Comment by JDH on Is there a statement independent from PA and does not...
This answer is correct for the question in the body, but not for the question in the title. The difference is that "not implying Con(PA)" is not the same as "not increasing the consistency strength"....
View ArticleComment by JDH on A transfinite epistemic logic puzzle: what numbers did...
Ah, I didn't think to keep a record of it. But I recall that he had had the idea of one character saying something like, "and no matter how many times I would say that, you still wouldn't know," which...
View ArticleComment by JDH on How can there be explicit polynomial equations for which...
I agree with the first part of your remark, which is why I made my comment as a suggestion on how to make a more precise statement. (I don't agree with the second part of your comment; my experience is...
View ArticleComment by JDH on How many sides does a circle have?
@Jared In your second quotation, you use quotation marks to criticize me, but those are not words that I said. Indeed, I wouldn't have, since my friends are not without knowledge—they are quite smart....
View ArticleComment by JDH on Are there complete Boolean algebras with no non-trivial...
Perhaps this is included in the examples you mention, but Donald Monk (link.springer.com/article/10.1007/BF02760942) provides a Boolean algebra whose endomorphisms all come from prime ideals. Since...
View ArticleComment by JDH on How is Pareto-domination semi-transitive?
A simpler counterexample: the equality relation $=$ is transitive, but not semi-transitive according to the definition you have given, if there are at least two points. After all, $a=b=c$ does not...
View ArticleComment by JDH on Quantifier complexity of the definition of continuity of...
Regarding the bounty, thanks very much! Yes, I agree those other cases remain open. I wonder whether a similar model-theoretic method can work...
View ArticleComment by JDH on Can we simplify the classification of infinite...
The main thing to say is that being infinite and having no injection from $\mathbb{N}$ is the same thing as being infinite but Dedekind finite.
View ArticleComment by JDH on Why doesn't "$V$ is a model of $ZF$" imply consistency of ZF?
For set-sized models, we can make such assertions about their theory, since we can define the satisfaction class, that is, for a set model $M$ we can define the satisfaction relation...
View ArticleComment by JDH on Explanation of Skolem's Paradox in Enderton's book
Some readers may find my essay on Skolem's paradox helpful: infinitelymore.xyz/p/skolems-paradox.
View ArticleComment by JDH on difference between maximal element and greatest element
But furthermore, greatest elements are always maximal, so there is no example the other way.
View ArticleAnswer by JDH for Prove that function is totally computable
Fix a computable enumeration of the graph of the function. On input $n$, wait for a pair $(n,y)$ to appear in the graph. When you find such a pair, output $y$. This is a computable procedure that...
View ArticleCan you answer my son's fourth-grade homework question: Which numbers are...
My son Horatio (nine years old, fourth grade) came home with some fun math homework exercises today. One of his problems was the following little question: I am thinking of a number...It is prime.The...
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What is the smallest digraph whose reflexive, symmetric, transitive closures...
For any given directed graph, we may consider the various closures of it with respect to reflexivity, symmetry, and transitivity, in any combination, like this:For the particular graph shown above,...
View ArticleAnswer by JDH for Provable soundness of finite fragments of ZFC
Consider any finite list $\Phi$ of axioms of ZFC and any other sentence $\phi$. By the Lévy-Montague reflection theorem, there is some rank-initial segment $V_\theta$ of the universe for which all the...
View ArticleAnswer by JDH for When is the closure of an open ball equal to the closed ball?
Here is a characterization that is straight from the definitions, but which it seems may be useful when verifying that a particular space has the property.For any metric space $(X,d)$, the following...
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Is this math game always winnable?
Kent Haines describes the game of Integer Solitaire, which I find to be excellent for young kids learning arithmetic. I'm sure they will be motivated by this game to get a lot of practice.Kent asks a...
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Answer by JDH for How to tell $i$ from $-i$?
This question relates directly to issues that are often discussed in the philosophy of mathematics.According to one of the standard accounts of structuralism, what mathematical objects are at bottom...
View ArticleAnswer by JDH for generate all possible theories compatible with axioms
Your idea that there is a computable tree of possible extensions of a given theory is completely right, in the case of arithmetic.If we start with any consistent computably axiomatizable theory $T$ of...
View ArticleAnswer by JDH for I am looking for class of math problems which are provable...
Noah gave a great answer, which is probably worth pursuing.But in terms of the question that you actually asked, one can give a complete, comprehensive answer.Namely, the set of statements $\varphi$...
View ArticleAnswer by JDH for Can we obtain a larger model of ZF or ZFC in this way?
