Finite chain condition - Variation of Martin's Axiom statement












1














In the following $k$ and $w$ will be cardinal numbers.



Consider the classical statement $MA(k)$:




For any partial order $P$ satisfying the countable chain condition (hereafter $ccc$) and any family $D$ of dense sets in $P$ such that $|D| = k$, there is a filter $F$ on $P$ such that $F cap d$ is non-empty for every $d$ in $D$




Let's generalize it to the statement $MA(w, k)$ that replaces the $ccc$ by any width $w$, stating:




For any partial order $P$ satisfying that every strong antichain is of cardinality $less$ than $w$ [...etc]




Eg. $MA(aleph_1, k) = MA(k)$



Of course, $MA(w, k)$ implies $MA(w', k')$ for every $w' geq w$ and $k' geq k$



Now, I was wondering why $MA$ was so specific about antichains being countable, so to motivate the classical definition I tried mapping $MA$'s validity for each $w$ and $k$ pair. So far I've got:





  • $MA(w, k)$ is true for all $k leq aleph_0$


  • $MA(aleph_1, 2^{aleph_0}) = MA(2^{aleph_0})$ is false, and then so it is for any $w$ and $k$ equal or greater


  • $MA(aleph_1, k) = MA(k)$ is independent from but consistent with $ZFC$ for every $aleph_1 leq k < 2^{aleph_0}$


  • $MA(aleph_2, aleph_1)$ is false, and then so it is for any $w$ and $k$ equal or greater


So $MA$ is no use stated for longer than countable antichains. But the case I can't figure out is for $w = aleph_0$ and $k > aleph_0$.



So the question is: Why $ccc$? What can be said about the validity of $MA$ when stated for posets sastisfying that every strong antichain is finite but given an uncountable number of dense sets? Does a filter always or never exist? Is it equivalent to the case $w = aleph_1$ (ie. is allowing arbitrarily long but finite antichains equivalent to allowing countable ones too)?










share|cite|improve this question
























  • You're missing/ignoring the requirement that the forcing notions must be separative.
    – Not Mike
    2 days ago






  • 1




    @NotMike I'm very new to this, and I don't understand why is that required in any sense for any of my statements/questions
    – Emilio Martinez
    2 days ago






  • 2




    Replacing "ccc" by uncountable cardinals is a very difficult problem. Recently James Cummings, Mirna Dzamonja, and Itay Neeman have proposed a nice generalization. You can find it on arXiv. Other than this, you'd venture well into generalized properness, which is depressingly convoluted and hard to understand, main works are by Shelah and Roslanowski.
    – Asaf Karagila
    2 days ago






  • 2




    I mean that there are a lot inherent difficulties in forcing axioms which do not revolve around $omega$ in some significant way.
    – Asaf Karagila
    2 days ago






  • 1




    The point is not to "just change ccc", but also to get something which is consistent and helpful in proving similar consequences to what MA provides us with.
    – Asaf Karagila
    5 hours ago
















1














In the following $k$ and $w$ will be cardinal numbers.



Consider the classical statement $MA(k)$:




For any partial order $P$ satisfying the countable chain condition (hereafter $ccc$) and any family $D$ of dense sets in $P$ such that $|D| = k$, there is a filter $F$ on $P$ such that $F cap d$ is non-empty for every $d$ in $D$




Let's generalize it to the statement $MA(w, k)$ that replaces the $ccc$ by any width $w$, stating:




For any partial order $P$ satisfying that every strong antichain is of cardinality $less$ than $w$ [...etc]




Eg. $MA(aleph_1, k) = MA(k)$



Of course, $MA(w, k)$ implies $MA(w', k')$ for every $w' geq w$ and $k' geq k$



Now, I was wondering why $MA$ was so specific about antichains being countable, so to motivate the classical definition I tried mapping $MA$'s validity for each $w$ and $k$ pair. So far I've got:





  • $MA(w, k)$ is true for all $k leq aleph_0$


  • $MA(aleph_1, 2^{aleph_0}) = MA(2^{aleph_0})$ is false, and then so it is for any $w$ and $k$ equal or greater


  • $MA(aleph_1, k) = MA(k)$ is independent from but consistent with $ZFC$ for every $aleph_1 leq k < 2^{aleph_0}$


  • $MA(aleph_2, aleph_1)$ is false, and then so it is for any $w$ and $k$ equal or greater


So $MA$ is no use stated for longer than countable antichains. But the case I can't figure out is for $w = aleph_0$ and $k > aleph_0$.



