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Todd Eisworth
Todd Eisworth
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In his book [1], Paul Larson remarks (Remark 1.1.22) that in L there is a function $h:\omega_1\rightarrow\omega_1$ such that for any countable elementary submodel $X$ of $V_\gamma$ (where $\gamma$ is the first strong limit cardinal), we have the order-type of $X\cap Ord$ is strictly less than $h(X\cap\omega_1)$.

What is known about the relationship between the existence of such a function and large cardinal phenomena?

Larson remarks that the existence of such a function is consistent with many large cardinals, and later in the book sketches a result of Velickovic showing that no such function exists in the presence of a precipitous ideal on $\omega_1$. What more is known? Happy to also learn results concerning other related functions, or be pointed to associated references.

[1] Larson, Paul B., The stationary tower. Notes on a course by W. Hugh Woodin, University Lecture Series 32. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3604-8/pbk). x, 132 p. (2004). ZBL1072.03031.

In his book [1], Paul Larson remarks (Remark 1.1.22) that in L there is a function $h:\omega_1\rightarrow\omega_1$ such that for any elementary submodel $X$ of $V_\gamma$ (where $\gamma$ is the first strong limit cardinal), we have the order-type of $X\cap Ord$ is strictly less than $h(X\cap\omega_1)$.

What is known about the relationship between the existence of such a function and large cardinal phenomena?

Larson remarks that the existence of such a function is consistent with many large cardinals, and later in the book sketches a result of Velickovic showing that no such function exists in the presence of a precipitous ideal on $\omega_1$. What more is known? Happy to also learn results concerning other related functions, or be pointed to associated references.

[1] Larson, Paul B., The stationary tower. Notes on a course by W. Hugh Woodin, University Lecture Series 32. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3604-8/pbk). x, 132 p. (2004). ZBL1072.03031.

In his book [1], Paul Larson remarks (Remark 1.1.22) that in L there is a function $h:\omega_1\rightarrow\omega_1$ such that for any countable elementary submodel $X$ of $V_\gamma$ (where $\gamma$ is the first strong limit cardinal), we have the order-type of $X\cap Ord$ is strictly less than $h(X\cap\omega_1)$.

What is known about the relationship between the existence of such a function and large cardinal phenomena?

Larson remarks that the existence of such a function is consistent with many large cardinals, and later in the book sketches a result of Velickovic showing that no such function exists in the presence of a precipitous ideal on $\omega_1$. What more is known? Happy to also learn results concerning other related functions, or be pointed to associated references.

[1] Larson, Paul B., The stationary tower. Notes on a course by W. Hugh Woodin, University Lecture Series 32. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3604-8/pbk). x, 132 p. (2004). ZBL1072.03031.

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Todd Eisworth
Todd Eisworth
  • 7.1k
  • 1
  • 23
  • 47

Consequences of existence of a certain function from $\omega_1$ to $\omega_1$

In his book [1], Paul Larson remarks (Remark 1.1.22) that in L there is a function $h:\omega_1\rightarrow\omega_1$ such that for any elementary submodel $X$ of $V_\gamma$ (where $\gamma$ is the first strong limit cardinal), we have the order-type of $X\cap Ord$ is strictly less than $h(X\cap\omega_1)$.

What is known about the relationship between the existence of such a function and large cardinal phenomena?

Larson remarks that the existence of such a function is consistent with many large cardinals, and later in the book sketches a result of Velickovic showing that no such function exists in the presence of a precipitous ideal on $\omega_1$. What more is known? Happy to also learn results concerning other related functions, or be pointed to associated references.

[1] Larson, Paul B., The stationary tower. Notes on a course by W. Hugh Woodin, University Lecture Series 32. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3604-8/pbk). x, 132 p. (2004). ZBL1072.03031.