# continuum

1. ### Discreet versus continuum

(sorry, don't know in which section this post goes) I just took a look at this: Continuity and Infinitesimals (Stanford Encyclopedia of Philosophy) and this https://en.wikipedia.org/wiki/Infinitesimal. A lot of it I can't understand, especially the math (meaning what is not explained with...
2. ### Continuum Mechanics/Elasticity help re: Polar Decomposition Theorem

That is a model question. b) is just stating the theorem, which is as follows in my notes: If a linear transformation F is invertible with det F > 0, then there exists unique symmetric positive-definite linear transformations U and V, and a unique proper orthogonal transofrmation R, such that...
3. ### real-valued-measurable cardinals and the cardinality of the continuum

Two results from the following web-site: Existence of probability measure defined on all subsets - MathOverflow which I am presuming quote the results correctly. (1) "Ulam also showed that successor cardinals like \aleph_1 cannot be real-valued measurable." (2) "Solovay …… showed that if...
4. ### cardinality of the Cantor set only uncountable or necessarily that of the continuum?

In the Wikipedia account of the Cantor tertiary set, it is remarked that the cardinality of the set is uncountable. Can this can be strengthened to it needing to be as large as the continuum? (This is not necessarily the same thing if we assume that the continuum hypothesis is false, so that...
5. ### Cardinality of the continuum

Hello, I need help with the following problem: Prove that Card(l^p) = c (cardinality of the continuum), p >= 1. I am not quite sure how to start the proof. I'd appreciate anyone's help. Thank you.
6. ### A continuum of homomorphic images

Let G be a finitely-generated (and thus countable) group and let H_i be a homomorphic image of G such that H_i \not\cong H_j for i \neq j with i, j \in I where I is some index set with cardinality equal to that of the reals. Such groups do exists (for example, non-elementary hyperbolic groups)...
7. ### Continuum Hypothesis

The author of my book stated The Continuum Hypothesis: There exists no set S such that \aleph_0<|S|<c. where c = |\mathbb{R}| He then went on raising a question: Is there as set S such that |S|>c? Following the question, he proved that |A|<|P(a)|, and at the end of the proof, he said the...
8. ### Maximizing over a continuum of possibilities

Let x be a random variable with density function f and [a,b] be the possible set of values of x. Let w(x) be a value associated with each value of x. I.e. w() is not known, and suppose that U(w(x)) is a C^{\infty} function. Let H(w(x)) = \int_{a}^b f(x)U(w(x))dx Find the derivative of H...
9. ### set of all finite subsets of a continuum has cardinality continuum

Hi. I'm learning about equivalent sets in a real analysis class and am struggling a bit with the abstractness of it. Would someone be able to explain in very basic language how to: Prove that the set of all finite subsets of a continuum has the cardinality of continuum. I guess I'm not...