Let $α$ and $β$ be ordinals. Let $(η_{ζ})_{ζinβ}$ be a $β$-sequence of ordinals. Then...
$begingroup$
Let $alpha$ and $beta$ be ordinals. Let $left(eta_{zeta}right)_{zetainbeta}$ be a sequence of ordinals indexed by $beta$. Define the sum $sum_{zetainbeta}eta_{zeta}$ as follows. Let $Z=bigcup_{zetainbeta}left(eta_{zeta}timesleft{ zetaright} right)$ and define a relation $sqsubseteq$ on $Z$ as follows. For all $zeta,zeta'inbeta$, $thetainalpha_{zeta}$, and $theta'inalpha_{zeta'}$, $left(theta,zetaright)sqsubseteqleft(theta',zeta'right)$ if and only if either $zeta<zeta'$ or $zeta=zeta'$ and $thetaleqtheta'$. It is easy to see that $sqsubseteq$ is a well-order. We define $sum_{zetainbeta}eta_{zeta}$ to be the unique ordinal which is order isomorphic to $left(Z,sqsubseteqright)$. This is the generalization of regular ordinal addition $+$.
Let $times$ denote ordinal multiplication. I have already proven that $times$ distributes over ordinal addition $+$, and now I am trying to prove this for a more general case. I want to prove that $$alphatimesleft(sum_{zetainbeta}eta_{zeta}right)=sum_{zetainbeta}left(alphatimeseta_{zeta}right).$$I am pretty sure that this result is true, but I am having a hard time proving it. I've tried proving this by induction on $beta$, but I am unable to show this for the limit ordinal step. Any ideas?
set-theory ordinals
asked Dec 5 '18 at 16:31
EigenfieldEigenfield
628518
$endgroup$
add a comment |
$begingroup$
Let $alpha$ and $beta$ be ordinals. Let $left(eta_{zeta}right)_{zetainbeta}$ be a sequence of ordinals indexed by $beta$. Define the sum $sum_{zetainbeta}eta_{zeta}$ as follows. Let $Z=bigcup_{zetainbeta}left(eta_{zeta}timesleft{ zetaright} right)$ and define a relation $sqsubseteq$ on $Z$ as follows. For all $zeta,zeta'inbeta$, $thetainalpha_{zeta}$, and $theta'inalpha_{zeta'}$, $left(theta,zetaright)sqsubseteqleft(theta',zeta'right)$ if and only if either $zeta<zeta'$ or $zeta=zeta'$ and $thetaleqtheta'$. It is easy to see that $sqsubseteq$ is a well-order. We define $sum_{zetainbeta}eta_{zeta}$ to be the unique ordinal which is order isomorphic to $left(Z,sqsubseteqright)$. This is the generalization of regular ordinal addition $+$.
Let $times$ denote ordinal multiplication. I have already proven that $times$ distributes over ordinal addition $+$, and now I am trying to prove this for a more general case. I want to prove that $$alphatimesleft(sum_{zetainbeta}eta_{zeta}right)=sum_{zetainbeta}left(alphatimeseta_{zeta}right).$$I am pretty sure that this result is true, but I am having a hard time proving it. I've tried proving this by induction on $beta$, but I am unable to show this for the limit ordinal step. Any ideas?
set-theory ordinals
asked Dec 5 '18 at 16:31
EigenfieldEigenfield
628518
$endgroup$
$begingroup$
Did you try to create an order isomorphism? What obstacles did you encounter?
