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Alec Rhea
Alec Rhea
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While reviewing some categorical versions of the axiom of choice, it occurred to me that none of the formulations I'm aware of actually reflect how I use choice in practice: pronounce that we 'choose an element $x$ of $X$' or something like this.

That is, while I understand that 'every surjection in ${\bf Set}$ splits' and 'every collection of nonempty sets has a choice function' etc. are ways to formalize choice that can be mildly contorted to give the way I used it above (surject onto a singleton and split, consider a collection containing one set, etc.), none of them straight up axiomatize my intuitive usage of choice. To this end:

Define simple choice (SC) to be the axiom asserting that every nonempty set $X$ comes equipped with a simple choice function $$f_X:1\to X,$$ where $1$ is the ordinal (and not some random singleton, to make things more concrete).

Using SC, when we say 'choose an element $x$ of $X$' we really mean 'look at $f_X(0)$', instead of the above more involved interpretations. For any nonempty collection of sets $Y$, I believe we can define a choice function $c_Y:Y\to\bigcup Y$ in the classical sense with $c_Y(X)\in X$ for all $X\in Y$ by defining $$c_Y(X)=f_X(0).$$

Further, while the classical versions introduce auxiliary contortions that are relatively harmless when working with sets, we suddenly have to use things like Scott's trick to get good behavior when working with the globalized versions of choice and proper classes -- the globalized version of SC seems to work in a more straightforward manner with proper classes.

Let'sWorking in $MK$, let's call global simple choice (GSC) the axiom asserting that every nonempty class $X$ comes equipped with a simple choice function $f_X:1\to X$. I believe this implies global choice in the same manner that it implies regular choice above, but if we have a proper class and want to select an element of it we don't need Scott's trick to intersect with some stage of the cumulative hierarchy anymore.

I'm certain that I'm not the first one to cook up something like this, so my questions are

What is wrong with this version of choice? Is it actually equivalent to the standard formulation? If it is equivalent, is this version of it less appealing for some reason that isn't occurring to me? Are there existing references to it somewhere?

Any assistance is appreciated.

While reviewing some categorical versions of the axiom of choice, it occurred to me that none of the formulations I'm aware of actually reflect how I use choice in practice: pronounce that we 'choose an element $x$ of $X$' or something like this.

That is, while I understand that 'every surjection in ${\bf Set}$ splits' and 'every collection of nonempty sets has a choice function' etc. are ways to formalize choice that can be mildly contorted to give the way I used it above (surject onto a singleton and split, consider a collection containing one set, etc.), none of them straight up axiomatize my intuitive usage of choice. To this end:

Define simple choice (SC) to be the axiom asserting that every nonempty set $X$ comes equipped with a simple choice function $$f_X:1\to X,$$ where $1$ is the ordinal (and not some random singleton, to make things more concrete).

Using SC, when we say 'choose an element $x$ of $X$' we really mean 'look at $f_X(0)$', instead of the above more involved interpretations. For any nonempty collection of sets $Y$, I believe we can define a choice function $c_Y:Y\to\bigcup Y$ in the classical sense with $c_Y(X)\in X$ for all $X\in Y$ by defining $$c_Y(X)=f_X(0).$$

Further, while the classical versions introduce auxiliary contortions that are relatively harmless when working with sets, we suddenly have to use things like Scott's trick to get good behavior when working with the globalized versions of choice and proper classes -- the globalized version of SC seems to work in a more straightforward manner with proper classes.

Let's call global simple choice (GSC) the axiom asserting that every nonempty class $X$ comes equipped with a simple choice function $f_X:1\to X$. I believe this implies global choice in the same manner that it implies regular choice above, but if we have a proper class and want to select an element of it we don't need Scott's trick to intersect with some stage of the cumulative hierarchy anymore.

I'm certain that I'm not the first one to cook up something like this, so my questions are

What is wrong with this version of choice? Is it actually equivalent to the standard formulation? If it is equivalent, is this version of it less appealing for some reason that isn't occurring to me? Are there existing references to it somewhere?

Any assistance is appreciated.

