# Tagged Questions

4answers
73 views

### How do I tell whether axiom of choice is used or not?

I am having a hard time understanding the Axiom of Choice(AC). Say I have an index set $A$ , and a collection of indexed sets {${V_\alpha}$}, where $\alpha$ is a member of $A$. Then, does the ...
1answer
235 views

### Does a nonlinear additive function on R imply a Hamel basis of R?

A function is additive if $f(x+y) = f(x) + f(y)$. Intuitively, it might seem that an additive function from R to R must be linear, specifically of the form $f(x) = kx$. But assuming the axiom of ...
2answers
288 views

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

Is there any measurable continuous differentiable analytic surjective function $f:\mathbb{R}\to\mathbb{C}$?
2answers
303 views

### What is a “linear set”

I'm reading "L'hypothèse du continu" by Sierpinski. He mentions many time "ensembles linéaires" or "linear sets" without defining this notion. Does anyone know what is the definition a such a set ? ...
3answers
717 views

### Curious facts about ordinal numbers

I have some notes about curious facts about ordinal numbers, for example that their addition is not commutative, multiplication is not distributive from the right hand side and that the exponent rule ...
1answer
128 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 ...
2answers
126 views

### Decimal expression of reals

Let $x>0$ be real. Then $A_1=\{n\in \mathbb{N}\mid x<n\}$ is nonempty since $\mathbb{R}$ is dedekind complete. Since $\mathbb{N}$ is well ordered, $A_1$ has a least element $k$. Thus $k-1$ is ...
3answers
594 views

### Is there a difference between allowing only countable unions/intersections, and allowing arbitrary (possibly uncountable) unions/intersections?

As in the title, I am asking if there is a difference between allowing set-theoretic operations over arbitrarily many sets, and restricting to only countably many sets. For example, the standard ...
2answers
152 views

### Measure on Baire space

Inspired by the first parenthetical sentence of Joel's answer to this question, I have the following question: is there any useful notion of measurability in the Baire space $\omega^\omega$? Some ...
2answers
952 views

### Continuity and the Axiom of Choice

In my introductory Analysis course, we learned two definitions of continuity. $(1)$ A function $f:E \to \mathbb{C}$ is continuous at $a$ if every sequence $(z_n) \in E$ such that $z_n \to a$ ...
3answers
254 views

### Integral of a function defined in the set of Surreal Numbers

Given ${\{C}\}\$ the set of all the $Surreal\ numbers$, is it possible to define the integral: $$\int_a^b{dxf(x)}$$where $$a\in{\{C}\},b\in{\{C}\},x\in{\{C}\}$$ Thanks
2answers
557 views

### What are the consequences if Axiom of Infinity is negated?

What mathematics can be built with standard ZFC with Axiom of Infinity replaced with its negation? Can the analysis be built? Is there special name for "ZFC without Infinity" set theory? I also ...
4answers
266 views

### Union of Uncountably Infinite Sets

How does one notationally describe the set which is the union of uncountably many other sets. For instance, for each x such that a < x < b, where a and b are real numbers, if there is assigned ...
0answers
166 views

### Can Egoroff's Theorem be Strengthened for A Sequence of Smooth Functions？

I have posted this question on MO, and I'll raise the question here in a more concrete way. To simplify the situaton, we assume the measure space we are dealing with right now is a closed interval $I$ ...
1answer
93 views

### Does the existence of an infimum imply that the set of lower bounds of a set is totally ordered?

Say we have a partially ordered set $(S,\preceq)$, and some subset $E\subseteq S$ such that $E$ is bounded below and $\inf E$ exists. My question is, since $S$ is not totally ordered is it possible to ...