So the question is: Why $ccc$? What can be said about the validity of $MA$ when stated for posets sastisfying that every strong antichain is finite but given an uncountable number of dense sets? Does a filter always or never exist? Is it equivalent to the case $w = aleph_1$ (ie. is allowing arbitrarily long but finite antichains equivalent to allowing countable ones too)?










share|cite|improve this question
























  • You're missing/ignoring the requirement that the forcing notions must be separative.
    – Not Mike
    2 days ago






  • 1




    @NotMike I'm very new to this, and I don't understand why is that required in any sense for any of my statements/questions
    – Emilio Martinez
    2 days ago






  • 2




    Replacing "ccc" by uncountable cardinals is a very difficult problem. Recently James Cummings, Mirna Dzamonja, and Itay Neeman have proposed a nice generalization. You can find it on arXiv. Other than this, you'd venture well into generalized properness, which is depressingly convoluted and hard to understand, main works are by Shelah and Roslanowski.
    – Asaf Karagila
    2 days ago






  • 2




    I mean that there are a lot inherent difficulties in forcing axioms which do not revolve around $omega$ in some significant way.
    – Asaf Karagila
    2 days ago






  • 1




    The point is not to "just change ccc", but also to get something which is consistent and helpful in proving similar consequences to what MA provides us with.
    – Asaf Karagila
    5 hours ago














1












1








1







In the following $k$ and $w$ will be cardinal numbers.



Consider the classical statement $MA(k)$:




For any partial order $P$ satisfying the countable chain condition (hereafter $ccc$) and any family $D$ of dense sets in $P$ such that $|D| = k$, there is a filter $F$ on $P$ such that $F cap d$ is non-empty for every $d$ in $D$




Let's generalize it to the statement $MA(w, k)$ that replaces the $ccc$ by any width $w$, stating:




For any partial order $P$ satisfying that every strong antichain is of cardinality $less$ than $w$ [...etc]




Eg. $MA(aleph_1, k) = MA(k)$



Of course, $MA(w, k)$ implies $MA(w', k')$ for every $w' geq w$ and $k' geq k$



Now, I was wondering why $MA$ was so specific about antichains being countable, so to motivate the classical definition I tried mapping $MA$'s validity for each $w$ and $k$ pair. So far I've got:





  • $MA(w, k)$ is true for all $k leq aleph_0$


  • $MA(aleph_1, 2^{aleph_0}) = MA(2^{aleph_0})$ is false, and then so it is for any $w$ and $k$ equal or greater


  • $MA(aleph_1, k) = MA(k)$ is independent from but consistent with $ZFC$ for every $aleph_1 leq k < 2^{aleph_0}$


  • $MA(aleph_2, aleph_1)$ is false, and then so it is for any $w$ and $k$ equal or greater


So $MA$ is no use stated for longer than countable antichains. But the case I can't figure out is for $w = aleph_0$ and $k > aleph_0$.



So the question is: Why $ccc$? What can be said about the validity of $MA$ when stated for posets sastisfying that every strong antichain is finite but given an uncountable number of dense sets? Does a filter always or never exist? Is it equivalent to the case $w = aleph_1$ (ie. is allowing arbitrarily long but finite antichains equivalent to allowing countable ones too)?










share|cite|improve this question















In the following $k$ and $w$ will be cardinal numbers.



Consider the classical statement $MA(k)$:




For any partial order $P$ satisfying the countable chain condition (hereafter $ccc$) and any family $D$ of dense sets in $P$ such that $|D| = k$, there is a filter $F$ on $P$ such that $F cap d$ is non-empty for every $d$ in $D$




Let's generalize it to the statement $MA(w, k)$ that replaces the $ccc$ by any width $w$, stating:




For any partial order $P$ satisfying that every strong antichain is of cardinality $less$ than $w$ [...etc]




Eg. $MA(aleph_1, k) = MA(k)$



Of course, $MA(w, k)$ implies $MA(w', k')$ for every $w' geq w$ and $k' geq k$



Now, I was wondering why $MA$ was so specific about antichains being countable, so to motivate the classical definition I tried mapping $MA$'s validity for each $w$ and $k$ pair. So far I've got:





  • $MA(w, k)$ is true for all $k leq aleph_0$


  • $MA(aleph_1, 2^{aleph_0}) = MA(2^{aleph_0})$ is false, and then so it is for any $w$ and $k$ equal or greater


  • $MA(aleph_1, k) = MA(k)$ is independent from but consistent with $ZFC$ for every $aleph_1 leq k < 2^{aleph_0}$


  • $MA(aleph_2, aleph_1)$ is false, and then so it is for any $w$ and $k$ equal or greater


So $MA$ is no use stated for longer than countable antichains. But the case I can't figure out is for $w = aleph_0$ and $k > aleph_0$.



