# Tagged Questions

**4**

votes

**2**answers

189 views

### What is the status of the extreme value theorem in forms of constructive mathematics, such as Smooth Infinitesimal Analysis?

In certain intuitionistic frameworks the extreme value theorem cannot be proved. Depending on the exact framework, counterexamples can be constructed as well; see for example pp. 294-295 in
...

**14**

votes

**2**answers

456 views

### Pathological behavior of Borel sets?

Usually in set theory, Borel sets are much more nicely behaved than arbitrary sets of reals. One reason for this is Borel determinacy, which immediately yields measurability, Baireness, and the ...

**16**

votes

**2**answers

457 views

### Is the notion of fixed point property for topological spaces an absolute notion?

Recall that a topological space $X$ has the fixed point property (FPP) if any continuous function $f: X\to X$ has a fixed point.
Is the notion of FPP for topological spaces an absolute notion? More ...

**30**

votes

**1**answer

911 views

### Is $π$ definable in $(\Bbb R,0,1,+,×,<,\exp)$?

(This question is originally from Math.SE, where it didn't receive any answers.)
Is there a first-order formula $\phi(x) $ with exactly one free variable $ x $ in the language of ordered fields ...

**5**

votes

**1**answer

365 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 ...

**12**

votes

**1**answer

738 views

### Is it necessary to use AC to solve this problem ?

Dear All,
As a routine application of Zorn's Lemma, one can show that there is a subset $A$ of $\mathbb{R}$ such that $A$ contains no arithmetic progression of length 3 but for any $x\not \in A$, ...

**2**

votes

**3**answers

261 views

### Construct a fixed-point set operator

How to find an uncountable set $S$, and construct an function $f : 2^S
\longrightarrow S$ such that for any $T \subseteq S$, $f \left( T \right) \in
T$?
for example, let $S =\mathbb{R}$, how can I ...

**18**

votes

**4**answers

1k views

### Is a random subset of the real numbers non-measurable? Is the set of measurable sets measurable?

One might say, "a random subset of $\mathbb{R}$ is not Lebesgue measurable" without really thinking about it. But if we unpack the standard definitions of all those terms (and work in ZFC), it's not ...

**6**

votes

**2**answers

692 views

### reverse mathematics strength of “Lipschitz functions are somewhere differentiable”

What is the reverse mathematics strength of
"For all Lipschitz functions $\; f : \mathbb{R} \to \mathbb{R} \;$, $\;$ there exists a real number $x$ such that $f$ is differentiable at $x$." ?
...

**4**

votes

**1**answer

276 views

### ordered fields with the bounded value property, without choice

In his answer to my question ordered fields with the bounded value property, Ali Enayat showed that if one assumes the countable axiom of choice, then there exists a non-Archimedean ordered field $F$ ...

**3**

votes

**2**answers

850 views

### a different nested intervals theorem

Is there any literature on (and a standard name for) the proposition that for any arbitrary-cardinality collection of closed intervals in the reals that is nested (in the sense that, given any two of ...

**3**

votes

**1**answer

644 views

### ordered fields with the bounded value property

Say that an ordered field $F$ satisfies the bounded value property if, for all $a < b$ in $F$ and for every continuous function $f$ from $[a,b]_F := ${$x \in F: a \leq x \leq b$} to $F$, there ...

**8**

votes

**3**answers

576 views

### truth vs. provability for ordered fields

In Propositions equivalent to the completeness of the real numbers I started by asking "Can anyone point me to a reasonably comprehensive article (or book chapter) explaining which basic theorems of ...

**7**

votes

**3**answers

1k views

### incompleteness in real analysis

Godel's theorem tells us that any sufficiently powerful consistent formal theory of the integers is incomplete; but what about formal theories of the real numbers? More precisely, what about theories ...

**13**

votes

**2**answers

1k views

### Propositions equivalent to the completeness of the real numbers

Can anyone point me to a reasonably comprehensive article (or book chapter) explaining which basic theorems of calculus are equivalent to the completeness axiom of the reals and which ones aren't?
...

**5**

votes

**2**answers

407 views

### Sturm chain analogue for exponential polynomials?

I'm going to define an exponential polynomial of degree $k$ as a function $f$ of the form
$f(x) = \sum_{i=1}^k c_ie^{\alpha_ix}$ ($\alpha_i$s real).
My first question is: is there an algorithm for ...

**3**

votes

**3**answers

603 views

### Monotone injection of an ordinal into $[0,1]$

This is related to my recent question and would provide a natural positive answer to Question 2. I am sure this must be known to experts.
Question: Is there a monotone injection $(\omega_1,<) ...

**4**

votes

**0**answers

769 views

### Proofs of Baire category theorem

I would like to have a list of proofs of the fact that the real line is not meager (also very useful would be a reference to such a list, if it already exists somewhere).
My motivation is the ...