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- how to prove uncountable infinite pigeonhole principle?
1 Can it be proven using the pigeonhole principle that if set A is an uncountable family of finite sets, it contains an uncountable subfamily all of whose elements have cardinality n? The idea is borrowed from here What is the Infinite Pigeonhole Principle?
- I have learned that 1 0 is infinity, why isnt it minus infinity?
An infinite number? Kind of, because I can keep going around infinitely However, I never actually give away that sweet This is why people say that 1 0 "tends to" infinity - we can't really use infinity as a number, we can only imagine what we are getting closer to as we move in the direction of infinity
- Is there a strictly-increasing infinite sequence of positive integers . . .
If I understand right, your question is whether there exists a strictly increasing infinite sequence $\left\ { a_n \right\}$ for which the infinite sequence $\left\ { T_n \right\}$ is a geometric progression
- real analysis - Why set of natural numbers is infinite, while each . . .
In his book Analysis Vol 1, author Terence Tao argues that while each natural number is finite, the set of natural numbers is infinite (though has not defined what infinite means yet) Using Peano
- Can a countable set contain uncountably many infinite subsets such that . . .
Can a countable set contain uncountably many infinite subsets such that the intersection of any two such distinct subsets is finite?
- Is there a shape with infinite volume but finite surface area?
Imagine a sphere outline in an infinite void If the area within the sphere outline is empty space, and the space outside is solid, it is a 3D shape of infinite volume, and since it continues infinitely, there is no outer edge of the shape to apply surface area to, meaning the surface area is a finite value, on the same spherical plane as the outline
- linear algebra - Is there a quick proof as to why the vector space of . . .
Your further question in the comments, whether a vector space over $\mathbb {Q}$ is finite dimensional if and only if the set of vectors is countable, has a negative answer If the vector space is finite dimensional, then it is a countable set; but there are infinite-dimensional vector spaces over $\mathbb {Q}$ that are countable as sets
- Is it viable to ask in an infinite set about the Cardinality?
Can you ask given an infinite set about its cardinality? Does an infinite set have a cardinality? So, for example, what would be the cardinality of $+\\infty$?
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