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Trevor Wilson
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Define a pointclass to be:

  • boldface inductive-like if it is $\mathbb{R}$-parameterized, has the scale property, and is closed under $\wedge$, $\vee$, $\forall^\mathbb{R}$, $\exists^\mathbb{R}$, and preimages by continuous functions, and

  • lightface inductive-like if it is $\omega$-parameterized, has the scale property, and is closed under $\wedge$, $\vee$, $\forall^\mathbb{R}$, $\exists^\mathbb{R}$, and preimages by continuous functions.

Given a lightface inductive-like pointclass $\Gamma$ we can define the boldface inductive-like pointclass ${\bf \Gamma} = \bigcup_{x \in \mathbb{R}} \Gamma(x)$ as usual (this does not have much to do with the particulars of "inductive-like.")

My question is, can we do the reverse? That is, given a boldface inductive-like pointclass ${\bf \Gamma}$ can we find a lightface inductive-like pointclass $\Gamma$ such that ${\bf \Gamma} = \bigcup_{x \in \mathbb{R}} \Gamma(x)$? (We wouldn't expect uniqueness; if $\Gamma$ works then so does $\Gamma(z)$ for any real $z$.)

I have heard that Howard Becker proved something like this, but I can't find it and I don't know what kind of pointclasses it was for (probably more general than "inductive-like," which is merely the case I happen to be interested in at the moment.)

For a motivating example, consider the case where $\mathsf{AD}$ holds and $\bf \Gamma$ is the pointclass of $\kappa$-Suslin sets. If $\bf \Gamma$ is closed under $\forall^\mathbb{R}$ then it is boldface inductive-like, but as far as I know there is no canonical choice of lightface pointclass that corresponds to it.

Define a pointclass to be:

  • boldface inductive-like if it is $\mathbb{R}$-parameterized, has the scale property, and is closed under $\wedge$, $\vee$, $\forall^\mathbb{R}$, $\exists^\mathbb{R}$, and preimages by continuous functions, and

  • lightface inductive-like if it is $\omega$-parameterized, has the scale property, and is closed under $\wedge$, $\vee$, $\forall^\mathbb{R}$, $\exists^\mathbb{R}$, and preimages by continuous functions.

Given a lightface inductive-like pointclass $\Gamma$ we can define the boldface inductive-like pointclass ${\bf \Gamma} = \bigcup_{x \in \mathbb{R}} \Gamma(x)$ as usual (this does not have much to do with the particulars of "inductive-like.")

My question is, can we do the reverse? That is, given a boldface inductive-like pointclass ${\bf \Gamma}$ can we find a lightface inductive-like pointclass $\Gamma$ such that ${\bf \Gamma} = \bigcup_{x \in \mathbb{R}} \Gamma(x)$? (We wouldn't expect uniqueness; if $\Gamma$ works then so does $\Gamma(z)$ for any real $z$.)

I have heard that Howard Becker proved something like this, but I can't find it and I don't know what kind of pointclasses it was for (probably more general than "inductive-like," which is merely the case I happen to be interested in at the moment.)

Define a pointclass to be:

  • boldface inductive-like if it is $\mathbb{R}$-parameterized, has the scale property, and is closed under $\wedge$, $\vee$, $\forall^\mathbb{R}$, $\exists^\mathbb{R}$, and preimages by continuous functions, and

  • lightface inductive-like if it is $\omega$-parameterized, has the scale property, and is closed under $\wedge$, $\vee$, $\forall^\mathbb{R}$, $\exists^\mathbb{R}$, and preimages by continuous functions.

Given a lightface inductive-like pointclass $\Gamma$ we can define the boldface inductive-like pointclass ${\bf \Gamma} = \bigcup_{x \in \mathbb{R}} \Gamma(x)$ as usual (this does not have much to do with the particulars of "inductive-like.")

My question is, can we do the reverse? That is, given a boldface inductive-like pointclass ${\bf \Gamma}$ can we find a lightface inductive-like pointclass $\Gamma$ such that ${\bf \Gamma} = \bigcup_{x \in \mathbb{R}} \Gamma(x)$? (We wouldn't expect uniqueness; if $\Gamma$ works then so does $\Gamma(z)$ for any real $z$.)

I have heard that Howard Becker proved something like this, but I can't find it and I don't know what kind of pointclasses it was for (probably more general than "inductive-like," which is merely the case I happen to be interested in at the moment.)

For a motivating example, consider the case where $\mathsf{AD}$ holds and $\bf \Gamma$ is the pointclass of $\kappa$-Suslin sets. If $\bf \Gamma$ is closed under $\forall^\mathbb{R}$ then it is boldface inductive-like, but as far as I know there is no canonical choice of lightface pointclass that corresponds to it.

added 120 characters in body
Source Link
Trevor Wilson
  • 5.5k
  • 29
  • 46

Define a pointclass to be:

  • boldface inductive-like if it is $\mathbb{R}$-parameterized, has the scale property, and is closed under $\wedge$, $\vee$, $\forall^\mathbb{R}$, $\exists^\mathbb{R}$, and preimages by continuous functions, and

  • lightface inductive-like if it is $\omega$-parameterized, has the scale property, and is closed under $\wedge$, $\vee$, $\forall^\mathbb{R}$, $\exists^\mathbb{R}$, and preimages by continuous functions.

