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| 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.
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May 15 |
revised |
Should every group be a monoid, or should no group be a monoid? fixed wrong "simplification" I applied to the information from my reference |
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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. |
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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. |
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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. |
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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. |
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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. |
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May 12 |
answered | A question about standard models |
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May 12 |
revised |
Definition(s) for variable binding in first-order logic noticed the similarities to bounded quantifiers, and modified text/defintions accordingly |
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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. |
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May 12 |
asked | Definition(s) for variable binding in first-order logic |
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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. |
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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. |
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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. |
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May 12 |
answered | Should every group be a monoid, or should no group be a monoid? |
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May 8 |
awarded | Caucus |
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May 5 |
accepted | Why is quantified propositional logic not part of first-order logic? |
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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. |
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May 5 |
revised |
Monadic second order logic without constants, functions and equality deleted 144 characters in body |
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May 5 |
revised |
Monadic second order logic without constants, functions and equality Replaced first-order comprehension axioms by comprehension axioms for first-order formulas |
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May 4 |
accepted | Monadic second order logic without constants, functions and equality |
