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Few days I asked this question (https://math.stackexchange.com/questions/2907733/simple-ordinal-question) on MSE. Summary of the question is that I defined a certain function over ordinals $x \mapsto \beta_x$ (where $x<\omega_1$) and asked about its relation to the function $x \mapsto \omega^{CK}_x$. And it seems they are "nearly" the same (I mean the values, not the definitions) with the former function being continuous while the latter function is discontinuous (it does obey continuity condition on some limit values). In particular, $\omega^{CK}_1=\beta_0$ and $\omega^{CK}_2=\beta_1$.

Since asking that question, another question has come to my mind. Before getting into detail I will just briefly state the statement of question: "Can we rigorously define an ordinal $\gamma$ which (intuitively) is to $\omega_1$ what $\omega^{CK}_2$ is to $\omega^{CK}_1$?"

Here is the main motivation for the question. The (fictional) idea is that suppose completely "fictitiously" that we indeed have some well-order (of $\mathbb{N}$) with order-type $\omega_1$. Now suppose someone asked me whether this given $\gamma$ (which we are trying to describe rigorously) is greater than say $\omega{_1}^{\omega_1}$? My answer would be yes. And my "reasoning" would be that if somehow one just had access to the fictitious well-order just described, then using that hypothetical orcale function there also exists an ordinary program that describes the well-order (of $\mathbb{N}$) with order-type $\omega{_1}^{\omega_1}$. By the same reasoning $\gamma$ has to be greater than, say, the first fixed point of $x \mapsto (\omega_1)^x$. And so on...

I hope the drift of the question is clear at this point. The question is that what would be a reasonable rigorous definition of $\gamma$ that also corresponds well with this intuition.

I also have a related side question. If we consider an ORM then it seems it would easily go beyond points such as $\beta_0$,$\beta_1$ or $\beta_{\omega_{CK}}$ etc. and probably also points far beyond it (I don't know the exact limit). And I suppose one of reasons is that it could easily decide whether a given (ordinary) program (with some oracle possibly) codes a well-order or not. Because corresponding to every well-order (of $\mathbb{N}$) with a certain order-type one can describe a "tree of all possible descents". The most important property of such a tree would be that it wouldn't contain any infinite path/branch. It seems to me that if one just narrows a few more properties for this "descent tree" well-enough, one can make them correspond exactly with well-orders (of $\mathbb{N}$). [I haven't thought about this in a fully thorough way so I would be happy to be corrected if I have made some mistake in this paragraph.]

Now my side question is that if we add an extra instruction of the form to $u=u+\omega_1$ (where $u$ is a variable) to the definition of ORMs, then what is the smallest point $p$ that such programs can't reach? My main reason for asking this side question it seems to me that $p \ge \gamma$ ($\gamma$ being the answer to the first part of this question). It also seems that luxury of using well-orders of $\mathbb{N}$ (described in previous paragraph) is unavailable to these programs.

My side question is "Is the inequality $p \ge \gamma$ strict or not?"

Few days I asked this question (https://math.stackexchange.com/questions/2907733/simple-ordinal-question) on MSE. Summary of the question is that I defined a certain function over ordinals $x \mapsto \beta_x$ (where $x<\omega_1$) and asked about its relation to the function $x \mapsto \omega^{CK}_x$. And it seems they are "nearly" the same (I mean the values, not the definitions) with the former function being continuous while the latter function is discontinuous (it does obey continuity condition on some limit values). In particular, $\omega^{CK}_1=\beta_0$ and $\omega^{CK}_2=\beta_1$.

Since asking that question, another question has come to my mind. Before getting into detail I will just briefly state the statement of question: "Can we rigorously define an ordinal $\gamma$ which (intuitively) is to $\omega_1$ what $\omega^{CK}_2$ is to $\omega^{CK}_1$?"

Here is the main motivation for the question. The (fictional) idea is that suppose completely "fictitiously" that we indeed have some well-order (of $\mathbb{N}$) with order-type $\omega_1$. Now suppose someone asked me whether this given $\gamma$ (which we are trying to describe rigorously) is greater than say $\omega{_1}^{\omega_1}$? My answer would be yes. And my "reasoning" would be that if somehow one just had access to the fictitious well-order just described, then using that hypothetical orcale function there also exists an ordinary program that describes the well-order (of $\mathbb{N}$) with order-type $\omega{_1}^{\omega_1}$. By the same reasoning $\gamma$ has to be greater than, say, the first fixed point of $x \mapsto (\omega_1)^x$. And so on...

