# Tagged Questions

148 views

### The word problem of the free left distributive algebra on one generator

A left distributive algebra is a set $A$ together with a binary operation, $\cdot$, satisfying $a\cdot(b\cdot c)=(a\cdot b)\cdot(a\cdot c)$. One important example of left distributive algebras arises ...
198 views

### Does forcing with recursively pointed perfect trees add a Turing degree that is minimal over $V$?

A tree $T$ on $\omega$ is recursively pointed if it is recursive in each of its branches. We can consider a variant of Sacks forcing where the conditions are recursively pointed perfect trees ordered ...
422 views

### Does Turing determinacy imply full determinacy?

The axiom of Turing determinacy is a weakening of the full axiom of determinacy, $AD$, in which only games with payoff sets which are $\equiv_T$-invariant are demanded to be determined. In "Turing ...
350 views

### Demuth's theorem in set theory

I am quite sure the following fact must have been known for set theorists, though I could not find it anywhere. If $r$ is random over $L$ and $x\in L[r]\setminus L$, then there must be some real ...
221 views

### Only admissibles start gaps in clockable ordinals

This is a question about ITTM model introduced by Hamkins et al. In this paper it is proven that no admissible ordinal is clockable, so it either starts or lies within a gap in clockable ordinals. I ...
613 views

### Why isn't this a computable description of the ordinal of ZF?

In a previous MO question, I was told by several commenters that (a) it's known that there exists a computable ordinal $\alpha_{ZF}$ that "encodes the strength of ZF set theory" (i.e., a least ...
183 views

### Would a non-constructible set become constructible if we had oracles of arbitrarily high cardinality for the halting problems of ordinal computers?

I still have trouble to grasp the concept of a non-constructible set, my intuition is that we could "avoid" the non-constructibility of many of them if we assume we have "ordinal computers" extended ...
605 views

### What set theoretical questions could never be answered by Turing machines of arbitrary cardinality?

Let us assume that there are Turing machines of arbitrary cardinality, by that I mean they can have input tapes of any arbitrarily high cardinality and compute for a number of steps also of ...
303 views

### “Hard” separation results in reverse mathematics (or similar)

This is a fairly broad question. In particular, I specify 5 questions (Q1, Q2.1, Q2.2, Q3, Q4) which for me all fall under one umbrella. Since this is unreasonably broad, I'm really interested in an ...
405 views

### Infinite Partitions of the Primes and Sums of Reciprocals (Revised)

I have revised my original post. The questions I asked there were not well-put or even thought through. I don't want to delete, however, since some of the comments may be of interest to other MO ...
247 views

### Sorting of countabe set [closed]

Let $X$ be a countable ordered set. My question is very simple - Can we sort $X$ in countable number of steps? When $X$ is finite, the answer is obviously yes. But what is the answer when $X$ is ...
159 views

### A well-behaved $A$ that is almost contained in every element of some filter for a countable arithmetically closed family $\mathfrak X$

The question has relevance for constructing Scott sets with certain extra desirable properties. Suppose that $\mathfrak X$ is a countable arithmetically closed family of subsets of $\mathbb N$: ...
287 views

### Concerning Silver's result

Jack Silver proved that if $x$ is a real so that every $x$-admissible ordinal is a cardinal in $L$, then $0^{\sharp}$ exists. I wonder whether various weaker or stronger versions of Silver's result ...
162 views

### A computability-theoretic preorder on reals

My question is about a fairly artificial preorder on functions from $\omega$ to $\omega$, which for simplicity I'll call "reals." For $r, s\in {}^\omega\omega$, write $r\le_E^*s$ if for each real $f$ ...
638 views

### Does an existence of large cardinals have implications in number theory or combinatorics?

Does an existence of large cardinals have implications in more down-to-earth fields like number theory, finite combinatorics, graph theory, Ramsey theory or computability theory? Are there any ...
739 views

### Can a Hamkins infinite time Turing Machine with infinite Super Turing jumps (from higher type oracles) get the power to decide $\Sigma_1^2$ sets?