There are several ways that one might answer this question.First, the method of forcing, which has been used to prove so many of the set-theoretic independence results, such as the independence of the...
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Can you provide a symmetric presentation of this partition lattice?
I have a question about the partition lattice on a set with four elements, shown here:Equivalently, this is the lattice of equivalence relations on that set.An observant person will notice that the...
View ArticleAnswer by JDH for The tree property for non-weakly compact $\kappa$
What you are looking for is the concept of Aronszajn tree. You can read about constructions of Aronszajn trees in any graduate level set theory text, and meanwhile, the Wikipedia page lists a summary...
View ArticleAnswer by JDH for Is it possible to formalize the idea that large cardinal...
This topic is both mathematically and philosophically rich. What precisely do we mean when we say that a set-theoretic principle is "restrictive"? Can we give a formal account of this notion that...
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What is the next number on the constructibility sequence? And what is the...
Let us systematically generate all constructible points in the plane. We begin with just two points, which specify the unit distance. With the straightedge, we may construct the line joining them. And...
View ArticleAnswer by JDH for Boolean algebras without atoms
It isn't true generally that all atomless Boolean algebras of the same cardinality are isomorphic, so we don't expect $\kappa$-categoricity, and the diversity of such Boolean algebras give rise to all...
View ArticleAnswer by JDH for Forcing Classes Into Sets
For any set $X$, we may consider the partial order $\mathbb{P}$ consisting of all finite partial functions from $\omega$ to $X$, ordered by extension. If $G\subset\mathbb{P}$ is $V$-generic for this...
View ArticleAnswer by JDH for Example of a dense open subset that is not of the form $U...
Counterexample in every dimension $n\geq 2$. Let $D$ be the complement of a point $p$ in any dimension $\newcommand\R{\mathbb{R}}\R^n$ with $n\geq 2$. This is a dense open set, but I claim it cannot be...
View ArticleWhat is the order-type of the set of natural numbers, when written in...
We are all familiar with the standard nomenclature for the smallishnatural numbers, such asone, two, three, ..., one hundred, one hundred one, ..., fifteen thousand two hundred forty-nine.I have in...
View ArticleAnswer by JDH for What is the order-type of the set of natural numbers, when...
Let us consider the digit-pronunciation naming system, by whichone simply pronounces the digits of a number in order, so that$7216$ is pronounced "seven two one six" and so on for any number.Thus, we...
View ArticleAnswer by JDH for Is there any way to find the number of real roots of a...
My understanding of the question is that an algorithm is sought that will use the input polynomial as a black-box for computational function evaluation, but without knowing the nature of the polynomial...
View ArticleAnswer by JDH for Proof that axiom of replacement won't generate a set going...
The replacement axiom is equivalent (over the other axioms of ZF) to the assertion that in every instance, the ranks of witnesses are bounded in the ordinals. Specifically, replacement is equivalent to...
View ArticleAnswer by JDH for Characterizing worldly cardinals as the supremum (by n) of...
The answer is negative. I claim that the supremum can see the$\omega$-cofinal sequence $\kappa_n$, where $\kappa_n$ is the least$\Sigma_n$-extendible cardinal.The main point is that if $\alpha$ is...
View ArticleMaking numbers from 2, 0, 1, 7. Also: are the the iterative factorials and...
My daughter (10 years old) was given the task by her math teacherto form as many numbers as she could using the numbers: 2, 0, 1, 7,exactly once each, and the operations of addition,...
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Answer by JDH for Is there an absolute notion of the infinite?
The finite/infinite distinction is not absolute. Indeed, different models of set theory can think vastly different things about the sizes of a set whose elements they have in common.For example, there...
View ArticleAnswer by JDH for Why is every limit cardinal a supremum of regular succesor...
All infinite successor cardinals are regular.To see that $\delta^+$ is regular, observe first that the union of $\delta$ many sets of size $\delta$ has size $\delta$. This is a generalization of the...
View ArticleAnswer by JDH for Doing an modal logic exercise. The tableau for both the...
The deduction is not valid.Since $\alpha$ and $\beta$ are not rigid descriptors, it could be that $\alpha=\beta$ in a world, but not in the accessible world where $P\alpha$ holds. So there is no reason...
View ArticleAnswer by JDH for Does satisfaction at all arithmetical sets of a...
Here is a different solution.The answer is no. Consider the statement $\varphi(T)=$"$T$ is not a truth predicate". That is, $T$ is not a set of sentences obeying the Tarski recursion, meaning that it...
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