I hope the drift of the question is clear at this point. The question is that what would be a reasonable rigorous definition of $\gamma$ that also corresponds well with this intuition.

Edit: Based upon suggestion in comments, I posted the side question separately: Relation of $\omega_{\omega_1+1}^{CK}$ to some other ordinals

Few days I asked this question (https://math.stackexchange.com/questions/2907733/simple-ordinal-question) on MSE. Summary of the question is that I defined a certain function over ordinals $x \mapsto \beta_x$ (where $x<\omega_1$) and asked about its relation to the function $x \mapsto \omega^{CK}_x$. And it seems they are "nearly" the same (I mean the values, not the definitions) with the former functions being continuous while the latter function is discontinuous (except obeying continuity condition on some limit values). In particular, $\omega^{CK}_1=\beta_0$ and $\omega^{CK}_2=\beta_1$.

Since asking that question, another question has come to my mind. Before getting into detail I will just briefly state the statement of question: "Can we rigorously define an ordinal $\gamma$ which (intuitively) is to $\omega_1$ what $\omega^{CK}_2$ is to $\omega^{CK}_1$?"

Here is the main motivation for the question. The (fictional) idea is that suppose completely "fictitiously" that we indeed have some well-order (of $\mathbb{N}$) with order-type $\omega_1$. Now suppose someone asked me whether this given $\gamma$ (which we are trying to describe rigorously) is greater than say $\omega{_1}^{\omega_1}$? My answer would be yes. And my "reasoning" would be that if somehow one just had access to the fictitious well-order just described, then using that hypothetical orcale function there also exists an ordinary program that describes the well-order (of $\mathbb{N}$) with order-type $\omega{_1}^{\omega_1}$. By the same reasoning $\gamma$ has to be greater than, say, the first fixed point of $x \mapsto (\omega_1)^x$. And so on...

I hope the drift of the question is clear at this point. The question is that what would be a reasonable rigorous definition of $\gamma$ that also corresponds well with this intuition.

I also have a related side question. If we consider an ORM then it seems it would easily go beyond points such as $\beta_0$,$\beta_1$ or $\beta_{\omega_{CK}}$ etc. and probably also points far beyond it (I don't know the exact limit). And I suppose one of reasons is that it could easily decide whether a given (ordinary) program (with some oracle possibly) codes a well-order or not. Because corresponding to every well-order (of $\mathbb{N}$) with a certain order-type one can describe a "tree of all possible descents". The most important property of such a tree would be that it wouldn't contain any infinite path/branch. It seems to me that if one just narrows a few more properties for this "descent tree" well-enough, one can make them correspond exactly with well-orders (of $\mathbb{N}$). [I haven't thought about this in a fully thorough way so I would be happy to be corrected if I have made some mistake in this paragraph.]

Now my side question is that if we add an extra instruction of the form to $u=u+\omega_1$ (where $u$ is a variable) to the definition of ORMs, then what is the smallest point $p$ that such programs can't reach? My main reason for asking this side question it seems to me that $p \ge \gamma$ ($\gamma$ being the answer to the first part of this question). It also seems that luxury of using well-orders of $\mathbb{N}$ (described in previous paragraph) is unavailable to these programs.

My side question is "Is the inequality $p \ge \gamma$ strict or not?"

Few days I asked this question (https://math.stackexchange.com/questions/2907733/simple-ordinal-question) on MSE. Summary of the question is that I defined a certain function over ordinals $x \mapsto \beta_x$ (where $x<\omega_1$) and asked about its relation to the function $x \mapsto \omega^{CK}_x$. And it seems they are "nearly" the same (I mean the values, not the definitions) with the former function being continuous while the latter function is discontinuous (it does obey continuity condition on some limit values). In particular, $\omega^{CK}_1=\beta_0$ and $\omega^{CK}_2=\beta_1$.

