Thomas Klimpel

1,794 reputation

1523

bio	website	jakitoimgeisterhaus.blogspot.…
	location	Munich, Germany
	age	36
visits	member for	1 year, 11 months
	seen	May 16 at 21:45
stats	profile views	354

My past research interests included differential algebraic equations, nonlinear analysis and relations between symmetries and structural properties.

I recently investigated hierarchical structures, starting from group cohomology, continuing with semi-group theory and ending with lattices and universal algebra.

	1,794 reputation	bio	website	jakitoimgeisterhaus.blogspot.…	visits	member for	1 year, 11 months
1523 badges		location	Munich, Germany		seen	May 16 at 21:45

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May 14	comment	Is second order logic even a logic? If I find time, I will try to write an answer arguing in favor of "trying to turn" monadic second-order logic into a "true" logic.
May 14	comment	Is second order logic even a logic? @user18921 With respect to references for higher-order logic, I liked "The Seven Virtues of Simple Type Theory". It explains higher-order logic, its standard and Henkin semantics, its deduction rules and many more related questions.
May 14	comment	Is second order logic even a logic? "The Road to Modern Logic-An Interpretation" has discussion on why certain topics related to "foundations" are considered part of logic while others are not. It argues that this has mostly historical reasons and cannot be justified a priory. With regards to the other part of the question, I think one should start by considering what one wants to do with monadic second-order logic, before deciding about the status of higher-order logic.
May 13	comment	A question about standard models @AsafKaragila It's good to know that there exists a countable standard model (granted standard models exists to begin with), and it's even more comforting to know that it can be as nice as the constructible universe (if I understood correctly). I wasn't aware of that.
May 12	comment	A question about standard models @AsafKaragila Perhaps the answer is not a perfect fit for the question, but it sort of addresses question (i) in my opinion. I intentionally wrote "A standard model is often...", because I also had the impression that the terminology "standard model" is used with slightly different meanings in different fields of mathematics. However, the notion described in my answer has relevance for set theory, both because of Gödel's constructible universe and because of Zermelo's refusal to replace his second order axiom of separation by a first order axiom scheme.
May 12	comment	Should every group be a monoid, or should no group be a monoid? @user18921 I own the fifth edition of the original German text. It comes with nearly complete solutions for all the exercises and very few errors. I really like its style and the way it goes deep instead of broad. But if you are just looking for further information with respect to my answer, I wouldn't recommend it. I had studied monoids, semigroups and universal algebra before even realizing that mathematical logic might be interesting. Ebbinghaus is a text that can (and should) be read from start to end, but even then you will make use of the "Symbol Index" more than once.
May 12	comment	Should every group be a monoid, or should no group be a monoid? @user18921 Such a structure is referred to as a universal Horn class by math.chapman.edu/~jipsen/structures/doku.php and similar references. The answer from Pece might also contain some truth. My own reference was "Mathematical Logic" from Ebbinghaus et al., but I simplified their statement a bit. They actually defined homomorphisms for the case that the term-interpretation is a model of the axioms, and then proved that this is the case if the axioms are universal Horn expressions.
May 12	comment	Should every group be a monoid, or should no group be a monoid? @user18921 Yes, $\forall x \exists y(x\geq y \land y\leq x)$ is a Horn expression. However, it is not a universal Horn expression.
May 12	comment	Should every group be a monoid, or should no group be a monoid? @user18921 A clause is "a disjunction of literals". A Horn expression is a conjunction of Horn clauses where some of the variables may be quantified (both universal or existential). A universal Horn expression is a Horn expression where no existential quantification occurs.
May 5	comment	Why is quantified propositional logic not part of first-order logic? I had to read "The Seven Virtues of Simple Type Theory" and then do some serious thinking before I could figure out the "The same holds for λ terms to define functions. There is no reason that they could not be included in first-order theories, and in fact they sometimes are, ..." part. It actually seems to be a nice system/language. I decided that the first-order restriction would mean that λ can only bind to normal variables or propositional-variables. Then (it seems) I only have to add an "if X then a else b" construct to get a nice expressive language.