While reviewing some categorical versions of the axiom of choice, it occurred to me that none of the formulations I'm aware of actually reflect how I use choice in practice: pronounce that we 'choose an element $x$ of $X$' or something like this.

That is, while I understand that 'every surjection in ${\bf Set}$ splits' and 'every collection of nonempty sets has a choice function' etc. are ways to formalize choice that can be mildly contorted to give the way I used it above (surject onto a singleton and split, consider a collection containing one set, etc.), none of them straight up axiomatize my intuitive usage of choice. To this end:

Define simple choice (SC) to be the axiom asserting that every nonempty set $X$ comes equipped with a simple choice function $$f_X:1\to X,$$ where $1$ is the ordinal (and not some random singleton, to make things more concrete).

Using SC, when we say 'choose an element $x$ of $X$' we really mean 'look at $f_X(0)$', instead of the above more involved interpretations. For any nonempty collection of sets $Y$, I believe we can define a choice function $c_Y:Y\to\bigcup Y$ in the classical sense with $c_Y(X)\in X$ for all $X\in Y$ by defining $$c_Y(X)=f_X(0).$$

Further, while the classical versions introduce auxiliary contortions that are relatively harmless when working with sets, we suddenly have to use things like Scott's trick to get good behavior when working with the globalized versions of choice and proper classes -- the globalized version of SC seems to work in a more straightforward manner with proper classes.

Working in $MK$, let's call global simple choice (GSC) the axiom asserting that every nonempty class $X$ comes equipped with a simple choice function $f_X:1\to X$. I believe this implies global choice in the same manner that it implies regular choice above, but if we have a proper class and want to select an element of it we don't need Scott's trick to intersect with some stage of the cumulative hierarchy anymore.

I'm certain that I'm not the first one to cook up something like this, so my questions are

What is wrong with this version of choice? Is it actually equivalent to the standard formulation? If it is equivalent, is this version of it less appealing for some reason that isn't occurring to me? Are there existing references to it somewhere?

Any assistance is appreciated.

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Alec Rhea
Alec Rhea
  • 10.1k
  • 3
  • 30
  • 88

A simple form of choice

While reviewing some categorical versions of the axiom of choice, it occurred to me that none of the formulations I'm aware of actually reflect how I use choice in practice: pronounce that we 'choose an element $x$ of $X$' or something like this.

That is, while I understand that 'every surjection in ${\bf Set}$ splits' and 'every collection of nonempty sets has a choice function' etc. are ways to formalize choice that can be mildly contorted to give the way I used it above (surject onto a singleton and split, consider a collection containing one set, etc.), none of them straight up axiomatize my intuitive usage of choice. To this end:

Define simple choice (SC) to be the axiom asserting that every nonempty set $X$ comes equipped with a simple choice function $$f_X:1\to X,$$ where $1$ is the ordinal (and not some random singleton, to make things more concrete).

Using SC, when we say 'choose an element $x$ of $X$' we really mean 'look at $f_X(0)$', instead of the above more involved interpretations. For any nonempty collection of sets $Y$, I believe we can define a choice function $c_Y:Y\to\bigcup Y$ in the classical sense with $c_Y(X)\in X$ for all $X\in Y$ by defining $$c_Y(X)=f_X(0).$$

Further, while the classical versions introduce auxiliary contortions that are relatively harmless when working with sets, we suddenly have to use things like Scott's trick to get good behavior when working with the globalized versions of choice and proper classes -- the globalized version of SC seems to work in a more straightforward manner with proper classes.

Let's call global simple choice (GSC) the axiom asserting that every nonempty class $X$ comes equipped with a simple choice function $f_X:1\to X$. I believe this implies global choice in the same manner that it implies regular choice above, but if we have a proper class and want to select an element of it we don't need Scott's trick to intersect with some stage of the cumulative hierarchy anymore.

I'm certain that I'm not the first one to cook up something like this, so my questions are

What is wrong with this version of choice? Is it actually equivalent to the standard formulation? If it is equivalent, is this version of it less appealing for some reason that isn't occurring to me? Are there existing references to it somewhere?

Any assistance is appreciated.