Since asking that question, another question has come to my mind. Before getting into detail I will just briefly state the statement of question: "Can we rigorously define an ordinal $\gamma$ which (intuitively) is to $\omega_1$ what $\omega^{CK}_2$ is to $\omega^{CK}_1$?"

Here is the main motivation for the question. The (fictional) idea is that suppose completely "fictitiously" that we indeed have some well-order (of $\mathbb{N}$) with order-type $\omega_1$. Now suppose someone asked me whether this given $\gamma$ (which we are trying to describe rigorously) is greater than say $\omega{_1}^{\omega_1}$? My answer would be yes. And my "reasoning" would be that if somehow one just had access to the fictitious well-order just described, then using that hypothetical orcale function there also exists an ordinary program that describes the well-order (of $\mathbb{N}$) with order-type $\omega{_1}^{\omega_1}$. By the same reasoning $\gamma$ has to be greater than, say, the first fixed point of $x \mapsto (\omega_1)^x$. And so on...

I hope the drift of the question is clear at this point. The question is that what would be a reasonable rigorous definition of $\gamma$ that also corresponds well with this intuition.

I also have a related side question. If we consider an ORM then it seems it would easily go beyond points such as $\beta_0$,$\beta_1$ or $\beta_{\omega_{CK}}$ etc. and probably also points far beyond it (I don't know the exact limit). And I suppose one of reasons is that it could easily decide whether a given (ordinary) program (with some oracle possibly) codes a well-order or not. Because corresponding to every well-order (of $\mathbb{N}$) with a certain order-type one can describe a "tree of all possible descents". The most important property of such a tree would be that it wouldn't contain any infinite path/branch. It seems to me that if one just narrows a few more properties for this "descent tree" well-enough, one can make them correspond exactly with well-orders (of $\mathbb{N}$). [I haven't thought about this in a fully thorough way so I would be happy to be corrected if I have made some mistake in this paragraph.]

Now my side question is that if we add an extra instruction of the form to $u=u+\omega_1$ (where $u$ is a variable) to the definition of ORMs, then what is the smallest point $p$ that such programs can't reach? My main reason for asking this side question it seems to me that $p \ge \gamma$ ($\gamma$ being the answer to the first part of this question). It also seems that luxury of using well-orders of $\mathbb{N}$ (described in previous paragraph) is unavailable to these programs.

My side question is "Is the inequality $p \ge \gamma$ strict or not?"

Few days I asked this question (https://math.stackexchange.com/questions/2907733/simple-ordinal-question) on MSE. Summary of the question is that I defined a certain function over ordinals $x \mapsto \beta_x$ (where $x<\omega_1$) and asked about its relation to the function $x \mapsto \omega^{CK}_x$. And it seems they are "nearly" the same (I mean the values, not the definitions) with the former functions being continuous while the latter function is discontinuous (except obeying continuity condition on some limit values). In particular, $\omega^{CK}_1=\beta_0$ and $\omega^{CK}_2=\beta_1$.

Since asking that question, another question has come to my mind. Before getting into detail I will just briefly state the statement of question: "Can we rigorously define an ordinal $\gamma$ which (intuitively) is to $\omega_1$ what $\omega^{CK}_2$ is to $\omega^{CK}_1$?"

Here is the main motivation for the question. The (fictional) idea is that suppose completely "fictitiously" that we indeed have some well-order (of $\mathbb{N}$) with order-type $\omega_1$. Now suppose someone asked me whether this given $\gamma$ (which we are trying to describe rigorously) is greater than say $\omega{_1}^{\omega_1}$? My answer would be yes. And my "reasoning" would be that if somehow one just had access to the fictitious well-order just described, then using that hypothetical orcale function there also exists a program that describes the well-order (of $\mathbb{N}$) with order-type $\omega{_1}^{\omega_1}$. By the same reasoning $\gamma$ has to be greater than, say, the first fixed point of $x \mapsto (\omega_1)^x$. And so on...

I hope the drift of the question is clear at this point. The question is that what would be a reasonable rigorous definition of $\gamma$ that also corresponds well with this intuition.

