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comment 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
5
comment 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).
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accepted Cardinal Arithmetic, foundations and constructive math
Nov
5
comment 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
5
comment 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
4
comment 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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comment Cardinal Arithmetic, foundations and constructive math
@Blass I think you are right in your first comment.
Nov
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comment 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
4
comment 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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comment Cardinal Arithmetic, foundations and constructive math
@Ricky, well GCH implies AC ...
Nov
4
comment 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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comment 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).
Jul
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comment A possible mistake in Kac's “Infinite Dimensional Lie Algebras”
Why are all integral points of X^{\prime} (defined as points where all simple roots give us integers) dense in the metric topology? I think it would be far from dense and actually discrete.