Proving $alpha+beta=sup{alpha+beta_{delta}:delta<gamma}$
For ordinals $alpha$, $beta$, $gamma$, if $gamma$ is a limit ordinal and $beta = sup{beta_{delta}:delta<gamma}$, why does below expression hold,
$$alpha+beta=alpha + sup{beta_{delta}:delta<gamma}=sup{alpha+beta_{delta}:delta<gamma}$$
Simply what I am asking is why,
$$alpha+beta=sup{alpha+beta_{delta}:delta<gamma}$$
holds? I tried to prove it by considering another $sup$ like $z$ then trying to prove that the $sup$ we've found is less that $z$. I don't know why and where in proof $gamma$ is limit ordinal is required?!
Edit I simply want to prove continuity property of ordinal addition. There was a question asked about what continuity of ordinals is here.
elementary-set-theory ordinals
edited Dec 13 '18 at 18:19
Andrés E. Caicedo
64.8k8158246
asked Dec 11 '18 at 5:36
FreeMind
9121133
add a comment |
For ordinals $alpha$, $beta$, $gamma$, if $gamma$ is a limit ordinal and $beta = sup{beta_{delta}:delta<gamma}$, why does below expression hold,
$$alpha+beta=alpha + sup{beta_{delta}:delta<gamma}=sup{alpha+beta_{delta}:delta<gamma}$$
Simply what I am asking is why,
$$alpha+beta=sup{alpha+beta_{delta}:delta<gamma}$$
holds? I tried to prove it by considering another $sup$ like $z$ then trying to prove that the $sup$ we've found is less that $z$. I don't know why and where in proof $gamma$ is limit ordinal is required?!
Edit I simply want to prove continuity property of ordinal addition. There was a question asked about what continuity of ordinals is here.
elementary-set-theory ordinals
edited Dec 13 '18 at 18:19
Andrés E. Caicedo
64.8k8158246
asked Dec 11 '18 at 5:36
FreeMind
9121133
What is $beta_delta$?
– Henno Brandsma
Dec 11 '18 at 5:37
What does hold for $gamma$ a limit, is $alpha + gamma = sup {alpha + beta: beta < gamma}$. Maybe that's what you mean? For some this is (part of) the definition of addition.
– Henno Brandsma
Dec 11 '18 at 5:39
@HennoBrandsma Yes, both say the same thing.
– FreeMind
Dec 11 '18 at 5:52
No they don’t say the same thing. But what’s your definition of addition?
– Henno Brandsma
Dec 11 '18 at 6:33
@HennoBrandsma My reference is Jech book
– FreeMind
Dec 11 '18 at 16:29
add a comment |
For ordinals $alpha$, $beta$, $gamma$, if $gamma$ is a limit ordinal and $beta = sup{beta_{delta}:delta<gamma}$, why does below expression hold,
$$alpha+beta=alpha + sup{beta_{delta}:delta<gamma}=sup{alpha+beta_{delta}:delta<gamma}$$
Simply what I am asking is why,
$$alpha+beta=sup{alpha+beta_{delta}:delta<gamma}$$
holds? I tried to prove it by considering another $sup$ like $z$ then trying to prove that the $sup$ we've found is less that $z$. I don't know why and where in proof $gamma$ is limit ordinal is required?!
Edit I simply want to prove continuity property of ordinal addition. There was a question asked about what continuity of ordinals is here.
elementary-set-theory ordinals
edited Dec 13 '18 at 18:19
Andrés E. Caicedo
64.8k8158246
asked Dec 11 '18 at 5:36
FreeMind
9121133
For ordinals $alpha$, $beta$, $gamma$, if $gamma$ is a limit ordinal and $beta = sup{beta_{delta}:delta<gamma}$, why does below expression hold,
$$alpha+beta=alpha + sup{beta_{delta}:delta<gamma}=sup{alpha+beta_{delta}:delta<gamma}$$
Simply what I am asking is why,
$$alpha+beta=sup{alpha+beta_{delta}:delta<gamma}$$
holds? I tried to prove it by considering another $sup$ like $z$ then trying to prove that the $sup$ we've found is less that $z$. I don't know why and where in proof $gamma$ is limit ordinal is required?!
