Do we know the consistency strength of the Singular Cardinal Hypothesis failing on an uncountable cofinality?

Suppose that $ \kappa$ is a strong limit cardinal. The singular cardinal hypothesis states $ 2^\kappa=\kappa^+$ . We know that the failure of SCH requires large cardinals, and in fact is equiconsistent with a measurable cardinal $ \kappa$ satisfying $ o(\kappa)=\kappa^{++}$ .

But this failure is at $ \aleph_\omega$ . Suppose we wanted more.

Suppose that we wanted the failure to happen on a couple isolated points. Well, it’s not hard to redo the standard constructions and get just that. But what happens when we have limit points?

Even more, by Silver’s theorem if SCH fails at $ \kappa>\operatorname{cf}(\kappa)>\omega$ , then there is a stationary subset of $ \kappa$ where SCH failed.

What would be the consistency strength when $ \kappa$ is a singular limit of singular cardinals, and SCH fails cofinally below $ \kappa$ ? What if we require $ \kappa$ to be of uncountable cofinality?

As a side question, what if $ \kappa$ , with uncountable cofinality, does satisfy SCH, but an unbounded subset (which has to be non-stationary, of course) of it does not?

Cardinal characteristics of the ideal of $\sigma$-continuity of the Pawlikowski function

A function $ f:X\to Y$ between topological spaces is called $ \sigma$ -continuous if there exists a countable cover $ \mathcal C$ of $ X$ such that for every $ C\in\mathcal C$ the restriction $ f{\restriction}C$ is continuous.

A typical example of a function (of the first Baire class) which is not $ \sigma$ -continuous is the Pawlikowski function $ P:(\omega+1)^\omega\to\omega^\omega$ (which is defined as the countable power $ P=f^\omega$ of a bijection $ f:\omega+1\to\omega$ ).

Let $ \mathcal I_P$ be the $ \sigma$ -ideal of subsets $ X$ of the compact metrizable space $ (\omega+1)^\omega$ such that $ P{\restriction}X$ is $ \sigma$ -continuous.

I am interesting in evaluating the standard cardinal characteristics $ \mathrm{add}(\mathcal I_P)$ , $ \mathrm{cov}(\mathcal I_P)$ , $ \mathrm{non}(\mathcal I_P)$ , $ \mathrm{cof}(\mathcal I_P)$ of the $ \sigma$ -ideal $ \mathcal I_P$ .

It seems that among these four cardinal characteristics only the covering number $ \mathrm{cov}(\mathcal I_P)$ was studied in the literature. In particular, Cichon, Morayne, Pawlikowski and Solecki proved that $ \mathrm{cov}(\mathcal I_P)\ge\mathrm{cov}(\mathcal M)$ where $ \mathcal M$ is the $ \sigma$ -ideal of meager subset in $ \mathbb R$ . Steprans gave a combinatorial description of the ideal $ \mathcal I_P$ and proved the consistency of the strict inequality $ \mathrm{cov}(\mathcal I_P)>\mathrm{cov}(\mathcal M)$ . Steprans also observed that for every $ A\in\mathcal I_P$ the image $ P(A)$ is a meager subset of $ \omega^\omega$ , which implies that $ \mathrm{cov}(\mathcal I_P)\ge\mathrm{cov}(\mathcal M)$ and $ \mathrm{non}(\mathcal I_P)\le\mathrm{non}(\mathcal M)$ . On the other hand, it can be shown that $ \mathrm{non}(\mathcal I_P)\ge\mathfrak p$ .

It is well-known that the strict inequality $ \mathfrak p<\mathrm{non}(\mathcal M)$ is consistent. In particular, according to Table 4 in the survey paper of Blass, this strict inequality holds in the random, Hechler, Laver, and Mathias forcing models.

Problem 1. Which of the inequalities $ \mathfrak p<\mathrm{non}(\mathcal I_P)$ and $ \mathrm{non}(\mathcal I_P)<\mathrm{non}(\mathcal M)$ is consistent?

Problem 2. What is the value of $ \mathrm{non}(\mathcal I_P)$ (and other cardinal characteristics of the ideal $ \mathcal I_P)$ in the random, Hechler, Laver, and Mathias forcing models?

Combinatorial cardinal characteristics of $\omega$

In a universe where the continuum hypothesis ($ CH$ ) fails or is undecidable we can ask about combinatorial cardinal characteristics of the continuum, i.e. cardinals larger than $ \omega$ but smaller than $ \mathfrak{c}$ for which theorems that are true at $ \omega$ but false at $ \mathfrak{c}$ still hold, but in a universe where $ CH$ is true we have $ (\omega,\mathfrak{c})=\emptyset$ so this study becomes vacuous.