So the question is: Why $ccc$? What can be said about the validity of $MA$ when stated for posets sastisfying that every strong antichain is finite but given an uncountable number of dense sets? Does a filter always or never exist? Is it equivalent to the case $w = aleph_1$ (ie. is allowing arbitrarily long but finite antichains equivalent to allowing countable ones too)?







set-theory first-order-logic order-theory cardinals forcing






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 6 hours ago

























asked 2 days ago









Emilio Martinez

1246




1246












  • You're missing/ignoring the requirement that the forcing notions must be separative.
    – Not Mike
    2 days ago






  • 1




    @NotMike I'm very new to this, and I don't understand why is that required in any sense for any of my statements/questions
    – Emilio Martinez
    2 days ago






  • 2




    Replacing "ccc" by uncountable cardinals is a very difficult problem. Recently James Cummings, Mirna Dzamonja, and Itay Neeman have proposed a nice generalization. You can find it on arXiv. Other than this, you'd venture well into generalized properness, which is depressingly convoluted and hard to understand, main works are by Shelah and Roslanowski.
    – Asaf Karagila
    2 days ago






  • 2




    I mean that there are a lot inherent difficulties in forcing axioms which do not revolve around $omega$ in some significant way.
    – Asaf Karagila
    2 days ago






  • 1




    The point is not to "just change ccc", but also to get something which is consistent and helpful in proving similar consequences to what MA provides us with.
    – Asaf Karagila
    5 hours ago


















  • You're missing/ignoring the requirement that the forcing notions must be separative.
    – Not Mike
    2 days ago






  • 1




    @NotMike I'm very new to this, and I don't understand why is that required in any sense for any of my statements/questions
    – Emilio Martinez
    2 days ago






  • 2




    Replacing "ccc" by uncountable cardinals is a very difficult problem. Recently James Cummings, Mirna Dzamonja, and Itay Neeman have proposed a nice generalization. You can find it on arXiv. Other than this, you'd venture well into generalized properness, which is depressingly convoluted and hard to understand, main works are by Shelah and Roslanowski.
    – Asaf Karagila
    2 days ago






  • 2




    I mean that there are a lot inherent difficulties in forcing axioms which do not revolve around $omega$ in some significant way.
    – Asaf Karagila
    2 days ago






  • 1




    The point is not to "just change ccc", but also to get something which is consistent and helpful in proving similar consequences to what MA provides us with.
    – Asaf Karagila
    5 hours ago
















You're missing/ignoring the requirement that the forcing notions must be separative.
– Not Mike
2 days ago




You're missing/ignoring the requirement that the forcing notions must be separative.
– Not Mike
2 days ago




1




1




@NotMike I'm very new to this, and I don't understand why is that required in any sense for any of my statements/questions
– Emilio Martinez
2 days ago




@NotMike I'm very new to this, and I don't understand why is that required in any sense for any of my statements/questions
– Emilio Martinez
2 days ago




2




2




Replacing "ccc" by uncountable cardinals is a very difficult problem. Recently James Cummings, Mirna Dzamonja, and Itay Neeman have proposed a nice generalization. You can find it on arXiv. Other than this, you'd venture well into generalized properness, which is depressingly convoluted and hard to understand, main works are by Shelah and Roslanowski.
– Asaf Karagila
2 days ago




Replacing "ccc" by uncountable cardinals is a very difficult problem. Recently James Cummings, Mirna Dzamonja, and Itay Neeman have proposed a nice generalization. You can find it on arXiv. Other than this, you'd venture well into generalized properness, which is depressingly convoluted and hard to understand, main works are by Shelah and Roslanowski.
– Asaf Karagila
2 days ago




2




2




I mean that there are a lot inherent difficulties in forcing axioms which do not revolve around $omega$ in some significant way.
– Asaf Karagila
2 days ago




I mean that there are a lot inherent difficulties in forcing axioms which do not revolve around $omega$ in some significant way.
– Asaf Karagila
2 days ago




1




1




The point is not to "just change ccc", but also to get something which is consistent and helpful in proving similar consequences to what MA provides us with.
– Asaf Karagila
5 hours ago




The point is not to "just change ccc", but also to get something which is consistent and helpful in proving similar consequences to what MA provides us with.
– Asaf Karagila
5 hours ago










1 Answer
1






active

oldest

votes


















3














The issue is that a poset where all strong antichains have finite width is trivial - at least, as far as forcing is concerned - and so the corresponding variant of Martin's Axiom is trivial too.





