Carl Mummert

Marshall University

science.marshall.edu/mummertc

Age: 36

I work in mathematical logic. My main areas of interest are arithmetic, reverse mathematics, computability, and proof theory.
1d
comment complete, finitely axiomatizable, theory with 3 countable models
Thanks for the comments, which make the question much more accessible to someone who hasn't thought about such things before.
2d
comment Is it meaningless to say $M\prec N$ for two proper class models?
But, in the metatheory, we can quantify over formulas. After all, the metatheory is exactly what establishes the set of formulas. And, for example, when we define $A \prec B$ for set models $A,B$ in the exact same metatheory, we don't have to make that a schema. So quantifying over formulas cannot be the real issue. The real issue seems to be an assumption that ZFC is not only the object theory we are studying, it is also the metatheory. And, of course, ZFC can't define satisfaction relations for class models, even though as a metatheory it can quantify over formulas.
2d
comment Is it meaningless to say $M\prec N$ for two proper class models?
There is some issue here, which I often find elusive to state precisely. It is certainly true that we can't expect to quantify over formulas internally, but then again $M \prec N$ is not an internal statement, it is a statement in the same metatheory that defines the set of formulas used to define $\prec$.
2d
comment Is it meaningless to say $M\prec N$ for two proper class models?
Of course, we can define a satisfaction predicate for formulas of ZFC relative to a proper class; we can do it in MK for example, or just in normal informal mathematics. We can't do it in ZFC, though. (Regarding the second paragraph.)
2d
comment Is it meaningless to say $M\prec N$ for two proper class models?
Kunen has a particular viewpoint in that text, which is that his metatheory is ZFC (or perhaps his object theory; another aspect of the book is that this is kept somewhat vague). Of course, it makes perfect sense to say $M \prec N$ when $M,N$ are proper classes; it means that $M$ is an elementary (class) submodel of $N$. So we all know what it means. But it cannot be formalized in ZFC.
2d
comment What is the pure essence of a definition of semantics?
What is your question, precisely? This post seems to be somewhat argumentative or polemical (the last two paragraphs in particular). On math.SE we encourage questions with an objective answer, rather than posts that simply present an argument.
2d
awarded Autobiographer
2d
comment How much set theory is necessary for serious logic?
I don't believe this answers the question that was asked.
Aug
27
comment Constructive proof of pigeonhole principle
I added some thoughts on this.
Aug
27
revised Constructive proof of pigeonhole principle
added 841 characters in body
Aug
27
comment Constructive proof of pigeonhole principle
Thanks for accepting - I assume that means you were able to formalize the proof? I have been thinking for a while about trying to learn Coq. Would you be willing to share the formalized proof with me? If so, could you email me at the address listed in my profile? I would deeply appreciate it. @max taldykin
Aug
27
comment Constructive proof of pigeonhole principle
(1) should be $0 \not = S(x)$
Aug
27
comment Constructive proof of pigeonhole principle
The facts about decidability follow from induction and the axioms (1) $x \not = S(x)$; (2) $x = 0 \lor (\exists y)[x = S(y)]$; (3) $x = y \leftrightarrow S(x) = S(y)$.
Aug
27
revised Constructive proof of pigeonhole principle
added 8 characters in body
Aug
27
answered Constructive proof of pigeonhole principle
Aug
27
comment Constructive proof of pigeonhole principle
Could you state exactly which pigeonhole principle you mean? There are several principles that could go by that name.
Aug
25
answered Showing unique prime factorization in first-order logic?
Aug
25
revised About the proof of the Heine-Borel Theorem
edited title
Aug
25
comment Find the limit of $ f(x)=\begin{cases}x^2&\text{$x\in\mathbb{Q}$}\\0&\text{$x\notin\mathbb{Q}$}\\\end{cases}$ for $x\rightarrow0$
@Danny Lim: the approach that one takes to these problems is somewhat different in Calculus than in more advanced classes. For this question, you want to use the "sandwich theorem" or "squeeze theorem" from the section of your book on limits.
Aug
25
comment Find the limit of $ f(x)=\begin{cases}x^2&\text{$x\in\mathbb{Q}$}\\0&\text{$x\notin\mathbb{Q}$}\\\end{cases}$ for $x\rightarrow0$
@Danny Lim: the fact that it's for a Calculus 1 class helps a lot, actually, in terms of giving an idea of the type of method that is likely desired. I voted to re-open the question just now.
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