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Cardinal Arithmetic, foundations and constructive math
Also above I said that my idea of constructivism is that it should be practically computable (I mentioned MP in addition to the law of excluded middle). I have no idea if this is even a topic of discussion in constructive/intuitionist circles but if you look at it from a programming point of view it is the most natural definition. |
Nov
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Cardinal Arithmetic, foundations and constructive math
Alright that example clarified things a bit. But I am fine with unprovabally finite subsets of a finite set. My initial thought after reading your statement was that, there is a freakish model where a finite set had a non-finite subset (that would is stupid but from my experience you can't just call something obviously stupid and move one hence my confusion). |
Nov
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accepted | Cardinal Arithmetic, foundations and constructive math |
Nov
5 |
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Cardinal Arithmetic, foundations and constructive math
Law of excluded middle is non-constructive. I have doubts about MP as well, if $A$ and $A \to B$ but it took a trillion years to get from $A$ to $B$ would that be constructive? I have realized that I use constructive in a much stronger form than anyone else here. For example, I have a class of graphs with $\chi = 5$ but the proof is classical, then I get a 5-coloring algorithm of order $O(|G|^{|G|})$. To my mind the algorithm is progress but still not constructive. In short being constructive without being computable in some sensible sense of the word does not make sense to me ... |
Nov
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Cardinal Arithmetic, foundations and constructive math
@Andrej: I am very sorry for the lack of clarity, I have not thought about these things and strangely enough I think I use constructive in a stronger form than anyone else here which just worsens the matters. The subset/Random Variable is not a real subset, so if the non-finite subsets of a finite set are only of that type then I am fine with the original statement. The next question is whether what I call real subset can be formalized in any sense or not ... |
Nov
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Cardinal Arithmetic, foundations and constructive math
@all: I call an axiom AX non-constructive if it has non-constructive consequences in everyday classical mathematics we use. I thought that was the normal understanding of the term. |
Nov
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Cardinal Arithmetic, foundations and constructive math
@Blass I think you are right in your first comment. |
Nov
4 |
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Cardinal Arithmetic, foundations and constructive math
@Andrej, @Guillaume: In those cases I would take issue with the way you define the subsets which depend on an odd black box. What you have is not a fixed subset, it can be different things in the future, it is essentially a random variable. |
Nov
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Cardinal Arithmetic, foundations and constructive math
@Noah, my understanding is that the model for V=L done by Godel is "constructive" but the theory itself (which is always much bigger than any single model ...) is not. |
Nov
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Cardinal Arithmetic, foundations and constructive math
@Ricky, well GCH implies AC ... |
Nov
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Cardinal Arithmetic, foundations and constructive math
Alright that was helpful but I still have some issues: (1) When you say there are models for your first list of statements, what is the base system of constructive math you are working with? (2) Throughout the whole discussion I am assuming a degree of reasonableness, an axiom stating the fact that a finite set can have non-finite subsets or that we have an embedding $\mathbf{R} \to \mathbf{N}$ are not resonable (3) I might be mistaken but I thought that intuitionists did not like Church's thesis ... |
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asked | Cardinal Arithmetic, foundations and constructive math |
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A possible mistake in Kac's “Infinite Dimensional Lie Algebras”
Does Kac answer email? Actually I found what seems to be a full proof in "Introduction to Kac-Moody Algebra" by Zhexian Wan (Proposition 5.6, page 98). |