I also have a related side question. If we consider an ORM then it seems it would easily go beyond points such as $\beta_0$,$\beta_1$ or $\beta_{\omega_{CK}}$ etc. and probably also points far beyond it (I don't know the exact limit). And I suppose one of reasons is that it could easily decide whether a given (ordinary) program (with some oracle possibly) codes a well-order or not. Because corresponding to every well-order (of $\mathbb{N}$) with a certain order-type one can describe a "tree of all possible descents". The most important property of such a tree would be that it wouldn't contain any infinite path/branch. It seems to me that if one just narrows a few more properties for this "descent tree" well-enough, one can make them correspond exactly with well-orders (of $\mathbb{N}$). [I haven't thought about this in a fully thorough way so I would be happy to be corrected if I have made some mistake in this paragraph.]

Now my side question is that if we add an extra instruction of the form to $u=u+\omega_1$ (where $u$ is a variable) to the definition of ORMs, then what is the smallest point $p$ that such programs can't reach? My main reason for asking this side question it seems to me that $p \ge \gamma$ ($\gamma$ being the answer to the first part of this question). It also seems that luxury of using well-orders of $\mathbb{N}$ (described in previous paragraph) is unavailable to these programs.

My side question is "Is the inequality $p \ge \gamma$ strict or not?"

Few days I asked this question (https://math.stackexchange.com/questions/2907733/simple-ordinal-question) on MSE. Summary of the question is that I defined a certain function over ordinals $x \mapsto \beta_x$ (where $x<\omega_1$) and asked about its relation to the function $x \mapsto \omega^{CK}_x$. And it seems they are "nearly" the same (I mean the values, not the definitions) with the former functions being continuous while the latter function is discontinuous (except obeying continuity condition on some limit values). In particular, $\omega^{CK}_1=\beta_0$ and $\omega^{CK}_2=\beta_1$.

Since asking that question, another question has come to my mind. Before getting into detail I will just briefly state the statement of question: "Can we rigorously define an ordinal $\gamma$ which (intuitively) is to $\omega_1$ what $\omega^{CK}_2$ is to $\omega^{CK}_1$?"

Here is the main motivation for the question. The (fictional) idea is that suppose completely "fictitiously" that we indeed have some well-order (of $\mathbb{N}$) with order-type $\omega_1$. Now suppose someone asked me whether this given $\gamma$ (which we are trying to describe rigorously) is greater than say $\omega{_1}^{\omega_1}$? My answer would be yes. And my "reasoning" would be that if somehow one just had access to the fictitious well-order just described, then using that hypothetical orcale function there also exists an ordinary program that describes the well-order (of $\mathbb{N}$) with order-type $\omega{_1}^{\omega_1}$. By the same reasoning $\gamma$ has to be greater than, say, the first fixed point of $x \mapsto (\omega_1)^x$. And so on...

I hope the drift of the question is clear at this point. The question is that what would be a reasonable rigorous definition of $\gamma$ that also corresponds well with this intuition.

I also have a related side question. If we consider an ORM then it seems it would easily go beyond points such as $\beta_0$,$\beta_1$ or $\beta_{\omega_{CK}}$ etc. and probably also points far beyond it (I don't know the exact limit). And I suppose one of reasons is that it could easily decide whether a given (ordinary) program (with some oracle possibly) codes a well-order or not. Because corresponding to every well-order (of $\mathbb{N}$) with a certain order-type one can describe a "tree of all possible descents". The most important property of such a tree would be that it wouldn't contain any infinite path/branch. It seems to me that if one just narrows a few more properties for this "descent tree" well-enough, one can make them correspond exactly with well-orders (of $\mathbb{N}$). [I haven't thought about this in a fully thorough way so I would be happy to be corrected if I have made some mistake in this paragraph.]

Now my side question is that if we add an extra instruction of the form to $u=u+\omega_1$ (where $u$ is a variable) to the definition of ORMs, then what is the smallest point $p$ that such programs can't reach? My main reason for asking this side question it seems to me that $p \ge \gamma$ ($\gamma$ being the answer to the first part of this question). It also seems that luxury of using well-orders of $\mathbb{N}$ (described in previous paragraph) is unavailable to these programs.

My side question is "Is the inequality $p \ge \gamma$ strict or not?"