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Please either define "exists" or formulate your claim without using the word :)
A Crash Course in the Mathematics Of Infinite Sets
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Re: A Crash Course in the Mathematics Of Infinite Sets
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#14[deleted]
Please either define "exists" or formulate your claim without using the word :)
I'll side with Poincaré and there is not much to say other than that.
"The objections to Cantor's work were occasionally fierce: Poincaré referred to his ideas as a "grave disease" infecting the discipline of mathematics" http://en.wikipedia.org/wiki/Georg_Cantor#cite_note-daub266-...
Re: A Crash Course in the Mathematics Of Infinite Sets
#15:/ does this have any practical application?
Re: A Crash Course in the Mathematics Of Infinite Sets
#16:/ does this have any practical application?
SQL is based on Set theory.
http://seanmehan.globat.com/blog/2011/12/20/set-theory-and-s...
Re: A Crash Course in the Mathematics Of Infinite Sets
#17I find infinite mathematical structures interesting from a philosophical point of view, because it’s not possible to reconstruct these concept in the physical world. In fact there is a mathematical-philosophical movement refusing infinite structures: http://en.wikipedia.org/wiki/Finitism
Re: A Crash Course in the Mathematics Of Infinite Sets
#18This seems to muddle cardinality. First he defines cardinality as "number of elements" and notes that infinite sets have infinite cardinality, but doesn't explain what that might mean. Then he introduces "same cardinality" as a new definition instead of letting it arise out of sameness of number. And next, without any new definition, starts referring to larger cardinality.
Re: A Crash Course in the Mathematics Of Infinite Sets
#19One minor note: When discussing the continuum hypothesis, Suber refers to ZF (without the axiom of choice) as "the closest thing we have to 'standard' set theory". However, he implicitly uses the axiom of choice, or at least the axiom of dependent choice (I didn't really read enough to see whether there are other uses); without this you can't prove that aleph_0 is the smallest infinite cardinal. So there is a mismatc…
Just to elaborate for those not familiar with the terminology, the axion of dependent choice is a weaker version of the axiom of choice. It asserts that you can pick a countable(denumerable using the terminology in the post) sequence from any infinite set. The axiom of choice asserts (informally) that given any collection of sets, you can form another set which consists of one element from each set in the collection. The axiom of choice asserts that this holds true even when the number of sets is uncountable (non-denumerable).
The axiom of choice and the continuum hypothesis are equivalent, meaning the assumption of either one implies the other. >
Re: A Crash Course in the Mathematics Of Infinite Sets
#20Earlier quoted context omitted.
Please either define "exists" or formulate your claim without using the word :)
I'll side with Poincaré and there is not much to say other than that. "The objections to Cantor's work were occasionally fierce: Poincaré referred to his ideas as a "grave disease" infecting the discipline of mathematics" http://en.wikipedia.org/wiki/Georg_Cantor#cite_note-daub266-...
Poincaré was a genius but math has moved forward since the 19th century.
Here's what I'd argue is the most important take away from set theory: for a given framework of mathematics, there are conjectures one can make that can be neither proved nor disproved. Perhaps the most high-profile of them is the continuum hypothesis.
Maybe P!=NP is such a conjecture. Maybe the existence of a solution to the Navier-Stokes equation is such a problem. For Poincaré, the very idea that there is math that can be neither proved nor disproved was not in his mental vocabulary.
On another note, Weil's celebrated proof of Fermat's last theorem relies on the existence of inaccessible cardinals[1], although I've heard it conjectured that they are not necessary.