FOM: Steve Simpson on K"onig's lemma, non-standard models etc.
neilt at hums62.cohums.ohio-state.edu
Fri Dec 5 08:24:17 EST 1997
Steve Simpson writes (in response to my question whether the existence
of non-standard models for full arithmetic is strictly weaker than the
completeness and compactness theorems):
The existence of a nonstandard model M of arithmetic
implies the existence of an omega-model of weak K"onig's lemma, namely
the Scott system of M. And the G"odel completeness and compactness
theorems hold in any such omega-model; in fact, they are equivalent to
weak K"onig's lemma, in the sense of reverse mathematics. All of this
is provable in weak systems, e.g. RCA_0.
Later he adds:
However, this is not quite an airtight answer to Neil's question,
because the existence of a nonstandard model of arithmetic does not
literally imply weak K"onig's lemma. It only implies the existence of
an omega-model of weak K"onig's lemma, as noted above. Nevertheless,
I would be willing to (somewhat loosely) summarize the precise results
above as follows: The existence of nonstandard models of arithmetic is
equivalent to the G"odel completeness and compactness theorems.
The not-completely-airtight answer has I think been too loosely
summarized. Trusting Steve's precise technical results, isn't their
correct summary only that
The existence of nonstandard models of arithmetic implies
the consistency of assuming that completeness and compactness hold.
Maybe I'm missing something here by not having all the succinct
technical phrases fully defined. Steve, could you clarify the position
and/or correct me?
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