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Timothy Chow
Timothy Chow
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Bjørn Kjos-Hanssen has answered the stated question but I think it would help to make a few clarifying comments.

Let BWQ denote the statement, "Every bounded infinite sequence from $\mathbb Q$ has an infinite Cauchy subsequence." What Friedman has shown is not that BWQ implies Con(PA). (I've gotten confused on this point myself in the past.) Instead, what he has shown is that Con(RCA_0 + BWQ) implies Con(PA). See Friedman's posts here and here for example.

If you're looking for other mundane-looking statements whose consistency implies the consistency of Con(PA)PA, then Simpson's book contains a number of candidates, e.g.,

  1. Every countable field is isomorphic to a subfield of a countable algebraically closed field.
  2. Every countable vector space over $\mathbb Q$ has a basis.
  3. Every countable commutative ring has a maximal ideal.

Note: One has to be a bit careful about what exactly "countable objects" are in the context of subsystems of second-order arithmetic (as opposed to straight set theory); this fine print is discussed in Simpson's book.

Bjørn Kjos-Hanssen has answered the stated question but I think it would help to make a few clarifying comments.

Let BWQ denote the statement, "Every bounded infinite sequence from $\mathbb Q$ has an infinite Cauchy subsequence." What Friedman has shown is not that BWQ implies Con(PA). (I've gotten confused on this point myself in the past.) Instead, what he has shown is that Con(RCA_0 + BWQ) implies Con(PA). See Friedman's posts here and here for example.

If you're looking for other mundane-looking statements whose consistency implies the consistency of Con(PA), then Simpson's book contains a number of candidates, e.g.,

  1. Every countable field is isomorphic to a subfield of a countable algebraically closed field.
  2. Every countable vector space over $\mathbb Q$ has a basis.
  3. Every countable commutative ring has a maximal ideal.

Note: One has to be a bit careful about what exactly "countable objects" are in the context of subsystems of second-order arithmetic (as opposed to straight set theory); this fine print is discussed in Simpson's book.

Bjørn Kjos-Hanssen has answered the stated question but I think it would help to make a few clarifying comments.

Let BWQ denote the statement, "Every bounded infinite sequence from $\mathbb Q$ has an infinite Cauchy subsequence." What Friedman has shown is not that BWQ implies Con(PA). (I've gotten confused on this point myself in the past.) Instead, what he has shown is that Con(RCA_0 + BWQ) implies Con(PA). See Friedman's posts here and here for example.

If you're looking for other mundane-looking statements whose consistency implies the consistency of PA, then Simpson's book contains a number of candidates, e.g.,

  1. Every countable field is isomorphic to a subfield of a countable algebraically closed field.
  2. Every countable vector space over $\mathbb Q$ has a basis.
  3. Every countable commutative ring has a maximal ideal.

Note: One has to be a bit careful about what exactly "countable objects" are in the context of subsystems of second-order arithmetic (as opposed to straight set theory); this fine print is discussed in Simpson's book.

Source Link
Timothy Chow
Timothy Chow
  • 82.7k
  • 26
  • 363
  • 587

Bjørn Kjos-Hanssen has answered the stated question but I think it would help to make a few clarifying comments.

Let BWQ denote the statement, "Every bounded infinite sequence from $\mathbb Q$ has an infinite Cauchy subsequence." What Friedman has shown is not that BWQ implies Con(PA). (I've gotten confused on this point myself in the past.) Instead, what he has shown is that Con(RCA_0 + BWQ) implies Con(PA). See Friedman's posts here and here for example.

If you're looking for other mundane-looking statements whose consistency implies the consistency of Con(PA), then Simpson's book contains a number of candidates, e.g.,

  1. Every countable field is isomorphic to a subfield of a countable algebraically closed field.
  2. Every countable vector space over $\mathbb Q$ has a basis.
  3. Every countable commutative ring has a maximal ideal.

Note: One has to be a bit careful about what exactly "countable objects" are in the context of subsystems of second-order arithmetic (as opposed to straight set theory); this fine print is discussed in Simpson's book.