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This answer started life as a nascent comment intended for the back-and-forth above, but it ballooned into what follows.

ZW, as I pointed out above, your current question does parallel your earlier question about CH, as do the (very good) answers in each case. From your further comments, though, I think I now have some idea why the answers haven't satisfied you; I'll take a stab at answering what I think's bothering you. (If I'm right, then it's a fairly simple matter, but just one that wouldn't be the initial guess as the issue on MO. And if I'm wrong about what you don't like, oh well; but I've genuinely tried to figure out why you're unhappy with the answers so far given.)

The answers given try to clarify a (very common and understandable) mathematical confusion that people can have about independence results, but your further comment:

My confusion is, people take V as the standard model. But why so?
suggests something else is at the heart of what's bothering you personally. And now looking at your original question about $CH$, it seems clear there as well:

suggests something else is at the heart of what's bothering you personally. And now looking at your original question about $CH$, it seems clear there as well:

OK, Cohen has constructed a model in which both ZFC and ~CH are true. Isn't this model an answer to the continuum problem? Hasn't he showed that it is indeed possible to construct a set with cardinality between that of the integers and that of the reals? Why is it still not considered sufficient to settle CH? Why is one model not enough? Why for all models? In other words, why do we have to answer whether "ZFC |- CH" instead of just "CH" itself?

So it seems that part of what you're not happy with is simply the (extra-mathematical, somewhat conventional) privileged position of $ZFC$ as a foundational theory for mathematics. (Again, if I'm wrong in ascribing such thoughts to you, my apologies.) And that's perfectly fair; plenty of people have taken issue with that status for myriad reasons.

So maybe you're really thinking: "Hey, Cohen constructed this model $\mathcal{M}\models ZFC + \neg CH$, and I think this $\mathcal{M}$ can be (or should be, or is) the mathematical universe we all work in." Well that's a perfectly acceptable way to think, but now you no longer have a purely mathematical pursuit on your hands (one reason, by the way, why myself and others generally would be expecting to answer the question the way they did), thanks to the privileged position $ZFC$ enjoys. Now you've also got a sociological (and dare I say philosophical) endeavor, namely that of convincing fellow mathematicians of the truth/efficacy/beauty/... of your favored universe.

Those who answered you were working under the accepted convention that "settling" a problem means either proving it in $ZFC$, or refuting it there, or establishing its independence from $ZFC$, and answered your initial queries accordingly (and accurately). If I'm right about what you're finding unsatisfactory here, then you now get to immerse yourself in the delights of the philosophy of mathematics. Enjoy! (And if I'm wrong, at least I've only wasted my own time.)

This answer started life as a nascent comment intended for the back-and-forth above, but it ballooned into what follows.

ZW, as I pointed out above, your current question does parallel your earlier question about CH, as do the (very good) answers in each case. From your further comments, though, I think I now have some idea why the answers haven't satisfied you; I'll take a stab at answering what I think's bothering you. (If I'm right, then it's a fairly simple matter, but just one that wouldn't be the initial guess as the issue on MO. And if I'm wrong about what you don't like, oh well; but I've genuinely tried to figure out why you're unhappy with the answers so far given.)

The answers given try to clarify a (very common and understandable) mathematical confusion that people can have about independence results, but your further comment:

My confusion is, people take V as the standard model. But why so?
suggests something else is at the heart of what's bothering you personally. And now looking at your original question about $CH$, it seems clear there as well:
OK, Cohen has constructed a model in which both ZFC and ~CH are true. Isn't this model an answer to the continuum problem? Hasn't he showed that it is indeed possible to construct a set with cardinality between that of the integers and that of the reals? Why is it still not considered sufficient to settle CH? Why is one model not enough? Why for all models? In other words, why do we have to answer whether "ZFC |- CH" instead of just "CH" itself?

So it seems that part of what you're not happy with is simply the (extra-mathematical, somewhat conventional) privileged position of $ZFC$ as a foundational theory for mathematics. (Again, if I'm wrong in ascribing such thoughts to you, my apologies.) And that's perfectly fair; plenty of people have taken issue with that status for myriad reasons.

So maybe you're really thinking: "Hey, Cohen constructed this model $\mathcal{M}\models ZFC + \neg CH$, and I think this $\mathcal{M}$ can be (or should be, or is) the mathematical universe we all work in." Well that's a perfectly acceptable way to think, but now you no longer have a purely mathematical pursuit on your hands (one reason, by the way, why myself and others generally would be expecting to answer the question the way they did), thanks to the privileged position $ZFC$ enjoys. Now you've also got a sociological (and dare I say philosophical) endeavor, namely that of convincing fellow mathematicians of the truth/efficacy/beauty/... of your favored universe.

Those who answered you were working under the accepted convention that "settling" a problem means either proving it in $ZFC$, or refuting it there, or establishing its independence from $ZFC$, and answered your initial queries accordingly (and accurately). If I'm right about what you're finding unsatisfactory here, then you now get to immerse yourself in the delights of the philosophy of mathematics. Enjoy! (And if I'm wrong, at least I've only wasted my own time.)

