Return to Answer

added 16 characters in body

Source Link

edited Nov 18, 2012 at 2:47

73.2k
8
191
290

To increase the power set of a regular cardinal $\kappa$, Easton used forcing conditions that are partial functions of size $<\kappa$. So the forcing is $\kappa$-closed and therefore adds no new subsets of any cardinals below $\kappa$. It therefore doesn't interfere with whatever he was trying to do with the power sets of those smaller cardinals. If he did the same thing with a singular $\kappa$, the forcing would be only cf$(\kappa)$-closed, not $\kappa$-closed. For example, if $\kappa=\aleph_\omega$, then the union of a countable chain of conditions (each of size $<\aleph_\omega$) could have size $\aleph_\omega$ and thus fail to be a condition. As a result, new subsets would be added at cardinals below $\kappa$ (but $\geq$ cf$(\kappa)$), thereby messing up whatever was supposed to happen with the power sets of those cardinals.

A decade later, Silver discovered that not only does Easton's method not work for singular cardinals (which Easton already knew), but there are non-trivial constraints on $2^\kappa$ for singular $\kappa$. In particular, a singular cardinal of uncountable cofinality cannot be the first place where GCH fails. Later, it was shown (I believe first by Magidor) that a singular cardinal of countable cofinality can be the first place where GCH fails, but a large cardinal was needed for the proof and, by a result of Jensen, large cardinals are unavoidable here. Work of Gitik has pinned down the exact large-cardinal strength of the negation of the singular cardinal hypothesis.

The bottom line here is that, in order to get anything like Easton's results for singular cardinals, one must use large cardinals, one must use considerably fancier forcing notions than Easton used, and even then, some manipulations of power sets of singular cardinals are outright impossible.

To increase the power set of a regular cardinal $\kappa$, Easton used forcing conditions that are partial functions of size $<\kappa$. So the forcing is $\kappa$-closed and therefore adds no new subsets of any cardinals below $\kappa$. It therefore doesn't interfere with whatever he was trying to do with the power sets of those smaller cardinals. If he did the same thing with a singular $\kappa$, the forcing would be only cf$(\kappa)$-closed, not $\kappa$-closed. For example, if $\kappa=\aleph_\omega$, then the union of a countable chain of conditions (each of size $<\aleph_\omega$) could have size $\aleph_\omega$ and thus fail to be a condition. As a result, new subsets would be added at cardinals below $\kappa$ (but $\geq$ cf$(\kappa)$), thereby messing up whatever was supposed to happen with the power sets of those cardinals.

A decade later, Silver discovered that not only does Easton's method not work for singular cardinals (which Easton already knew), but there are non-trivial constraints on $2^\kappa$ for singular $\kappa$. In particular, a singular cardinal of uncountable cofinality cannot be the first place where GCH fails. Later, it was shown (I believe first by Magidor) that a singular cardinal of countable cofinality can be the first place where GCH fails, but a large cardinal was needed for the proof and, by a result of Jensen, large cardinals are unavoidable here. Work of Gitik has pinned down the exact large-cardinal strength of the singular cardinal hypothesis.

The bottom line here is that, in order to get anything like Easton's results for singular cardinals, one must use large cardinals, one must use considerably fancier forcing notions than Easton used, and even then, some manipulations of power sets of singular cardinals are outright impossible.

To increase the power set of a regular cardinal $\kappa$, Easton used forcing conditions that are partial functions of size $<\kappa$. So the forcing is $\kappa$-closed and therefore adds no new subsets of any cardinals below $\kappa$. It therefore doesn't interfere with whatever he was trying to do with the power sets of those smaller cardinals. If he did the same thing with a singular $\kappa$, the forcing would be only cf$(\kappa)$-closed, not $\kappa$-closed. For example, if $\kappa=\aleph_\omega$, then the union of a countable chain of conditions (each of size $<\aleph_\omega$) could have size $\aleph_\omega$ and thus fail to be a condition. As a result, new subsets would be added at cardinals below $\kappa$ (but $\geq$ cf$(\kappa)$), thereby messing up whatever was supposed to happen with the power sets of those cardinals.

A decade later, Silver discovered that not only does Easton's method not work for singular cardinals (which Easton already knew), but there are non-trivial constraints on $2^\kappa$ for singular $\kappa$. In particular, a singular cardinal of uncountable cofinality cannot be the first place where GCH fails. Later, it was shown (I believe first by Magidor) that a singular cardinal of countable cofinality can be the first place where GCH fails, but a large cardinal was needed for the proof and, by a result of Jensen, large cardinals are unavoidable here. Work of Gitik has pinned down the exact large-cardinal strength of the negation of the singular cardinal hypothesis.

The bottom line here is that, in order to get anything like Easton's results for singular cardinals, one must use large cardinals, one must use considerably fancier forcing notions than Easton used, and even then, some manipulations of power sets of singular cardinals are outright impossible.

Source Link

created Nov 18, 2012 at 1:51

73.2k
8
191
290

To increase the power set of a regular cardinal $\kappa$, Easton used forcing conditions that are partial functions of size $<\kappa$. So the forcing is $\kappa$-closed and therefore adds no new subsets of any cardinals below $\kappa$. It therefore doesn't interfere with whatever he was trying to do with the power sets of those smaller cardinals. If he did the same thing with a singular $\kappa$, the forcing would be only cf$(\kappa)$-closed, not $\kappa$-closed. For example, if $\kappa=\aleph_\omega$, then the union of a countable chain of conditions (each of size $<\aleph_\omega$) could have size $\aleph_\omega$ and thus fail to be a condition. As a result, new subsets would be added at cardinals below $\kappa$ (but $\geq$ cf$(\kappa)$), thereby messing up whatever was supposed to happen with the power sets of those cardinals.

A decade later, Silver discovered that not only does Easton's method not work for singular cardinals (which Easton already knew), but there are non-trivial constraints on $2^\kappa$ for singular $\kappa$. In particular, a singular cardinal of uncountable cofinality cannot be the first place where GCH fails. Later, it was shown (I believe first by Magidor) that a singular cardinal of countable cofinality can be the first place where GCH fails, but a large cardinal was needed for the proof and, by a result of Jensen, large cardinals are unavoidable here. Work of Gitik has pinned down the exact large-cardinal strength of the singular cardinal hypothesis.

The bottom line here is that, in order to get anything like Easton's results for singular cardinals, one must use large cardinals, one must use considerably fancier forcing notions than Easton used, and even then, some manipulations of power sets of singular cardinals are outright impossible.