Apr 26	comment	The right way to motivate lattice theory in a combinatorics class A nice reference list can be found at the end of this blog post. Vijay Garg's "Lattice Theory with Applications" is good for practical application of lattice theory. For your specific question, "Combinatorics: The Rota Way" will be more helpful. Möbius functions just require posets, but if you restrict yourself to (semi)lattices, things like the "Lindström–Wilf determinantal formula" become available. But I don't see the point to advertise Möbius functions on lattices instead of Möbius functions on posets.
Apr 22	comment	Infinite sets don't exist!? [continued] Regarding "Do modern texts on set theory bend over backwards to say precisely what is and what is not an infinite set?", don't confuse this with the axiom of infinity from ZFC. Of course, there are some sets which are explicitly blessed as finite, and other sets which are explicitly blessed as infinite. However, the "modern texts on set theory" I consulted didn't bother to discuss the problem of how to define when an arbitrary set is finite or infinite. There are texts which discuss this problem, and it turns out that different sensible definitions lead to different answers!
Apr 22	comment	Infinite sets don't exist!? @Arthur: I have now spend considerable time searching the internet to determine whether Wildberger is (considered to be) a crackpot or not. His essay may be provocative and controversial, but the general opinion seems to be that it is unwarranted to compare him to infamous people like Wolfgang Mückenheim or John Gabriel (should I know these?). There is also general agreement that his opinions and conclusions are "wrong", but not in a "dishonest" or "stupid" way as suggested here.
Apr 21	comment	Infinite sets don't exist!? Do you really think MSE is a place for "debate, arguments and extended discussion"? The "majority opinion" is one thing, but the stack-exchange framework explicitly tries to minimize such things, probably for good reasons. Anyway, I was just explaining why I downvoted.
Apr 21	comment	Infinite sets don't exist!? Wildberger is a professional mathematician, so he knows that he has to accept "some version" of infinity. I'm not sure he intents to use a straw-man, when he writes: "The ‘Axioms’ are first of all unintelligible unless you are already a trained mathematician." And he is criticising set theory mainly in its role as foundation of mathematics. You write "he is using an layperson's interpretation of the axiom", but Wildberger has a point that even the undergraduate student is in this layperson's position. Carl Mummert answer that it's a "tongue-in-cheek essay" is good, I really laughed out loud.
Apr 21	comment	Infinite sets don't exist!? -1 On meta, you wrote things like "poor fit for this site ... to argue specifically with a well-known crankish essay, a task that is generally considered fruitless" and "Furthermore, giving serious responses to it is arguably counter-productive to begin with, by implying the original essay actually merits a direct response". I think your answer contains "debate, arguments and extended discussion", all these things why you correctly explained that such a question should be closed. Why on earth then do you write such an answer???
Apr 21	comment	Infinite sets don't exist!? [continued] However, the feeling that using "potential infinite" instead of "actual infinite" would create more problems and paradoxes than it's worth is probably correct. But while Hilbert's vision that "actual infinite" could be justified by finitistic methods was disproved by Gödel in a certain sense, the same question for the "potential infinite" never received the same attention. However, the "actual infinite" is also problematic, not because it could be self-contradictory, but because some non-sense theorems will become true (like doubling the sphere), if the axioms are mighty.
Apr 21	comment	Infinite sets don't exist!? @ArthurFischer Wildberger probably takes issue that infinite mathematical sets don't correspond "sufficiently well" to non-finite sets in the physical world. I like to think of a bottle with fluid as a nice model for a "compact" (i.e. quasi finite) physical set, and of a toy ballon with gas as a nice model for a "non-compact" physical set. The problem is, neither finite sets nor infinite sets are a good model for the toy ballon. If we keep filling gas into it, it will explode, but we can't say exactly when. A "potential infinite" is a better model than an "actual infinite" for this fact.
Apr 7	comment	Monadic second order logic without constants, functions and equality Sorry, I have edited the question and slightly modified its meaning, because I initially got the comprehension axioms badly wrong. This has probably invalidated large parts of your answer, which is why such edits are frowned upon. However, I have read quite a bit of the sep article on second order logic now, and had the impression that it was really necessary to "fix" the bad mistakes in the question.
Apr 7	comment	Monadic second order logic without constants, functions and equality Sorry, I have edited the question and slightly modified its meaning, because I got both the comprehension axioms and the compatibility scheme wrong. To simplify things, I restricted the comprehension axioms to first order (hoping that I get at least the first order comprehension axioms right).