Does a similar phenomenon occur at the countable level in a universe without choice? Specifically, are there properties which are true for finite sets but false for $ \omega$ which are still true for the cardinal of an amorphous set, like divisibility as suggested here by Fran├žois G. Dorais?

In a universe without choice we have the existence of amorphous sets and we can ask about their ‘amorphous cardinals’ which are incomparable with $ \omega$ (thank you Asaf for the correction) and may satisfy nice theorems, but in a universe with choice there are no infinite sets whose cardinality is incomparable with $ \omega$ so this study becomes vacuous in similar fashion to the uncountable case.

A possible candidate for characteristics smaller than $ \omega$ could come from theorems in finite group theory that become false for countable groups, since it is possible to have a group structure on an unbounded amorphous cardinal as constructed by Asaf Karagila here.

There is an article behind a paywall published in 2010 that appears to touch on these matters but I can’t access it; if anyone is familiar with its contents and willing to give a brief exposition it would be greatly appreciated.

Free skew fileds over sets of different cardinal

Let $ K$ be a field and let $ X$ be a set. Denote by $ \mathcal D_K(X)$ the free skew $ K$ -field on $ X$ .

Assume that $ |X|\ne |Y|$ . Is it true that $ \mathcal D_K(X)$ and $ \mathcal D_K(Y)$ are not isomorphic?

There are several constructions of $ \mathcal D_K(X)$ . The standard one is the following. Let $ F_X$ be a free group on $ X$ . Fix a bi-invariant total order $ \le$ on $ F_X$ and let $ K_{\le}((F_X))$ be the Malcev-Neumann ring of formal series (it consists of formal series having well-ordered support). One proves that $ K_{\le}((F_X))$ is a skew field. Then $ \mathcal D_K(X)$ is isomorphic to the division closure of the group algebra $ K[F_X]$ in $ K_{\le}((F_X))$ .

Python implementation of approximating the chance a particle is at a location after n steps in the cardinal directions

Recently, I became very interested in a probability practice problem in my textbook for my class. I decided to implement it in code and I think I got most of it implemented. Right now, I’m hoping to see if there is any possible way that I can improve upon it.

The question reads as follows:

A particle starts at (0, 0) and moves in one unit independent steps with equal probabilities of 1/4 in each of the four directions: north, south, east, and west. Let S equal the east-west position and T the north-south position after n steps. 

The code (and more information) can be found here in the GitHub repository.

The code is right here: simulation.py

""" A particle starts at (0, 0) and moves in one unit independent steps with equal probabilities of 1/4 in each of the four directions: north, south, east, and west. Let S equal the east-west position and T the north-south position after n steps. """  from random import choice  import numpy as np  from options import Options  # Directions (K -> V is initial of direction -> (dx, dy) directions = {     'S': (0, -1),     'N': (0, 1),     'E': (1, 0),     'W': (-1, 0)     }   def get_direction():     """     Get a random direction. Each direction has a 25% chance of occurring.      :returns: the chosen directions changes in x and y     """     dirs = "NSEW"     return directions[choice(dirs)]   def change_position(curr_pos, change_in_pos):     """     Updates the current location based on the change in position.      :returns: the update position (x, y)     """     return curr_pos[0] + change_in_pos[0], curr_pos[1] + change_in_pos[1]   def increment_counter(counter, end_pos):     """     Increments the provided counter at the given location.      :param counter: an numpy ndarray with the number of all ending locations in the simulation.     :param end_pos: the ending position of the last round of the simulation.     :returns: the updated counter.     """     counter[end_pos[1], end_pos[0]] += 1     return counter   def get_chance_of_positions(n):     """     Gets the approximated chance the particle ends at a given location.      Starting location is in the center of the output.      :param n: The number of steps the simulation is to take.     :returns: the number of iterations and an ndarray with the approximated chance the particle would be at each location.     """     # The starting position starts at n, n so that we can pass in the location     # into the counter without worrying about negative numbers.     starting_pos = (n, n)      options = Options.get_options()      total_num_of_sims = options.num_of_rounds      counter = np.zeros(shape=(2 * n + 1, 2 * n + 1))      for j in range(total_num_of_sims):         curr_pos = starting_pos         for i in range(n):             change_in_pos = get_direction()             curr_pos = change_position(curr_pos, change_in_pos)         counter = increment_counter(counter, curr_pos)      chances = np.round(counter / total_num_of_sims, decimals=n + 1)     return total_num_of_sims, chances 

