Earlier quoted context omitted.
Cantor did work on cardinalities, presumably.
He practically invented the field.
There's an implicit "because" at the front there, I know about Cantor's influence quite well ;)
List of Statements Independent of ZFC
11–20 of 108 posts
Re: List of Statements Independent of ZFC
#12Why is it always zfc plus optionally something else? Is there anything other than zfc that creates an interesting starting point?
Re: List of Statements Independent of ZFC
#13Why is it always zfc plus optionally something else? Is there anything other than zfc that creates an interesting starting point?
These are excellent questions! And yes, there is!
For example, I highly recommend the paper Rethinking set theory by Tom Leinster:
https://arxiv.org/abs/1212.6543
It highlights Lawvere set theory. The paper won this year's Chauvenet Prize:
https://www.maa.org/programs-and-communities/member-communit...
Re: List of Statements Independent of ZFC
#14Earlier quoted context omitted.
What does this have to do with Cantor?
ZFC was created to avoid the paradoxes inherent in Cantor's set theory work.
I think it's fairer to say that ZFC was a response to paradoxes in naïve set theory of the sort discovered by Russell than in response to Cantor; see, e.g.,
> In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox.
(https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory). Cantor discovered a rigorous basis for the intuitively paradoxical situation of two sets where it seems that one is manifestly bigger, but they are actually the same size; but I think that this work is best viewed as a resolution of the paradox, not a paradox itself.
Re: List of Statements Independent of ZFC
#15Hopefully this will clear at least some of the Gödelian misconceptions that pop up from time to time here
How? Misunderstandings about Gödel's theorem are unlikely to be due to a misunderstanding of the axioms of set theory (more likely of logical issues such as the technical meaning of 'incompleteness'), let alone to a lack of understanding of what statements are independent of a common set of axioms.
Re: List of Statements Independent of ZFC
#16A very interesting discussion about this topic that also includes many examples and references is available on MathOverflow: What are some reasonable-sounding statements that are independent of ZFC? https://mathoverflow.net/questions/1924/what-are-some-reason... With the top voted result currently being: "If a set X is smaller in cardinality than another set Y, then X has fewer subsets than Y." As is also mentioned i…
WTF I'm pretty sure I can prove that
Re: List of Statements Independent of ZFC
#17Why is it always zfc plus optionally something else? Is there anything other than zfc that creates an interesting starting point?
Re: List of Statements Independent of ZFC
#18Hopefully this will clear at least some of the Gödelian misconceptions that pop up from time to time here
Which Gödel misconceptions?
Re: List of Statements Independent of ZFC
#19A very interesting discussion about this topic that also includes many examples and references is available on MathOverflow: What are some reasonable-sounding statements that are independent of ZFC? https://mathoverflow.net/questions/1924/what-are-some-reason... With the top voted result currently being: "If a set X is smaller in cardinality than another set Y, then X has fewer subsets than Y." As is also mentioned i…
WTF I'm pretty sure I can prove that
>I'm pretty sure I can prove that
For infinite sets of arbitrarily large cardinality?
Re: List of Statements Independent of ZFC
#20A very interesting discussion about this topic that also includes many examples and references is available on MathOverflow: What are some reasonable-sounding statements that are independent of ZFC? https://mathoverflow.net/questions/1924/what-are-some-reason... With the top voted result currently being: "If a set X is smaller in cardinality than another set Y, then X has fewer subsets than Y." As is also mentioned i…
WTF I'm pretty sure I can prove that
It is easy to prove for finite sets (just count) but much harder for infinite ones. For example, which has more subsets, the integers or the positive integers? How about the integers and the reals?
If you answered "the integers" to the first question, you aren't thinking about this right, as the integers and the positive integers have the same number of elements to start with. source: https://en.wikipedia.org/wiki/Aleph_number