To see that such forcings are trivial, the key point is the following:




Let $mathbb{P}$ be a poset with an element $p$ such that every $q_1,q_2le p$ have a common extension $r$ (that is, $p$ doesn't bound any nontrivial strong antichain). Then there is a $mathbb{P}$-generic filter in the ground model already.




The proof is simple: let $G$ be the set of all conditions compatible with $p$.



Now suppose $mathbb{P}$ is a poset where every strong antichain is finite. I claim that $mathbb{P}$ has such a "trivializing" element $p$. For if not, we can inductively define a map $t$ from $2^{<omega}$ to $mathbb{P}$ such that:




  • $sigmaprectauimplies t(sigma)ge t(tau)$.


  • $t(sigma0)perp t(sigma1)$.



But then the set $${t(0), t(10), t(110), t(1110), ...}$$ forms an infinite strong antichain in $mathbb{P}$.



In fact, we can do even better (since having only finite strong antichains is preserved by passing from $mathbb{P}$ to $mathbb{P}_{le s}$):




If $mathbb{P}$ has only finite strong antichains, then the set of "trivializing" $p$ is dense in $mathbb{P}$; so every $mathbb{P}$-generic filter is already in the ground model. (And consequently, we have "${bf MA(aleph_0,infty)}$.")




So such a poset really is trivial (in the sense of forcing), not just "possibly trivial," as is the corresponding variant of Martin's Axiom.






share|cite|improve this answer























  • Thanks. It was my intuition that the problem would trivilize like that, but I hadn't yet matured the subject enough to prove it.
    – Emilio Martinez
    2 days ago











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1 Answer
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3














The issue is that a poset where all strong antichains have finite width is trivial - at least, as far as forcing is concerned - and so the corresponding variant of Martin's Axiom is trivial too.





To see that such forcings are trivial, the key point is the following:




Let $mathbb{P}$ be a poset with an element $p$ such that every $q_1,q_2le p$ have a common extension $r$ (that is, $p$ doesn't bound any nontrivial strong antichain). Then there is a $mathbb{P}$-generic filter in the ground model already.




The proof is simple: let $G$ be the set of all conditions compatible with $p$.



Now suppose $mathbb{P}$ is a poset where every strong antichain is finite. I claim that $mathbb{P}$ has such a "trivializing" element $p$. For if not, we can inductively define a map $t$ from $2^{<omega}$ to $mathbb{P}$ such that:




  • $sigmaprectauimplies t(sigma)ge t(tau)$.


  • $t(sigma0)perp t(sigma1)$.



But then the set $${t(0), t(10), t(110), t(1110), ...}$$ forms an infinite strong antichain in $mathbb{P}$.



In fact, we can do even better (since having only finite strong antichains is preserved by passing from $mathbb{P}$ to $mathbb{P}_{le s}$):




If $mathbb{P}$ has only finite strong antichains, then the set of "trivializing" $p$ is dense in $mathbb{P}$; so every $mathbb{P}$-generic filter is already in the ground model. (And consequently, we have "${bf MA(aleph_0,infty)}$.")




So such a poset really is trivial (in the sense of forcing), not just "possibly trivial," as is the corresponding variant of Martin's Axiom.






share|cite|improve this answer























  • Thanks. It was my intuition that the problem would trivilize like that, but I hadn't yet matured the subject enough to prove it.
    – Emilio Martinez
    2 days ago
















3














The issue is that a poset where all strong antichains have finite width is trivial - at least, as far as forcing is concerned - and so the corresponding variant of Martin's Axiom is trivial too.





To see that such forcings are trivial, the key point is the following:




Let $mathbb{P}$ be a poset with an element $p$ such that every $q_1,q_2le p$ have a common extension $r$ (that is, $p$ doesn't bound any nontrivial strong antichain). Then there is a $mathbb{P}$-generic filter in the ground model already.




The proof is simple: let $G$ be the set of all conditions compatible with $p$.



Now suppose $mathbb{P}$ is a poset where every strong antichain is finite. I claim that $mathbb{P}$ has such a "trivializing" element $p$. For if not, we can inductively define a map $t$ from $2^{<omega}$ to $mathbb{P}$ such that:




  • $sigmaprectauimplies t(sigma)ge t(tau)$.


  • $t(sigma0)perp t(sigma1)$.



But then the set $${t(0), t(10), t(110), t(1110), ...}$$ forms an infinite strong antichain in $mathbb{P}$.



