# Tagged Questions

Reverse mathematics is the study of which axioms are required to prove mathematical theorems. This study is carried out by using formal theories of arithmetic, particularly subsystems of second-order arithmetic. Similar results in the context of set theory, for example those related to the axiom of ...

23 views

### Self-Similar Reverse-Sum Sequences

When a number is added to its reverse (digits in reverse order), sum it. For example: 102 + 201 = 303 508 + 805 = 1313 246 + 642 = 888 Given all numbers between 0 and 10^9, discover ...
15 views

### Proof of Borel-Wadge determinacy without using Borel determinacy?

It's easy to prove Borel-Wadge determinacy from Borel determinacy. But it's often said that Borel-Wadge determinacy is 'much weaker' than the latter. This is then argued by showing models in which the ...
42 views

### Ramsey theorems for the naturals and for general infinite sets

In reverse mathematics and in recursion theory, the infinite Ramsey theorems are usually stated in terms of coloring of $[\Bbb N]^n$. How do these (not) imply the Ramsey theorems for general infinite ...
131 views

### Applications of the Mean Value Theorem (but not Mean Value Inequality)

The mean value theorem, found in every calculus textbook since the time of Cauchy (or before), says the following: (MVT) Suppose $f : [a,b] \to \mathbb{R}$ is continuous on $[a,b]$ and ...
54 views

### Omega-model of WWKL consisting of random reals

I've been trying to show, as an exercise, that over $\mathrm{RCA_0}$ weak weak Kőnig's lemma (WWKL) does not imply weak Kőnig' lemma (WKL). I've been working on it by constructing an $\omega$-model ...
21 views

### non-trivial conditions for integrability of $| d^2u/ dr d\theta |$ on an annulus

Let $u \in C^\infty (A)$ be a smooth bounded function on annulus $A = \{ (r,\theta) | \varepsilon \le r < 1 \}$, converging to a continuous function on the outer boundary (smooth on the inner). I ...
12 views

### Limit of the derivative and LUB

Let $(k,+,.,0,1,<)$ be an ordered field. In the folowing definitions, all numbers and notions are derived from the ordered field structure of $k$, and $a < c$ are generic elements of $k$. ...
14 views

### Proof of determinacy result without power set?

The wiki page about 'Determinacy' contains the following fragment: "For every integer n, ZFC\P proves determinacy in the nth level of the difference hierarchy of $\Pi_3^0$ sets (...)" (Here P ...
28 views

92 views

### $\omega-$model for $RCA_0$ and Proof of Ramsey's Theorem in $ACA_0$

These are two questions I encounter in Reverse Mathematics recently. In the characterization of the $\omega-model$ $M$ for $RCA_0$, the necessary and sufficient condition is $M$ is non-empty and ...
118 views

### Ring Theory and Induction

Andrew Boucher has developed a theory called General Arithmetic (GA): http://www.andrewboucher.com/papers/ga.pdf GA is a sub-theory of Peano Arithmetic (PA). If we add an induction schema (IND) to ...
129 views

### Existence of sequence of rational numbers in $RCA_0$

In subsystem $RCA_0$ of second order arithmetics, existence of arbitrary functions is not guaranteed since some may not be defined using $\Delta_1^0$ comprehension. In Simpson's book, when real ...
140 views

209 views

### How does adding the full second order induction scheme affect the consistency strength of subsystems of second order arithmetic?

Following on from my question about $\omega$-models, I'm interested in the interaction between subsystems of second order arithmetic with restricted induction such as $\mathsf{RCA}_0$ and those which ...
Richard Shore, in his 2010 paper in the Bulletin of Symbolic Logic, 'Reverse Mathematics: The Playground of Logic', writes that Obviously, if an $\omega$-model $\mathcal{M}$ (those with $M = \... 1answer 80 views ### Determinacy of Negation of Class of Formula First, this question is strictly in the context of Reverse Mathematics where various set comprehension and various axiom of choice may not be available. Question: Over$\text{RCA}_0$, if one has$\...
In Simpson's book, a well-ordered set $X$ is a linear ordering such that there are no functions $f : \mathbb{N} \rightarrow X$ which is decreasing. However, a familiar definition of well-ordering is ...