This answer started life as a nascent comment intended for the back-and-forth above, but it ballooned into what follows.

ZW, as I pointed out above, your current question does parallel your earlier question about CH, as do the (very good) answers in each case. From your further comments, though, I think I now have some idea why the answers haven't satisfied you; I'll take a stab at answering what I think's bothering you. (If I'm right, then it's a fairly simple matter, but just one that wouldn't be the initial guess as the issue on MO. And if I'm wrong about what you don't like, oh well; but I've genuinely tried to figure out why you're unhappy with the answers so far given.)

The answers given try to clarify a (very common and understandable) mathematical confusion that people can have about independence results, but your further comment:

My confusion is, people take V as the standard model. But why so?

suggests something else is at the heart of what's bothering you personally. And now looking at your original question about $CH$, it seems clear there as well:

OK, Cohen has constructed a model in which both ZFC and ~CH are true. Isn't this model an answer to the continuum problem? Hasn't he showed that it is indeed possible to construct a set with cardinality between that of the integers and that of the reals? Why is it still not considered sufficient to settle CH? Why is one model not enough? Why for all models? In other words, why do we have to answer whether "ZFC |- CH" instead of just "CH" itself?

So it seems that part of what you're not happy with is simply the (extra-mathematical, somewhat conventional) privileged position of $ZFC$ as a foundational theory for mathematics. (Again, if I'm wrong in ascribing such thoughts to you, my apologies.) And that's perfectly fair; plenty of people have taken issue with that status for myriad reasons.

So maybe you're really thinking: "Hey, Cohen constructed this model $\mathcal{M}\models ZFC + \neg CH$, and I think this $\mathcal{M}$ can be (or should be, or is) the mathematical universe we all work in." Well that's a perfectly acceptable way to think, but now you no longer have a purely mathematical pursuit on your hands (one reason, by the way, why myself and others generally would be expecting to answer the question the way they did), thanks to the privileged position $ZFC$ enjoys. Now you've also got a sociological (and dare I say philosophical) endeavor, namely that of convincing fellow mathematicians of the truth/efficacy/beauty/... of your favored universe.

Those who answered you were working under the accepted convention that "settling" a problem means either proving it in $ZFC$, or refuting it there, or establishing its independence from $ZFC$, and answered your initial queries accordingly (and accurately). If I'm right about what you're finding unsatisfactory here, then you now get to immerse yourself in the delights of the philosophy of mathematics. Enjoy! (And if I'm wrong, at least I've only wasted my own time.)

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Ed Dean
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This answer started life as a nascent comment intended for the back-and-forth above, but it ballooned into what follows.

ZW, as I pointed out above, your current question does parallel your earlier question about CH, as do the (very good) answers in each case. From your further comments, though, I think I now have some idea why the answers haven't satisfied you; I'll take a stab at answering what I think's bothering you. (If I'm right, then it's a fairly simple matter, but just one that wouldn't be the initial guess as the issue on MO. And if I'm wrong about what you don't like, oh well; but I've genuinely tried to figure out why you're unhappy with the answers so far given.)

The answers given try to clarify a (very common and understandable) mathematical confusion that people can have about independence results, but your further comment:

My confusion is, people take V as the standard model. But why so?
suggests something else is at the heart of what's bothering you personally. And now looking at your original question about $CH$, it seems clear there as well:
OK, Cohen has constructed a model in which both ZFC and ~CH are true. Isn't this model an answer to the continuum problem? Hasn't he showed that it is indeed possible to construct a set with cardinality between that of the integers and that of the reals? Why is it still not considered sufficient to settle CH? Why is one model not enough? Why for all models? In other words, why do we have to answer whether "ZFC |- CH" instead of just "CH" itself?

So it seems that part of what you're not happy with is simply the (extra-mathematical, somewhat conventional) privileged position of $ZFC$ as a foundational theory for mathematics. (Again, if I'm wrong in ascribing such thoughts to you, my apologies.) And that's perfectly fair; plenty of people have taken issue with that status for myriad reasons.

So maybe you're really thinking: "Hey, Cohen constructed this model $\mathcal{M}\models ZFC + \neg CH$, and I think this $\mathcal{M}$ can be (or should be, or is) the mathematical universe we all work in." Well that's a perfectly acceptable way to think, but now you no longer have a purely mathematical pursuit on your hands (one reason, by the way, why myself and others generally would be expecting to answer the question the way they did), thanks to the privileged position $ZFC$ enjoys. Now you've also got a sociological (and dare I say philosophical) endeavor, namely that of convincing fellow mathematicians of the truth/efficacy/beauty/... of your favored universe.

Those who answered you were working under the accepted convention that "settling" a problem means either proving it in $ZFC$, or refuting it there, or establishing its independence from $ZFC$, and answered your initial queries accordingly (and accurately). If I'm right about what you're finding unsatisfactory here, then you now get to immerse yourself in the delights of the philosophy of mathematics. Enjoy! (And if I'm wrong, at least I've only wasted my own time.)