plot.py

import matplotlib.pyplot as plt from mpl_toolkits.axes_grid1 import make_axes_locatable  from simulation import get_chance_of_positions as get_chance from options import Options  def plot(n):     """     Saves the plots of the chance of positions from simulation.py     :param n: the number of steps the simulation will take.     """     num_of_iterations, counter = get_chance(n)      fig = plt.figure()      ax = fig.add_subplot(111)     ax.set_title("Position Color Map (n = {})".format(n))     ax.set_xlabel("Number of iterations: {}".format(int(num_of_iterations)))      plt.imshow(counter, cmap=plt.get_cmap('Blues_r'))      ax.set_aspect('equal')      divider = make_axes_locatable(ax)     cax = divider.append_axes("right", size="5%", pad=0.05)      plt.colorbar(orientation='vertical', cax=cax)     fig.savefig('plots/plot-{:03}.png'.format(n), dpi=fig.dpi)   def main():     """     The main function for this file. Makes max_n plots of the simulation.     """     options = Options.get_options()      for n in range(1, options.num_of_sims + 1):         plot(n)   if __name__ == "__main__":     main() 

options.py

import argparse  class Options:       options = None       @staticmethod     def generate_options():         arg_parser = argparse.ArgumentParser()         arg_parser.add_argument('-N', '--num-of-rounds',                         type = int,                         required = False,                         default = 10**5,                         help = "The number of rounds to run in each simulation. Should be a big number. Default is 1E5")         arg_parser.add_argument('-n', '--num-of-sims',                         type = int,                         required = False,                         default = 10,                         help = "The number of simulations (and plots) to run. Default is 10.")         return arg_parser.parse_args()       @staticmethod     def get_options():         if Options.options is None:             Options.options = Options.generate_options()         return Options.options 

I would love for some recommendations on PEP 8, design, and “Pythonic” code (code that Python expects and will thus be better optimized, such as list comprehension and numpy optimizations, wherever they may be.)

What is the cardinal of the set of Uniform Continous functions with domain in $\mathbb{R}$?

I learned in my topology class that $ \mathbb{R} \cong \mathfrak{C}(\mathbb{R})$ The latter being the set of continous functions with domain in $ \mathbb{R}$ .

Since if a function is uniformly continous then it is continous, the set of all uniform function must have cardinal $ \leq \mathfrak{c}$ .

But continous functions need not be uniformly continous; does this show that there is no bijection from continous functions to uniformly continous functions?

Im kind of lost with all this cardinality stuff, its not intuitive at all.

Cardinal exponentiation without generalized continuum hypothesis

First I have to confess that I don’t know about set theory language.

Let $ A$ and $ B$ be infinite cardinals with $ A>B$ .

My question is: $ A^B=A$ ? (without assuming generalized continuum hypothesis)

Remark: assuming generalized continuum hypothesis (GCH briefly), this can be proved by the following (at least for unlimit cardinal).

Sps $ A$ is a unlimit cardinal. Then $ A=2^C$ for some $ C\ge B$ by GCH. Therefore $ A^B = (2^C)^B=2^{CB}=2^C=A$ .

Unfortunately, I don’t know how to prove for limit cardinal case. Please somebody help me!

Does measurability of cardinal $\kappa$ imply measurability of $2^\kappa$?

A cardinal $ \kappa$ is real-valued measurable if there is a $ \kappa$ -additive probability measure on $ 2^\kappa$ which vanishes on singletons. The existence of measurable $ \kappa$ is independent of ZFC.

Question: if $ \kappa$ is assumed to be real-valued measurable, does it necessarily follow that $ 2^\kappa$ is real-valued measurable?

Regarding the definition of a Reinhardt cardinal being inconsistent with choice

Consider the following comment Noah Schweber made to Embiggening regarding his (Embiggening’s) mathstackexchange question, “Could Reinhardt cardinals exist even if all cardinals below it were well-orderable”:

When we say that a Reinhardt cardinal is inconsistent with choice, we mean that if $ M$ is a model of $ ZF$ , $ \kappa$ $ \in$ $ M$ , and $ M$ $ \vDash$ $ \kappa$ is a Reinhardt cardinal,” then there is some set in $ M$ which is not well-orderable in $ M$ . But this set won’t itself be $ \kappa$ .”

It would seem reasonable to assume that this definition of a Reinhardt cardinal being inconsistent with choice would also hold for $ NGB$ .

My question is simply this: if such a model of $ NGB$ were to exist, in what way would Choice have to fail in order for $ \kappa$ in “$ \kappa$ is a Reinhardt cardinal” to be well-orderable? What sort of models of $ NGB$ would satisfy these constraints?