Given a lightface inductive-like pointclass $\Gamma$ we can define the boldface inductive-like pointclass ${\bf \Gamma} = \bigcup_{x \in \mathbb{R}} \Gamma(x)$ as usual (this does not have much to do with the particulars of "inductive-like.")

My question is, can we do the reverse? That is, given a boldface inductive-like pointclass ${\bf \Gamma}$ can we find a lightface inductive-like pointclass $\Gamma$ such that ${\bf \Gamma} = \bigcup_{x \in \mathbb{R}} \Gamma(x)$? (We wouldn't expect uniqueness; if $\Gamma$ works then so does $\Gamma(z)$ for any real $z$.)

I have heard that Howard Becker proved something like this, but I can't find it and I don't know what kind of pointclasses it was for (perhapsprobably more general than "inductive-like.," which is merely the case I happen to be interested in at the moment.)

Define a pointclass to be:

  • boldface inductive-like if it is $\mathbb{R}$-parameterized and closed under $\wedge$, $\vee$, $\forall^\mathbb{R}$, $\exists^\mathbb{R}$, and preimages by continuous functions, and

  • lightface inductive-like if it is $\omega$-parameterized and closed under $\wedge$, $\vee$, $\forall^\mathbb{R}$, $\exists^\mathbb{R}$, and preimages by continuous functions.

Given a lightface inductive-like pointclass $\Gamma$ we can define the boldface inductive-like pointclass ${\bf \Gamma} = \bigcup_{x \in \mathbb{R}} \Gamma(x)$ as usual (this does not have much to do with the particulars of "inductive-like.")

My question is, can we do the reverse? That is, given a boldface inductive-like pointclass ${\bf \Gamma}$ can we find a lightface inductive-like pointclass $\Gamma$ such that ${\bf \Gamma} = \bigcup_{x \in \mathbb{R}} \Gamma(x)$? (We wouldn't expect uniqueness; if $\Gamma$ works then so does $\Gamma(z)$ for any real $z$.)

I have heard that Howard Becker proved something like this, but I can't find it and I don't know what kind of pointclasses it was for (perhaps more general than "inductive-like.")

Define a pointclass to be:

  • boldface inductive-like if it is $\mathbb{R}$-parameterized, has the scale property, and is closed under $\wedge$, $\vee$, $\forall^\mathbb{R}$, $\exists^\mathbb{R}$, and preimages by continuous functions, and

  • lightface inductive-like if it is $\omega$-parameterized, has the scale property, and is closed under $\wedge$, $\vee$, $\forall^\mathbb{R}$, $\exists^\mathbb{R}$, and preimages by continuous functions.

Given a lightface inductive-like pointclass $\Gamma$ we can define the boldface inductive-like pointclass ${\bf \Gamma} = \bigcup_{x \in \mathbb{R}} \Gamma(x)$ as usual (this does not have much to do with the particulars of "inductive-like.")

My question is, can we do the reverse? That is, given a boldface inductive-like pointclass ${\bf \Gamma}$ can we find a lightface inductive-like pointclass $\Gamma$ such that ${\bf \Gamma} = \bigcup_{x \in \mathbb{R}} \Gamma(x)$? (We wouldn't expect uniqueness; if $\Gamma$ works then so does $\Gamma(z)$ for any real $z$.)

I have heard that Howard Becker proved something like this, but I can't find it and I don't know what kind of pointclasses it was for (probably more general than "inductive-like," which is merely the case I happen to be interested in at the moment.)

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Trevor Wilson
  • 5.5k
  • 29
  • 46

Obtaining a lightface pointclass from a boldface one

Define a pointclass to be:

  • boldface inductive-like if it is $\mathbb{R}$-parameterized and closed under $\wedge$, $\vee$, $\forall^\mathbb{R}$, $\exists^\mathbb{R}$, and preimages by continuous functions, and

  • lightface inductive-like if it is $\omega$-parameterized and closed under $\wedge$, $\vee$, $\forall^\mathbb{R}$, $\exists^\mathbb{R}$, and preimages by continuous functions.

Given a lightface inductive-like pointclass $\Gamma$ we can define the boldface inductive-like pointclass ${\bf \Gamma} = \bigcup_{x \in \mathbb{R}} \Gamma(x)$ as usual (this does not have much to do with the particulars of "inductive-like.")

My question is, can we do the reverse? That is, given a boldface inductive-like pointclass ${\bf \Gamma}$ can we find a lightface inductive-like pointclass $\Gamma$ such that ${\bf \Gamma} = \bigcup_{x \in \mathbb{R}} \Gamma(x)$? (We wouldn't expect uniqueness; if $\Gamma$ works then so does $\Gamma(z)$ for any real $z$.)

I have heard that Howard Becker proved something like this, but I can't find it and I don't know what kind of pointclasses it was for (perhaps more general than "inductive-like.")