Skip to main content
MathOverflow

Return to Question

added 154 characters in body
Source Link
Zuhair Al-Johar
Zuhair Al-Johar
  • 11.3k
  • 1
  • 13
  • 47

Harvey Friedman is well known for investigating concrete mathematical statements that requires strong assumptions, i.e. those that can only be interpreted in a strong extension of $\text{ZF(C)}$. My first question:

  1. Is there any known example of a concrete mathematical statement that requires the strength of an extension of $\text{ZF}$ that can prove the existence of a Reinhardt cardinal?

My second question is:

  1. If we suppose such an example had been put forth, would that constitute a reason to reject choiceChoice?

The idea is that if 1 is true, then this might open the door for many examples of concrete mathematical statements that can only be interpreted in extensions of $\text{ZF}$ that can prove the existence of a Reinhardt cardinal or higher cardinals, but by then this would necessarily meanit is known that all such extensions negate choiceChoice. And by then choiceChoice would appear as a restrictive assumption, since it would restrict interpret-ability of concrete mathematical statements, which is the primary goal of foundation after all.

  1. What makes us believe in an axiom? is it its formalization ability (i.e. its interpret-ability strength) or its intuitive appeal? so in a situation like the above, which would we favor?

Harvey Friedman is well known for investigating concrete mathematical statements that requires strong assumptions, i.e. those that can only be interpreted in a strong extension of $\text{ZF(C)}$. My first question:

  1. Is there any known example of a concrete mathematical statement that requires the strength of an extension of $\text{ZF}$ that can prove the existence of a Reinhardt cardinal?

My second question is:

  1. If we suppose such an example had been put forth, would that constitute a reason to reject choice?

The idea is that if 1 is true, then this might open the door for many examples of concrete mathematical statements that can only be interpreted in extensions of $\text{ZF}$ that can prove the existence of a Reinhardt cardinal or higher cardinals, but by then this would necessarily mean that all such extensions negate choice. And by then choice would appear as a restrictive assumption, since it would restrict interpret-ability of concrete mathematical statements, which is the primary goal of foundation after all.

Harvey Friedman is well known for investigating concrete mathematical statements that requires strong assumptions, i.e. those that can only be interpreted in a strong extension of $\text{ZF(C)}$. My first question:

  1. Is there any known example of a concrete mathematical statement that requires the strength of an extension of $\text{ZF}$ that can prove the existence of a Reinhardt cardinal?

My second question is:

  1. If we suppose such an example had been put forth, would that constitute a reason to reject Choice?

The idea is that if 1 is true, then this might open the door for many examples of concrete mathematical statements that can only be interpreted in extensions of $\text{ZF}$ that can prove the existence of a Reinhardt cardinal or higher cardinals, but it is known that all such extensions negate Choice. And by then Choice would appear as a restrictive assumption, since it would restrict interpret-ability of concrete mathematical statements, which is the primary goal of foundation after all.

  1. What makes us believe in an axiom? is it its formalization ability (i.e. its interpret-ability strength) or its intuitive appeal? so in a situation like the above, which would we favor?
Source Link
Zuhair Al-Johar
Zuhair Al-Johar
  • 11.3k
  • 1
  • 13
  • 47

Concrete mathematical statements in relation to Choice versus Reinhardt cardinals?

Harvey Friedman is well known for investigating concrete mathematical statements that requires strong assumptions, i.e. those that can only be interpreted in a strong extension of $\text{ZF(C)}$. My first question:

  1. Is there any known example of a concrete mathematical statement that requires the strength of an extension of $\text{ZF}$ that can prove the existence of a Reinhardt cardinal?

My second question is:

  1. If we suppose such an example had been put forth, would that constitute a reason to reject choice?

The idea is that if 1 is true, then this might open the door for many examples of concrete mathematical statements that can only be interpreted in extensions of $\text{ZF}$ that can prove the existence of a Reinhardt cardinal or higher cardinals, but by then this would necessarily mean that all such extensions negate choice. And by then choice would appear as a restrictive assumption, since it would restrict interpret-ability of concrete mathematical statements, which is the primary goal of foundation after all.