Edit I simply want to prove continuity property of ordinal addition. There was a question asked about what continuity of ordinals is here.
elementary-set-theory ordinals
elementary-set-theory ordinals
edited Dec 13 '18 at 18:19
Andrés E. Caicedo
64.8k8158246
asked Dec 11 '18 at 5:36
FreeMind
9121133
edited Dec 13 '18 at 18:19
Andrés E. Caicedo
64.8k8158246
asked Dec 11 '18 at 5:36
FreeMind
9121133
edited Dec 13 '18 at 18:19
Andrés E. Caicedo
64.8k8158246
edited Dec 13 '18 at 18:19
Andrés E. Caicedo
64.8k8158246
edited Dec 13 '18 at 18:19
Andrés E. Caicedo
64.8k8158246
64.8k8158246
asked Dec 11 '18 at 5:36
FreeMind
9121133
asked Dec 11 '18 at 5:36
FreeMind
9121133
asked Dec 11 '18 at 5:36
FreeMind
9121133
9121133
What is $beta_delta$?
– Henno Brandsma
Dec 11 '18 at 5:37
What does hold for $gamma$ a limit, is $alpha + gamma = sup {alpha + beta: beta < gamma}$. Maybe that's what you mean? For some this is (part of) the definition of addition.
– Henno Brandsma
Dec 11 '18 at 5:39
@HennoBrandsma Yes, both say the same thing.
– FreeMind
Dec 11 '18 at 5:52
No they don’t say the same thing. But what’s your definition of addition?
– Henno Brandsma
Dec 11 '18 at 6:33
@HennoBrandsma My reference is Jech book
– FreeMind
Dec 11 '18 at 16:29
add a comment |
What is $beta_delta$?
– Henno Brandsma
Dec 11 '18 at 5:37
What does hold for $gamma$ a limit, is $alpha + gamma = sup {alpha + beta: beta < gamma}$. Maybe that's what you mean? For some this is (part of) the definition of addition.
– Henno Brandsma
Dec 11 '18 at 5:39
@HennoBrandsma Yes, both say the same thing.
– FreeMind
Dec 11 '18 at 5:52
No they don’t say the same thing. But what’s your definition of addition?
– Henno Brandsma
Dec 11 '18 at 6:33
@HennoBrandsma My reference is Jech book
– FreeMind
Dec 11 '18 at 16:29
What is $beta_delta$?
– Henno Brandsma
Dec 11 '18 at 5:37
What is $beta_delta$?
– Henno Brandsma
Dec 11 '18 at 5:37
What does hold for $gamma$ a limit, is $alpha + gamma = sup {alpha + beta: beta < gamma}$. Maybe that's what you mean? For some this is (part of) the definition of addition.
– Henno Brandsma
Dec 11 '18 at 5:39
What does hold for $gamma$ a limit, is $alpha + gamma = sup {alpha + beta: beta < gamma}$. Maybe that's what you mean? For some this is (part of) the definition of addition.
– Henno Brandsma
Dec 11 '18 at 5:39
@HennoBrandsma Yes, both say the same thing.
– FreeMind
Dec 11 '18 at 5:52
@HennoBrandsma Yes, both say the same thing.
– FreeMind
Dec 11 '18 at 5:52
No they don’t say the same thing. But what’s your definition of addition?
– Henno Brandsma
Dec 11 '18 at 6:33
No they don’t say the same thing. But what’s your definition of addition?
– Henno Brandsma
Dec 11 '18 at 6:33
@HennoBrandsma My reference is Jech book
– FreeMind
Dec 11 '18 at 16:29
@HennoBrandsma My reference is Jech book
– FreeMind
Dec 11 '18 at 16:29
add a comment |
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What is $beta_delta$?
– Henno Brandsma
Dec 11 '18 at 5:37
What does hold for $gamma$ a limit, is $alpha + gamma = sup {alpha + beta: beta < gamma}$. Maybe that's what you mean? For some this is (part of) the definition of addition.
– Henno Brandsma
Dec 11 '18 at 5:39
@HennoBrandsma Yes, both say the same thing.
– FreeMind
Dec 11 '18 at 5:52
No they don’t say the same thing. But what’s your definition of addition?
– Henno Brandsma
Dec 11 '18 at 6:33
@HennoBrandsma My reference is Jech book
– FreeMind
Dec 11 '18 at 16:29