Hamkins showed that his infinite time Turing machine has the power to decide some $\Delta_2^1$ sets. I wonder if some modifications of the machine could be made to reach level $\Sigma_1^2$ sets, or, ...
560 views

### Polynomial-time algorithm to compare numbers in Conway chained arrow notation

I am looking for a polynomial-time algorithm which, given a character string containing two numbers in Conway's chained arrow notation for large numbers, indicates whether the first number is less ...
375 views

### Definition of HYP in $L_{\omega_1^{CK}}[a]$?

The structure $L_{\omega_1^{CK}}$ consists of only HYP sets (I believe) and HYP in this structure is the same as the actual hyperaritmetic sets. Now if I move to the structure $L_{\omega_1^{CK}}[a]$ ...
167 views

### $\omega$-models of $\mathbf{\Sigma^1_1}-DC$ and $\mathbf{\Delta^1_1}-CA$

So what is needed to demonstrate something (say like $L_{\omega_1^{CK}}[a]$ is a $\omega$-models of $\mathbf{\Sigma^1_1}$-$DC$ or $\mathbf{\Delta^1_1}$-$CA$? It's not like I don't understand what the ...
289 views

### Second-order undecidability

Hi, The idea of undecidability in computability theory seems to be along the lines of: There can't be an effective procedure, that decides all instances of input, but a single instance can still be ...
257 views

### Various definitions of recursion from ordinal machines

Background: I'm trying to get an intuitive understanding of Î±-recursion and related concepts in higher recursion theory. Once nice book is Peter Hinman's Recursion-Theoretic Hierarchies, available ...
143 views

### Alternate proof of van de Wiele's theorem in E-recursion

Hello, all I'm currently trying to understand $E$-recursion theory, which is a generalization of classical recursion theory to arbitrary sets. One of the difficulties I'm having with understanding ...
202 views

### Companion of the pointclass of inductive sets

This question is about the notion of a companion for a Spector class, as defined in Moschovakis's book Elementary Induction on Abstract Structures. I am interested in Spector classes on $\mathbb{R}$, ...
13k views

### How large is TREE(3) ?

Friedman, in http://www.math.osu.edu/~friedman.8/pdf/EnormousInt112201.pdf, shows that TREE(3) is much larger than n(4), itself bounded below by $A^{A(187195)}(3)$ (where $A$ is the Ackerman ...
497 views

### Countable admissible ordinals

Jensen claimed that for any finite increasing sequence countable admissible ordinals $\omega= \alpha_0<\alpha_1\cdots <\alpha_n$, there is a real $x$ so that, for each $m\leq n$, $\alpha_m$ is ...
272 views

304 views

### How does the Constructibility Degree of a real compare with its Turing Degree?

Specifically, is it the case that (for $a,b\in\omega^\omega$) $a$ $\leq_T$ $b$ implies $a$ $\leq_c$ $b$? I suspect it might be trivial, but not knowing much Recursion Theory, it's hard to see how it ...
304 views

### An eventually different function adding no Solovay real nor dominating function?

Definitions I believe set theorists have studied all of the following three notions in the context of forcing extensions of a model of ZFC, $M$ (hopefully the terminology is the standard one). A ...
4k views

### Succinctly naming big numbers: ZFC versus Busy-Beaver

Years ago, I wrote an essay called Who Can Name the Bigger Number?, which posed the following challenge: You have fifteen seconds. Using standard math notation, English words, or both, name a single ...
1k views

### Finding the largest integer describable with a string of symbols of predefined length

(This question is motivated by the reading of the article Large numbers and unprovable theorems by Joel Spencer, which can be found at ...
232 views

### When do cofinal chains of universal codings of the integers exist?

Universal codings of integers A (binary) coding of the integers is a prefix-free code of the natural numbers, whose codewords are non-decreasing in size. A coding is universal if it is short enough ...
610 views

### Do all uncountable sets contain elements with infinite Kolmogorov complexity?

Otherwise, if all the elements in a set can be represented by a at most n symbols (finite Kolmogorov complexity), I could count them by creating a n dimensional pairing function. Or atleast, that is ...
1k views

### Axiom of Computable Choice versus Axiom of Choice

What would be the consequence of requiring that any choice function be computable; i.e. using as the foundational basis ZF + ACC? Does it make a difference if we admit definable functions? I guess I ...
2k views

### What is a universal function?

This question stems from Dick Lipton's recent blog post on the Axiom of Choice. I asked there but got no takers. I promise I'm not an inept Googler, but I couldn't find a satisfactory answer. I ...
310 views

### Properties of collections (functions) that make them proper classes (uncomputable)

There are collections too big to be a set, e.g. the collection of all sets (in ZFC), and there are collections that cannot be sets for "pure" logical reasons, e.g. the collection of sets that do not ...
4k views

### Which graphs are Cayley graphs?

Every group presentation determines the corresponding Cayley graph, which has a node for each group element, and arrows labeled with the generators to get from one group element to another. My main ...
Background Assuming ZFC is consistent, then by downward LĂ¶wenheimâ€“Skolem, there is a countable model (M,$\in$) of ZFC. Since the universe M is countable, we may as well think of it as actually being ...