In fact, we can do even better (since having only finite strong antichains is preserved by passing from $mathbb{P}$ to $mathbb{P}_{le s}$):




If $mathbb{P}$ has only finite strong antichains, then the set of "trivializing" $p$ is dense in $mathbb{P}$; so every $mathbb{P}$-generic filter is already in the ground model. (And consequently, we have "${bf MA(aleph_0,infty)}$.")




So such a poset really is trivial (in the sense of forcing), not just "possibly trivial," as is the corresponding variant of Martin's Axiom.






share|cite|improve this answer























  • Thanks. It was my intuition that the problem would trivilize like that, but I hadn't yet matured the subject enough to prove it.
    – Emilio Martinez
    2 days ago














3












3








3






The issue is that a poset where all strong antichains have finite width is trivial - at least, as far as forcing is concerned - and so the corresponding variant of Martin's Axiom is trivial too.





To see that such forcings are trivial, the key point is the following:




Let $mathbb{P}$ be a poset with an element $p$ such that every $q_1,q_2le p$ have a common extension $r$ (that is, $p$ doesn't bound any nontrivial strong antichain). Then there is a $mathbb{P}$-generic filter in the ground model already.




The proof is simple: let $G$ be the set of all conditions compatible with $p$.



Now suppose $mathbb{P}$ is a poset where every strong antichain is finite. I claim that $mathbb{P}$ has such a "trivializing" element $p$. For if not, we can inductively define a map $t$ from $2^{<omega}$ to $mathbb{P}$ such that:




  • $sigmaprectauimplies t(sigma)ge t(tau)$.


  • $t(sigma0)perp t(sigma1)$.



But then the set $${t(0), t(10), t(110), t(1110), ...}$$ forms an infinite strong antichain in $mathbb{P}$.



In fact, we can do even better (since having only finite strong antichains is preserved by passing from $mathbb{P}$ to $mathbb{P}_{le s}$):




If $mathbb{P}$ has only finite strong antichains, then the set of "trivializing" $p$ is dense in $mathbb{P}$; so every $mathbb{P}$-generic filter is already in the ground model. (And consequently, we have "${bf MA(aleph_0,infty)}$.")




So such a poset really is trivial (in the sense of forcing), not just "possibly trivial," as is the corresponding variant of Martin's Axiom.






share|cite|improve this answer














The issue is that a poset where all strong antichains have finite width is trivial - at least, as far as forcing is concerned - and so the corresponding variant of Martin's Axiom is trivial too.





To see that such forcings are trivial, the key point is the following:




Let $mathbb{P}$ be a poset with an element $p$ such that every $q_1,q_2le p$ have a common extension $r$ (that is, $p$ doesn't bound any nontrivial strong antichain). Then there is a $mathbb{P}$-generic filter in the ground model already.




The proof is simple: let $G$ be the set of all conditions compatible with $p$.



Now suppose $mathbb{P}$ is a poset where every strong antichain is finite. I claim that $mathbb{P}$ has such a "trivializing" element $p$. For if not, we can inductively define a map $t$ from $2^{<omega}$ to $mathbb{P}$ such that:




  • $sigmaprectauimplies t(sigma)ge t(tau)$.


  • $t(sigma0)perp t(sigma1)$.



But then the set $${t(0), t(10), t(110), t(1110), ...}$$ forms an infinite strong antichain in $mathbb{P}$.



In fact, we can do even better (since having only finite strong antichains is preserved by passing from $mathbb{P}$ to $mathbb{P}_{le s}$):




If $mathbb{P}$ has only finite strong antichains, then the set of "trivializing" $p$ is dense in $mathbb{P}$; so every $mathbb{P}$-generic filter is already in the ground model. (And consequently, we have "${bf MA(aleph_0,infty)}$.")




So such a poset really is trivial (in the sense of forcing), not just "possibly trivial," as is the corresponding variant of Martin's Axiom.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited 2 days ago

























answered 2 days ago









Noah Schweber

122k10148284




122k10148284












  • Thanks. It was my intuition that the problem would trivilize like that, but I hadn't yet matured the subject enough to prove it.
    – Emilio Martinez
    2 days ago


















  • Thanks. It was my intuition that the problem would trivilize like that, but I hadn't yet matured the subject enough to prove it.
    – Emilio Martinez
    2 days ago
















Thanks. It was my intuition that the problem would trivilize like that, but I hadn't yet matured the subject enough to prove it.
– Emilio Martinez
2 days ago




Thanks. It was my intuition that the problem would trivilize like that, but I hadn't yet matured the subject enough to prove it.
– Emilio Martinez
2 days ago


















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