Proving $alpha+beta=sup{alpha+beta_{delta}:delta<gamma}$
$begingroup$
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
65.8k8160251
asked Dec 11 '18 at 5:36
FreeMindFreeMind
9381133
$endgroup$
add a comment |
$begingroup$
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
65.8k8160251
asked Dec 11 '18 at 5:36
FreeMindFreeMind
9381133
$endgroup$
$begingroup$
What is $beta_delta$?
$endgroup$
– Henno Brandsma
Dec 11 '18 at 5:37
$begingroup$
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.
$endgroup$
– Henno Brandsma
Dec 11 '18 at 5:39
$begingroup$
@HennoBrandsma Yes, both say the same thing.
$endgroup$
– FreeMind
Dec 11 '18 at 5:52
$begingroup$
No they don’t say the same thing. But what’s your definition of addition?
$endgroup$
– Henno Brandsma
Dec 11 '18 at 6:33
$begingroup$
@HennoBrandsma My reference is Jech book
$endgroup$
– FreeMind
Dec 11 '18 at 16:29
add a comment |
$begingroup$
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
65.8k8160251
asked Dec 11 '18 at 5:36
FreeMindFreeMind
9381133
$endgroup$
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
65.8k8160251
asked Dec 11 '18 at 5:36
FreeMindFreeMind
9381133
edited Dec 13 '18 at 18:19
Andrés E. Caicedo
65.8k8160251
asked Dec 11 '18 at 5:36
FreeMindFreeMind
9381133
edited Dec 13 '18 at 18:19
Andrés E. Caicedo
65.8k8160251
edited Dec 13 '18 at 18:19
Andrés E. Caicedo
65.8k8160251
edited Dec 13 '18 at 18:19
Andrés E. Caicedo
65.8k8160251
65.8k8160251
asked Dec 11 '18 at 5:36
FreeMindFreeMind
9381133
asked Dec 11 '18 at 5:36
FreeMindFreeMind
9381133
asked Dec 11 '18 at 5:36
FreeMindFreeMind
9381133
9381133
$begingroup$
What is $beta_delta$?
$endgroup$
– Henno Brandsma
Dec 11 '18 at 5:37
$begingroup$
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.
$endgroup$
– Henno Brandsma
Dec 11 '18 at 5:39
$begingroup$
@HennoBrandsma Yes, both say the same thing.
$endgroup$
– FreeMind
Dec 11 '18 at 5:52
$begingroup$
No they don’t say the same thing. But what’s your definition of addition?
$endgroup$
– Henno Brandsma
Dec 11 '18 at 6:33
$begingroup$
@HennoBrandsma My reference is Jech book
$endgroup$
– FreeMind
Dec 11 '18 at 16:29
add a comment |
$begingroup$
What is $beta_delta$?
$endgroup$
– Henno Brandsma
Dec 11 '18 at 5:37
$begingroup$
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.
$endgroup$
– Henno Brandsma
Dec 11 '18 at 5:39
$begingroup$
@HennoBrandsma Yes, both say the same thing.
$endgroup$
– FreeMind
Dec 11 '18 at 5:52
$begingroup$
No they don’t say the same thing. But what’s your definition of addition?
$endgroup$
– Henno Brandsma
Dec 11 '18 at 6:33
$begingroup$
@HennoBrandsma My reference is Jech book
$endgroup$
– FreeMind
Dec 11 '18 at 16:29
$begingroup$
What is $beta_delta$?
$endgroup$
– Henno Brandsma
Dec 11 '18 at 5:37
$begingroup$
What is $beta_delta$?
$endgroup$
– Henno Brandsma
Dec 11 '18 at 5:37
$begingroup$
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.
$endgroup$
– Henno Brandsma
Dec 11 '18 at 5:39
$begingroup$
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.
$endgroup$
– Henno Brandsma
Dec 11 '18 at 5:39
$begingroup$
@HennoBrandsma Yes, both say the same thing.
$endgroup$
– FreeMind
Dec 11 '18 at 5:52
$begingroup$
@HennoBrandsma Yes, both say the same thing.
$endgroup$
– FreeMind
Dec 11 '18 at 5:52
$begingroup$
No they don’t say the same thing. But what’s your definition of addition?
$endgroup$
– Henno Brandsma
Dec 11 '18 at 6:33
$begingroup$
No they don’t say the same thing. But what’s your definition of addition?
$endgroup$
– Henno Brandsma
Dec 11 '18 at 6:33
$begingroup$
@HennoBrandsma My reference is Jech book
$endgroup$
– FreeMind
Dec 11 '18 at 16:29
$begingroup$
@HennoBrandsma My reference is Jech book
$endgroup$
– FreeMind
Dec 11 '18 at 16:29
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3034935%2fproving-alpha-beta-sup-alpha-beta-delta-delta-gamma%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3034935%2fproving-alpha-beta-sup-alpha-beta-delta-delta-gamma%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
What is $beta_delta$?
$endgroup$
– Henno Brandsma
Dec 11 '18 at 5:37
$begingroup$
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.
$endgroup$
– Henno Brandsma
Dec 11 '18 at 5:39
$begingroup$
@HennoBrandsma Yes, both say the same thing.
$endgroup$
– FreeMind
Dec 11 '18 at 5:52
$begingroup$
No they don’t say the same thing. But what’s your definition of addition?
$endgroup$
– Henno Brandsma
Dec 11 '18 at 6:33
$begingroup$
@HennoBrandsma My reference is Jech book
$endgroup$
– FreeMind
Dec 11 '18 at 16:29