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Oct
5 |
comment |
A question about simple closed curves in 3-dimensional Euclidean space
I assume that C is the image of a continuous mapping whose domain is the closed unit interval. Thanks to both of you for very nice solutions of this problem. |
Oct
5 |
accepted | A question about simple closed curves in 3-dimensional Euclidean space |
Oct
4 |
asked | A question about simple closed curves in 3-dimensional Euclidean space |
Sep
22 |
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Can non-computable real numbers be defined without making use of any notions from computability theory
My statement of the modified Lebesgue covering problem is not quite correct. It should read "What is the minimum (or greatest lower bound) of the set of diameters of plane convex sets that cover every plane set having diameter 1" |
Sep
22 |
comment |
Can non-computable real numbers be defined without making use of any notions from computability theory
Thanks for this reference. I will try to formulate some of these examples as geometric problems with specific real number solutions. |
Sep
21 |
asked | Can non-computable real numbers be defined without making use of any notions from computability theory |
Sep
7 |
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Is every computable real primitively recursively computable?
This is a very nice approach which looks as if it would work, although I do not know if I would be able to fill out all the details of a rigorous proof. It would seem, then, that we could define an ascending chain of subsets of the set of computable functions-starting with the set of primitive recursive functions- and classify a computable real number r by the smallest set in the chain containing a function that determines a Cauchy sequence of rational numbers which converges to r. |
Sep
7 |
accepted | Is every computable real primitively recursively computable? |
Sep
6 |
asked | Is every computable real primitively recursively computable? |
Jul
26 |
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A question about sentences undecidable in Peano's Arithmetic
I am really trying to avoid theories in which second order axioms are "represented" by first order axiom schemes. These are actually first order theories. |
Jul
26 |
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A question about sentences undecidable in Peano's Arithmetic
I am really trying to avoid theories in which second order axioms are "represented |
Jul
25 |
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A question about sentences undecidable in Peano's Arithmetic
The question at the end of my previous comment is not quite what I meant to ask. I meant to ask whether every consistent and axiomatizable extension of PA-even if based upon higher order logic-has non-standard models which contain "infinite" positive integers. I am not too sure about exactly what Godel's incompleteness theorem and model theory have to say regarding consistent and axiomatizable extensions of PA that are not based upon first order logic. |
Jul
24 |
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A question about sentences undecidable in Peano's Arithmetic
Many thanks for your answers which have given me alot to think about. I thought that Hilbert and Bernays presented the first formalized and axiomatizable version of what I call SOA. You do not mention whether any of the interesting examples of sentences such as the Paris-Harrington theorem or Goodstein's theorem-which have been shown to be undecidable in PA-are also undecidable in SOA. Finally, does every axiomatizable theory-even if based upon higher order logic-have non-standard models which contain "infinite" positive integers? |
Jul
24 |
accepted | A question about sentences undecidable in Peano's Arithmetic |
Jul
23 |
asked | A question about sentences undecidable in Peano's Arithmetic |
May
5 |
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A question about simple closed curves in finite dimensional Euclidean spaces
Thanks for the clarification |
May
3 |
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A question about simple closed curves in finite dimensional Euclidean spaces
Many thanks for such a complete answer. I am not too familiar with homology theory so it may take me awhile to fully understand your proof. |
May
3 |
accepted | A question about simple closed curves in finite dimensional Euclidean spaces |
May
2 |
asked | A question about simple closed curves in finite dimensional Euclidean spaces |
May
1 |
comment |
A question about Cantor's Power Set theorem without the Axiom of Choice
You are right. I was finally able to digest all the steps of your proof. Proving theorems about all infinite cardinal numbers can be quite tricky when the Axiom of Choice is not available. |