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List of Statements Independent of ZFC

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Re: List of Statements Independent of ZFC

#21

Post by ur-whale »

triska wrote: post #3

A 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…

So, does that mean that one can assume this to be true and build a perfectly consistent theory, or conversely assume it to be false (with - say - at least one counter-example) and build another perfectly consistent theory?

Re: List of Statements Independent of ZFC

#22

Post by [deleted] »

ur-whale wrote: post #21
triska wrote: post #3

A 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…

So, does that mean that one can assume this to be true and build a perfectly consistent theory, or conversely assume it to be false (with - say - at least one counter-example) and build another perfectly consistent theory?

[deleted]

Re: List of Statements Independent of ZFC

#23

Post by karatinversion »

ur-whale wrote: post #21
triska wrote: post #3

A 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…

So, does that mean that one can assume this to be true and build a perfectly consistent theory, or conversely assume it to be false (with - say - at least one counter-example) and build another perfectly consistent theory?

Yes, this is the exact meaning of “independent from” here.

Re: List of Statements Independent of ZFC

#25

Post by 0-_-0 »

The_rationalist wrote: post #16
triska wrote: post #3

A 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 know that feeling. It never lasts long.

Re: List of Statements Independent of ZFC

#26

Post by The_rationalist »

correct_horse wrote: post #20
The_rationalist wrote: post #16

Earlier quoted context omitted.

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/Al…

Another example of why mathematics are a wrong abstraction to be optimally useful. Mathematics should have bounds in the same way as our universe has bounds. Any theorem that has a different behavior if something is infinite doesn't matter at all and is a waste of time for real engineers who solve things in the real world. The niche of mathematics that describe things beyond what our universe has to offer should be an optional extension to the mathematical framework instead of being the default, in order to not pollute discussions.

Re: List of Statements Independent of ZFC

#30

Post by new2628 »

The_rationalist wrote: post #26
correct_horse wrote: post #20

Earlier quoted context omitted.

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/Al…

Another example of why mathematics are a wrong abstraction to be optimally useful. Mathematics should have bounds in the same way as our universe has bounds. Any theorem that has a different behavior if something is infinite doesn't matter at all and is a waste of time for real engineers who solve things in the real world. The niche of mathematics that describe things beyond what our universe has to offer should be a…

This is a fine and defensible (although not mainstream) viewpoint with several adherents among mathematicians and various levels of success in making it formal and precise.

See e.g. https://en.wikipedia.org/wiki/Finitism

Doron Zeilberger would like to pat you on the back.

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