$endgroup$
– Alberto Takase
Dec 5 '18 at 18:06
add a comment |
$begingroup$
Let $alpha$ and $beta$ be ordinals. Let $left(eta_{zeta}right)_{zetainbeta}$ be a sequence of ordinals indexed by $beta$. Define the sum $sum_{zetainbeta}eta_{zeta}$ as follows. Let $Z=bigcup_{zetainbeta}left(eta_{zeta}timesleft{ zetaright} right)$ and define a relation $sqsubseteq$ on $Z$ as follows. For all $zeta,zeta'inbeta$, $thetainalpha_{zeta}$, and $theta'inalpha_{zeta'}$, $left(theta,zetaright)sqsubseteqleft(theta',zeta'right)$ if and only if either $zeta<zeta'$ or $zeta=zeta'$ and $thetaleqtheta'$. It is easy to see that $sqsubseteq$ is a well-order. We define $sum_{zetainbeta}eta_{zeta}$ to be the unique ordinal which is order isomorphic to $left(Z,sqsubseteqright)$. This is the generalization of regular ordinal addition $+$.
Let $times$ denote ordinal multiplication. I have already proven that $times$ distributes over ordinal addition $+$, and now I am trying to prove this for a more general case. I want to prove that $$alphatimesleft(sum_{zetainbeta}eta_{zeta}right)=sum_{zetainbeta}left(alphatimeseta_{zeta}right).$$I am pretty sure that this result is true, but I am having a hard time proving it. I've tried proving this by induction on $beta$, but I am unable to show this for the limit ordinal step. Any ideas?
set-theory ordinals
asked Dec 5 '18 at 16:31
EigenfieldEigenfield
628518
$endgroup$
Let $alpha$ and $beta$ be ordinals. Let $left(eta_{zeta}right)_{zetainbeta}$ be a sequence of ordinals indexed by $beta$. Define the sum $sum_{zetainbeta}eta_{zeta}$ as follows. Let $Z=bigcup_{zetainbeta}left(eta_{zeta}timesleft{ zetaright} right)$ and define a relation $sqsubseteq$ on $Z$ as follows. For all $zeta,zeta'inbeta$, $thetainalpha_{zeta}$, and $theta'inalpha_{zeta'}$, $left(theta,zetaright)sqsubseteqleft(theta',zeta'right)$ if and only if either $zeta<zeta'$ or $zeta=zeta'$ and $thetaleqtheta'$. It is easy to see that $sqsubseteq$ is a well-order. We define $sum_{zetainbeta}eta_{zeta}$ to be the unique ordinal which is order isomorphic to $left(Z,sqsubseteqright)$. This is the generalization of regular ordinal addition $+$.
Let $times$ denote ordinal multiplication. I have already proven that $times$ distributes over ordinal addition $+$, and now I am trying to prove this for a more general case. I want to prove that $$alphatimesleft(sum_{zetainbeta}eta_{zeta}right)=sum_{zetainbeta}left(alphatimeseta_{zeta}right).$$I am pretty sure that this result is true, but I am having a hard time proving it. I've tried proving this by induction on $beta$, but I am unable to show this for the limit ordinal step. Any ideas?
set-theory ordinals
set-theory ordinals
asked Dec 5 '18 at 16:31
EigenfieldEigenfield
628518
asked Dec 5 '18 at 16:31
EigenfieldEigenfield
628518
asked Dec 5 '18 at 16:31
EigenfieldEigenfield
628518
asked Dec 5 '18 at 16:31
EigenfieldEigenfield
628518
asked Dec 5 '18 at 16:31
EigenfieldEigenfield
628518
628518
$begingroup$
Did you try to create an order isomorphism? What obstacles did you encounter?
$endgroup$
– Alberto Takase
Dec 5 '18 at 18:06
add a comment |
$begingroup$
Did you try to create an order isomorphism? What obstacles did you encounter?
$endgroup$
– Alberto Takase
Dec 5 '18 at 18:06
$begingroup$
Did you try to create an order isomorphism? What obstacles did you encounter?
$endgroup$
– Alberto Takase
Dec 5 '18 at 18:06
$begingroup$
Did you try to create an order isomorphism? What obstacles did you encounter?
$endgroup$
– Alberto Takase
Dec 5 '18 at 18:06
add a comment |
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$begingroup$
Did you try to create an order isomorphism? What obstacles did you encounter?
$endgroup$
– Alberto Takase
Dec 5 '18 at 18:06