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 $\varphi$...
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