, as a single real number, a set of real numbers. When dealing with only rational numbers, it would be easy ...
10
votes
3answers
1k views

Axiom of choice, non-measurable sets, countable unions

I have been looking through several mathoverflow posts, especially these ones http://mathoverflow.net/questions/32720/non-borel-sets-without-axiom-of-choice , ...
20
votes
1answer
3k views

Cardinality of Borel sigma algebra

It seems it's well known that if a sigma algebra is generated by countably many sets, then the cardinality of it is either finite or $c$ (the cardinality of continuum). But it seems hard to prove it, ...
2
votes
1answer
358 views

Formalising real numbers in set theory

If I understand correctly, the real numbers can be formalised in a first-order, like in ZF. However, such a formalisation is not strong enough to ensure that all models of the reals are isomorphic. ...
16
votes
4answers
1k views

Why does Cantor's diagonal argument yield uncomputable numbers?

As everyone knows, the set of real numbers is uncountable. The most ubiquitous proof of this fact uses Cantor's diagonal argument. However, I was surprised to learn about a gap in my perception of the ...
2
votes
3answers
493 views

Set Theory as it Relates to Number Systems?

I've been referred to this website, hopefully you have the background in set theory to help me out here. Got two questions, the first is on number systems arising out of set theory and the second ...
4
votes
1answer
176 views

Where in the analytic hierarchy does V=L start having consequences?

I note that the ordinals of L are the same as V, so I guess that it has no $\Pi_1^1$ consequences. On the other hand Wikipedia tells me that it asserts the existance of a $\Delta_2^1$ non-measurable ...