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| bio | website | jakitoimgeisterhaus.blogspot.… |
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| location | Munich, Germany | |
| age | 36 | |
| visits | member for | 1 year, 11 months |
| seen | May 16 at 21:45 | |
| stats | profile views | 353 |
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.
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May 4 |
accepted | Paradox: Any set theory without universe set is not a model of itself |
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May 1 |
answered | Which languages are preferable to study for a mathematician? |
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Apr 26 |
awarded | Nice Question |
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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. |
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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! |
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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. |
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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. |
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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. |
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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??? |
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Apr 21 |
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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. |
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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. |
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Apr 18 |
answered | Monadic second order logic without constants, functions and equality |
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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. |
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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). |
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Apr 7 |
revised |
Monadic second order logic without constants, functions and equality Tried to fix comprehension axioms by restricting to first order. Also tried to fix the formula expression compatibility of E with the language. |
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Apr 7 |
comment |
Monadic second order logic without constants, functions and equality I wrote "Here I still assume equality to be part of the language", so your statement "In fact, there is no formula $\varphi(x,y)$ such that in any Henkin model..." seems wrong to me. Why not just take $\varphi(x,y):=(x=y)$? So I ask myself whether your statement "No. It is evident..." really refers to the logical system intended by the question (i.e. a system where equality is part of the language). It certainly isn't evident to me, but that may just be a consequence of the fact that I have problems to get the comprehension axioms right. |
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Apr 6 |
revised |
Monadic second order logic without constants, functions and equality edited body |
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Apr 6 |
asked | Monadic second order logic without constants, functions and equality |
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Apr 6 |
accepted | Are there simple models of Euclid's postulates that violate Pasch's theorem or Pasch's axiom? |
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Apr 5 |
comment |
Are there simple models of Euclid's postulates that violate Pasch's theorem or Pasch's axiom? @StevenStadnicki OK, I see. I guess that's the reason why "Pasch's theorem" cannot be derived, but is less important than "Pasch's axiom". |
